Abstract
Random graph models with limited choice have been studied extensively with the goal of understanding the mechanism of the emergence of the giant component. One of the standard models are the Achlioptas random graph processes on a fixed set of \(n\) vertices. Here at each step, one chooses two edges uniformly at random and then decides which one to add to the existing configuration according to some criterion. An important class of such rules are the bounded-size rules where for a fixed \(K\ge 1\), all components of size greater than \(K\) are treated equally. While a great deal of work has gone into analyzing the subcritical and supercritical regimes, the nature of the critical scaling window, the size and complexity (deviation from trees) of the components in the critical regime and nature of the merging dynamics has not been well understood. In this work we study such questions for general bounded-size rules. Our first main contribution is the construction of an extension of Aldous’s standard multiplicative coalescent process which describes the asymptotic evolution of the vector of sizes and surplus of all components. We show that this process, referred to as the standard augmented multiplicative coalescent (AMC) is ‘nearly’ Feller with a suitable topology on the state space. Our second main result proves the convergence of suitably scaled component size and surplus vector, for any bounded-size rule, to the standard AMC. This result is new even for the classical Erdős–Rényi setting. The key ingredients here are a precise analysis of the asymptotic behavior of various susceptibility functions near criticality and certain bounds from Bhamidi et al. (The barely subcritical regime. Arxiv preprint, 2012) on the size of the largest component in the barely subcritical regime.
1 Introduction
The profusion of empirical data for real-world networks has provided an impetus for research into mathematical models which explain the various observed statistics such as scale-free degree distributions, small world properties and clustering. A range of mathematical models has been proposed, both static and dynamic, with the aim of understanding the structural properties of these real-world networks and their evolution over time. One particular direction of significant research is focused on understanding the effect of choice in the evolution of random network models (see [27] and references therein). More precisely, suppose that at time \(t=0\) we start with the empty configuration on \([n]:=\{1,2,\ldots , n\}\) vertices. At each discrete step \(k=0,1,2,\ldots \), we choose two edges \((e_1(k),e_2(k))\) uniformly at random amongst all \(\left( \small {\begin{array}{l} n\\ 2\\ \end{array}}\right) \) edges and decide whether the graph at instant \((k+1)\), denoted by \({\mathbf{G}}_n(k+1)\), is \({\mathbf{G}}_n(k)\cup e_1(k)\) or \({\mathbf{G}}_n(k)\cup e_2(k)\) according to some pre-specified rule that takes into account suitable properties of the chosen edges with respect to the present configuration \({\mathbf{G}}_n(k)\). Speeding up time by a factor of \(n\) and abusing notation, for \(t\ge 0\) write, \({\mathbf{G}}_n(t) = {\mathbf{G}}_n(\lfloor nt/2 \rfloor )\). Then the basic goal is to understand the effect of the rule governing the edge formations in the evolution of various characteristics of the network such as, the size of the largest component, the vector of sizes of all components, component complexities, etc. Three prototypical examples to keep in mind are as follows:
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(a)
Erdős–Rényi random graph: At each stage include edge \(e_1\) and ignore \(e_2\). This results in the classical Erdős–Rényi random graph evolution. For a component \(\mathcal {C}\), define \(|\mathcal {C}|\) for the size (number of vertices) of the component. Well known results [13, 16] say that the critical time for the emergence of a giant component for this model is \(1\), namely for \(t<1\) the size of the largest component \(|\mathcal {C}^{\scriptscriptstyle \mathbf ER}_1(t)| = O(\log {n})\) while for \(t> 1\), the size of the largest component \(|\mathcal {C}^{\scriptscriptstyle \mathbf ER}_1(t)| = \Theta (n)\).
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(b)
Bohman–Frieze (BF) process: This was the first rigorously analyzed example of a rule that delayed the emergence of the giant component through limited choice [11]. Here the rule is to use the first edge if it connects two singletons (vertices which have no connections at the present time), otherwise use the second edge. It has been shown [19, 27] that there is a critical time \(t_c^{\scriptscriptstyle \mathbf{BF}} \approx 1.176\) when the largest component transitions from \(O(\log {n})\) to \(\Theta (n)\).
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(c)
General bounded-size rules (BSR): The BF process corresponds to a choice rule which treats all components with size greater than one in an identical fashion. It is a special case of the general family of models, referred to as bounded-size rules. Here one fixes \(K\ge 1\) and then the rule for attachment is invariant on components of size greater than \(K\). We postpone a precise description to Sect. 2.2. General bounded-size rules were analyzed in [27] where it was shown that there exists a (rule dependent) critical time \(t_c\) such that for \(t< t_c\), the largest component \(|\mathcal {C}^{\scriptscriptstyle \mathbf{{BSR}}}_1(t)| = O(\log {n})\) when \(t< t_c\) while \(|\mathcal {C}^{\scriptscriptstyle \mathbf{{BSR}}}_1(t)| = \Theta (n)\) for \(t> t_c\).
Thus as time transitions from below to above \(t_c\), a giant component (of the same order as the network) emerges. Motivated by recent results on the Erdős–Rényi random graph components at criticality [2] as well as general rules such as the (unbounded-size) product rule [1], there has been a renewed interest in understanding the precise nature of the emergence of the giant component as well as structural properties of components near \(t_c\) for classes of rules which incorporate limited choice in their evolution. Define the surplus or complexity of a component \({\mathbf{spls}}(\mathcal {C})\) as
If a component were a tree, its surplus would be zero, thus this is a measure of the deviation of the component from a tree. Write \(\mathcal {C}_i(t)\) for the \(i\)th largest component and \(\xi _i(t) := {\mathbf{spls}}(\mathcal {C}_i(t))\) for the surplus of the component \(\mathcal {C}_i(t)\). For any of the rules above and a fixed \(t\ge 0\), consider the vector of component sizes and associated surplus \((|\mathcal {C}_i(t)|, \xi _i(t): i\ge 1)\). In the context of the Erdős–Rényi random graph process, precise fine-scale results are known about the nature of the emergence of the giant component as time \(t\) transitions through the scaling window around \(t_c^{\scriptscriptstyle \mathbf ER}=1\). More precisely, for fixed \(\lambda \in {\mathbb {R}}\) write
Then Aldous in [4] showed:
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(a)
The process \(({\bar{\varvec{C}}}^{\scriptscriptstyle {\mathbf{ER}}}(\lambda ): -\infty < \lambda < \infty )\) converges to a Markov process called the standard multiplicative coalescent.
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(b)
For fixed \(\lambda \in {\mathbb {R}}\), the rescaled component sizes and the corresponding surplus \(({\bar{\varvec{C}}}^{\scriptscriptstyle {\mathbf{ER}}}(\lambda ), {\bar{\varvec{Y}}}^{\scriptscriptstyle {\mathbf{ER}}}(\lambda ))\) converge jointly to a limiting random process described by excursions above zero of an inhomogeneous reflected Brownian motion \(\hat{W}_\lambda \) and a counting process \(\hat{N}_{\lambda }\) with intensity function \(\hat{W}_\lambda (\cdot )\).
We give a precise description of these results in Sect. 2.3.1. Obtaining similar results on critical asymptotics for general inhomogeneous Markovian models such as the bounded-size rules requires new ideas. These rules lack a simple description for the dependence between edges making the direct use of the component exploration and associated random walk construction, the major workhorse in understanding random graph models at criticality [4, 9, 10, 21, 26], intractable. Thus it is nontrivial to identify the critical scaling window for such processes, let alone distributional asymptotics for the component sizes and surplus. In the current work we develop a different machinery that allows us to identify the critical scaling window for all bounded-size rules. Furthermore, denoting the suitably scaled component sizes and surplus processes as \(({\bar{\varvec{C}}}^{\scriptscriptstyle (n)}(\lambda ), {\bar{\varvec{Y}}}^{\scriptscriptstyle (n)}(\lambda ))\), our results describe the joint asymptotic behavior of
for \(-\infty < \lambda _1< \lambda _2< \cdots < \lambda _m < \infty \). The starting point of our work is the construction of a Markov process that is associated with the inhomogeneous reflected Brownian motion \(\{\hat{W}_{\lambda }\}_{\lambda \in {\mathbb {R}}}\) and the associated counting process \(\{\hat{N}_{\lambda }\}_{\lambda \in {\mathbb {R}}}\) which we refer to as the augmented multiplicative coalescent (AMC). The main result of this work shows that AMC is the characterizing process for the universality class that includes, in addition to critically scaled components and surplus vectors for Erdős–Rényi graphs, analogous processes for all bounded-size rules. More precisely, our contributions are as follows.
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(a)
In Theorem 3.1 we show the existence and “near” Feller property of a Markov process \({\varvec{Z}}(\lambda )\), \(-\infty < \lambda < \infty \), called the augmented multiplicative coalescent, which tracks the evolution of both component sizes and surplus edges over the critical window. Aldous’s standard multiplicative coalescent corresponds to the first coordinate of this process. Identifying the correct state space and topology that is suitable for obtaining the Feller property for this process turns out to be particularly delicate (see Remark 4.15). The (near) Feller property plays a key role in analyzing the joint distribution, at multiple time instants, of the component sizes and surplus for bounded-size rules in the critical scaling window. In proving the existence of the standard augmented process, a key role is played by Theorem 5.1 which is a generalization of a result of Aldous for the component sizes of an inhomogeneous random graph, to a setting where one considers joint distributions of component sizes and surplus. We believe that this result is of broader significance and can be used to analyze the distribution of surplus in the critical regime for various other inhomogeneous random graph models, e.g. the rank-1 inhomogeneous random graphs [12].
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(b)
In Theorems 3.2 we analyze susceptibility functions (sums of moments of component sizes) associated with a general bounded-size rule. Spencer and Wormald [27] showed that these susceptibility functions converge to limiting monotonically increasing deterministic functions which are finite only for \(t< t_c\) and explode for \(t> t_c\). Theorem 3.2(i) uses a dynamic random graph process with immigration and attachment to show that these limiting functions for all bounded-size rules have the same critical exponents as the Erdős–Rényi random graph process. Theorem 3.2(ii) and (iii) shows that the susceptibility functions are close to their deterministic analogs in a strong sense even as \(t\uparrow t_c\) when the limiting functions explode.
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(c)
The analysis of the susceptibility functions gives rise to (rule dependent) constants \(\alpha , \beta > 0\) which describe the nature of the explosion of the limiting susceptibility functions as \(t\uparrow t_c\). For a given bounded-size rule we consider the rescaled process \(\{{{\bar{\varvec{Z}}}}^{\scriptscriptstyle (n)}(\lambda ): -\infty < \lambda < \infty \} \) where \({{\bar{\varvec{Z}}}}^{\scriptscriptstyle (n)}(\lambda ) = ({\bar{\varvec{C}}}^{\scriptscriptstyle (n)}(\lambda ),{\bar{\varvec{Y}}}^{\scriptscriptstyle (n)}(\lambda ) )\) with \({\bar{\varvec{C}}}^{\scriptscriptstyle (n)}(\lambda )\) denoting the rescaled component sizes and \({\bar{\varvec{Y}}}^{\scriptscriptstyle (n)}(\lambda )\) denoting the surplus of these components, namely
$$\begin{aligned} {\bar{\varvec{C}}}^{\scriptscriptstyle (n)}(\lambda )&:= \left( \frac{\beta ^{1/3}}{n^{2/3}}\left| \mathcal {C}_i\left( t_c+\frac{\alpha \beta ^{2/3}}{n^{1/3}}\lambda \right) \right| :i\ge 1\right) \quad \hbox {and} \\ {\bar{\varvec{Y}}}^{\scriptscriptstyle (n)}(\lambda )&= \left( \xi _i\left( t_c+\frac{\alpha \beta ^{2/3}}{n^{1/3}}\lambda \right) :i\ge 1\right) . \end{aligned}$$Using the Feller property proved in Theorem 3.1, Theorem 5.1, results on the susceptibility functions, and bounds on the size of the largest component in the barely subcritical regime from [8], we show the convergence of the finite-dimensional distributions of this process to that of the augmented multiplicative coalescent \(\{{\varvec{Z}}(\lambda ):-\infty < \lambda <\infty \}\), namely for any set of times \(-\infty < \lambda _1< \lambda _2< \cdots < \lambda _m < \infty \),
$$\begin{aligned} \left( {\bar{\varvec{Z}}}^{\scriptscriptstyle (n)}(\lambda _1), \ldots , {\bar{\varvec{Z}}}^{\scriptscriptstyle (n)}(\lambda _m)\right) \mathop {\longrightarrow }\limits ^{d}\left( \varvec{Z}(\lambda _1), \ldots , \varvec{Z}(\lambda _m)\right) . \end{aligned}$$The result in particular identifies the critical scaling window for all bounded-size rules as well as the asymptotic joint distributions of component sizes and surplus for any fixed \(\lambda \), implying that such rules belong to the same universality class as the Erdős–Rényi random graph process. The convergence for the joint distribution of the surplus and the component sizes for multiple time points \(\lambda \) in the critical scaling window for Erdős–Rényi random graph process is regarded as a “folk theorem”—our results, in particular, give a rigorous proof of this statement.
The paper is organized as follows. In Sect. 2 we introduce some common notation, give a precise description of bounded-size rules, and give an informal description of the augmented multiplicative coalescent. Section 3 contains the statements of our main results. Sections 4 and 5 are devoted to proving the existence and near Feller property of the AMC. In particular Sect. 5 contains the proof of Theorem 3.1. Section 6 studies the asymptotics of the susceptibility functions associated with general bounded-size rules and proves Theorems 3.2. Finally in Sect. 7 we complete the proof of Theorem 3.3.
2 Definitions and notation
2.1 Notation
We collect some common notation and conventions used in this work. A graph \({\mathbf{G}}=(\mathcal {V}, \mathcal {E})\) consists of a vertex set \(\mathcal {V}\) and an edge set \(\mathcal {E}\), where \(\mathcal {V}\) is a subset of some type space \(\mathcal {X}\). For a finite set \(A\) write \(|A|\) for its cardinality. A graph \({\mathbf{G}}\) with no vertices and edges will be called a null graph. For graphs \({\mathbf{G}}_1, {\mathbf{G}}_2\), if \({\mathbf{G}}_1\) is a subgraph of \({\mathbf{G}}_2\) we shall write this as \({\mathbf{G}}_1 \subset {\mathbf{G}}_2\). The number of vertices in a connected component \(\mathcal {C}\) of a graph \({\mathbf{G}}\) will be called the size of the component and will be denoted by \(|\mathcal {C}|\). Let \(\mathcal {G}\) be the set of all possible graphs \((\mathcal {V}, \mathcal {E})\) on a given type space \(\mathcal {X}\). When \(\mathcal {V}\) is finite, we will consider \(\mathcal {G}\) to be endowed with the discrete topology and the corresponding Borel sigma field and refer to a random element of \(\mathcal {G}\) as a random graph.
We use \(\mathop {\longrightarrow }\limits ^{\scriptscriptstyle {\mathbb P}}\) and \(\mathop {\longrightarrow }\limits ^{d}\) to denote convergence in probability and in distribution respectively. All the unspecified limits are taken as \(n \rightarrow \infty \). Given a sequence of events \(\{E_n\}_{n\ge 1}\), we say \(E_n\) occurs with high probability (whp) if \(\mathbb {P}\{E_n\} \rightarrow 1\). The notation \(O,\Omega ,\Theta \) and \(o\) are defined in the usual manner. More precisely, given a sequence of random variables \(\{\xi _n\}_{n\ge 1}\) and a function \(f(n)\), we say \(\xi _n = O (f)\) if there is a constant \(C\) such that \(\xi _n \le C f(n)\) whp, and we say \(\xi _n = \Omega (f)\) if there is a constant \(C\) such that \(\xi _n \ge Cf(n)\) whp. Say that \(\xi _n = \Theta (f)\) if \(\xi _n = O(f)\) and \(\xi _n = \Omega (f)\). We say \(\xi _n = o (f)\) if \(\xi _n/f(n) \mathop {\longrightarrow }\limits ^{\scriptscriptstyle {\mathbb P}}0\).
For a Polish space \(S\), \(\mathcal {D}([0,T]:S)\) (resp. \(\mathcal {D}([0,\infty ):S)\)) denote the space of right continuous functions with left limits (RCLL) from \([0,T]\) (resp. \([0,\infty )\)) equipped with the usual Skorohod topology. For a RCLL function \(f:[0,\infty ) \rightarrow \mathbb {R}\), we write \(\Delta f(t) = f(t) - f(t-)\), \(t >0\). Suppose that \((S, \mathcal {S})\) is a measurable space and we are given a partial ordering on \(S\). We say the \(S\) valued random variable \(\xi \) stochastically dominates \(\tilde{\xi }\), and write \(\xi \ge _d \tilde{\xi }\) if there exists a coupling between the two random variables on a common probability space such that \(\xi ^* \ge \tilde{\xi }^*\) a.s., where \(\xi ^*=_d \xi \) and \(\tilde{\xi }^* =_d \tilde{\xi }\). For probability measures \(\mu , \tilde{\mu }\) on \(S\), we say \(\mu \) stochastically dominates \(\tilde{\mu }\), and write \(\mu \ge _d \tilde{\mu }\) if \(\xi \ge _d \tilde{\xi }\) where \(\xi \) has distribution \(\mu \) and \(\tilde{\xi }\) has distribution \(\tilde{\mu }\). Two examples of \(S\) relevant to this work are \(\mathcal {D}([0,T]: \mathbb {R})\) and \(\mathcal {D}([0,T]: \mathcal {G})\) with the natural associated partial ordering. Given a metric space \(S\), we denote by \(\mathcal {B}(S)\) the Borel \(\sigma \)-field on \(S\) and by \(\text{ BM }(S), C_b(S), \mathcal {P}(S)\), the space of bounded (Borel) measurable functions, continuous and bounded function, and probability measures, on \(S\), respectively. The set of nonnegative integers will be denoted by \(\mathbb {N}_0\).
2.2 Bounded-size rules (BSR)
We now define the general class of rules that will be analyzed in this paper. Much of the notation follows [27] which provides a comprehensive analysis of the sub and supercritical regime.
Fix \(K \in \mathbb {N}\) and let \(\Omega _0=\{\varpi \}\) and \(\Omega _K =\{1,2,\ldots , K, \varpi \}\) for \(K \ge 1\), where \(\varpi \) will represent components of size greater than \(K\). Given a graph \({\mathbf{G}}\) and a vertex \(v\in {\mathbf{G}}\), write \(\mathcal {C}_v({\mathbf{G}})\) for the component that contains \(v\). Let
For a quadruple of vertices \(v_1, v_2, v_3, v_4\), write \(\varvec{v} = (v_1,v_2,v_3,v_4)\) and let \(c(\varvec{v}) = (c(v_1), c(v_2), c(v_3), c(v_4))\). Fix \(F\subseteq \Omega _K^4\). We now define the random graph process \(\{\mathbf{{BSR}}^{\scriptscriptstyle (n)}(k)\}_{k\ge 0}\) on the vertex set \([n]\) evolving through a \(F\)-bounded-size rule (\(F\)-BSR) as follows. Define \(\mathbf{{BSR}}^{\scriptscriptstyle (n)}(0) = \mathbf {0}_n\), the graph on \([n]\) with no edges. Having defined \(\mathbf{{BSR}}^{\scriptscriptstyle (n)}(k)\) for \(k \ge 0\), \(\mathbf{{BSR}}^{\scriptscriptstyle (n)}(k+1)\) is constructed as follows: Choose four vertices \((v_1(k), v_2(k), v_3(k), v_4(k))\) uniformly at random amongst all possible \(n^4\) vertices uniformly at random and let
Denote the function \(c(\varvec{v})\) associated with \(\mathbf{{BSR}}^{\scriptscriptstyle (n)}(k)\) as \(c_k(\varvec{v})\). Define
These rules are called bounded-size rules since they treat all components of size greater than \(K\) identically. Concrete examples of such rules include Erdős–Rényi random graph (here \(K=0\), \(F=\Omega _0^4 =\{\varpi ,\varpi ,\varpi ,\varpi \}\)) and Bohman–Frieze process (here \(K=1\), \(F=\{(1,1,\alpha , \beta ): \alpha , \beta \in \Omega _1\}\)).
Continuous time formulation \(\{\mathbf{{BSR}}^{\scriptscriptstyle (n)}(t)\}_{t\ge 0}\): It will be more convenient to work in continuous time. For every quadruple of vertices \(\varvec{v} = (v_1, v_2, v_3, v_4)\in [n]^4\), let \(\mathcal {P}_{\varvec{v}}\) be a Poisson process with rate \(\frac{1}{2n^3}\), independent between quadruples. The continuous time random graph process \(\{\mathbf{{BSR}}^{\scriptscriptstyle (n)}(t)\}_{t\ge 0}\) is constructed recursively as follows. We denote the function \(c(v)\) [resp. \(c(\varvec{v})\)] associated with \(\mathbf{{BSR}}^{\scriptscriptstyle (n)}(t-)\) as \(c_{t-}(v)\) [resp. \(c_{t-}(\varvec{v})\)]. Given \(\mathbf{{BSR}}^{\scriptscriptstyle (n)}(t-)\), and that for some \(\varvec{v} \in [n]^4\), \(\mathcal {P}_{\varvec{v}}\) has a point at the time instant \(t\), we define
The rationale behind this scaling for the rate of the Poisson point process is that the total rate of adding edges is
Thus with this scaling, for the \(F\)-BSR rule corresponding to the Erdős–Rényi evolution, the giant component emerges at time \(t=1\). To simplify notation, when there is no scope for confusion, we will suppress \(n\) in the notation. For example, we write \(\mathbf{{BSR}}_t:= \mathbf{{BSR}}^{\scriptscriptstyle (n)}(t)\).
Denote \(\mathcal {C}_i^{\scriptscriptstyle (n)}(t)\) for the \(i\)th largest component of \(\mathbf{{BSR}}_t\). The work of Spencer and Wormald (see [27]) shows that for any given BSR, there exists a (model dependent) critical time \(t_c>0\) such that for \(t < t_c\), \(|\mathcal {C}_1^{\scriptscriptstyle (n)}(t)|= O(\log n)\) and for \(t > t_c\), \(|\mathcal {C}_1^{\scriptscriptstyle (n)}(t)| \sim f(t)n\) where \(f(t) >0\).
Along with the size of the components, another key quantity of interest is the complexity of components. Recall the definition of the surplus of a component from (1.1), and denote \(\xi _i^{\scriptscriptstyle (n)}(t) :={\mathbf{spls}}(\mathcal {C}_i^{\scriptscriptstyle (n)}(t))\) for the surplus of the component \(\mathcal {C}_i^{\scriptscriptstyle (n)}(t)\). We will be interested in the joint vector of ordered component sizes and corresponding surplus
2.3 Augmented multiplicative coalescent
2.3.1 The multiplicative coalescent
Let \(l^2 = \{x = (x_1,x_2,\ldots ): \sum _i x_i^2< \infty \}\). Then \(l^2\) is a separable Hilbert space with the inner product \(\langle x, y\rangle = \sum _{i=1}^{\infty }x_iy_i\), \(x=(x_i), y= (y_i) \in l^2\). Let
Then \(l^2_{\downarrow }\) is a closed subset of \(l^2\) which we equip with the metric inherited from \(l^2\). In [4] Aldous introduced a \(l^2_{\downarrow }\) valued continuous time Markov process, called the standard multiplicative coalescent, that can be used to describe the asymptotic behavior of suitably scaled component size vector in Erdős–Rényi random graph evolution, near criticality. Subsequently, similar results have been shown to hold for other random graph models (see [6, 7] and references therein). We now give a brief description of this Markov process.
Fix \(x=(x_i)_{i \in \mathbb {N}}\). Let \(\{\xi _{i,j}, i,j \in \mathbb {N}\}\) be a collection of independent rate one Poisson processes. Given \(t \ge 0\), consider the random graph with vertex set \(\mathbb {N}\) in which there exist \(\xi _{i,j}([0,t x_i x_j/2]) + \xi _{j,i}([0,t x_i x_j/2]) \) edges between \((i,j)\), \(1 \le i < j < \infty \), and there are \(\xi _{i,i}([0,t x_i^2/2])\) self-loops at vertex \(i \in \mathbb {N}\). The volume of a component \(\mathcal {C}\) of this graph is defined to be
Let \(X_i(x,t)\) be the volume of the \(i\)th largest (by volume) component. It can be shown that \(X(x,t) = (X_i(x,t), i \ge 1) \in l^2_{\downarrow }\), a.s. (see Lemma 20 in [4]). Define
as \(T_tf(x) = \mathbb {E}(f(X(x,t)))\). It is easily checked that \((T_t)_{t\ge 0}\) satisfies the semigroup property \(T_{t+s} = T_tT_s\), \(s,t\ge 0\), and [4] shows that \((T_t)\) is Feller in the sense that \(T_t(C_b(l^2_{\downarrow })) \subset C_b(l^2_{\downarrow })\) for all \(t \ge 0\). The paper [4] also shows that the semigroup \((T_t)\) along with an initial distribution \(\mu \in \mathcal {P}(l^2_{\downarrow })\) determines a Markov process with values in \(l^2_{\downarrow }\) and RCLL sample paths. Denoting by \(P^{\mu }\) the probability distribution of this Markov process on \(\mathcal {D}([0,\infty ): l^2_{\downarrow })\), the Feller property says that \(\mu \mapsto P^{\mu }\) is a continuous map. One special choice of initial distribution for this Markov process is particularly relevant for the study of asymptotics of random graph models. We now describe this distribution. Let \(\{W(t)\}_{t\ge 0}\) be a standard Brownian motion, and for a fixed \(\lambda \in {\mathbb {R}}\), define
Let \(\hat{W}_{\lambda }\) denote the reflected version of \(W_{\lambda }\), i.e.,
An excursion of \(\hat{W}_\lambda \) is an interval \((l,u) \subset [0,+\infty )\) such that \(\hat{W}_\lambda (l)=\hat{W}_\lambda (u)=0\) and \(\hat{W}_\lambda (t)>0\) for all \(t \in (l,u)\). Define \(u-l\) as the length of the excursion. Order the lengths of excursions of \(\hat{W}_\lambda \) as
and write \(\varvec{X}^*(\lambda ) = (X^*_i(\lambda ):i\ge 1).\) Then \(\varvec{X}^*(\lambda )\) defines a \(l^2_{\downarrow }\) valued random variable (see Lemma 25 in [4]) and let \(\mu _{\lambda }\) be its probability distribution. Using the Feller property and connections with certain inhomogeneous random graph models, the paper [4] shows that \(\mu _{\lambda }T_t = \mu _{\lambda +t}\), for all \(\lambda \in {\mathbb {R}}\) and \(t \ge 0\), where for \(\mu \in \mathcal {P}(l^2_{\downarrow })\), \(\mu T_t \in \mathcal {P}(l^2_{\downarrow })\) is defined in the usual way: \(\mu T_t(A) = \int T_t(\mathbb {1}_A)(x) \mu (dx)\), \(A \in \mathcal {B}(l^2_{\downarrow })\). Using this consistency property one can determine a unique probability measure \(\mu _\mathrm{MC} \in \mathcal {P}(\mathcal {D}((-\infty , \infty ): l^2_{\downarrow }))\) such that, denoting the canonical coordinate process on \(\mathcal {D}((-\infty , \infty ): l^2_{\downarrow })\) by \(\{\pi _t\}_{-\infty < t < \infty }\),
where \(\pi _{t+ \cdot }\) is the process \(\{\pi _{t+s}\}_{s\ge 0}\). The measure \(\mu _\mathrm{MC}\) is known as the standard multiplicative coalescent. This measure plays a central role in characterizing asymptotic distribution of component size vectors in the critical window for random graph models [4, 6, 7].
2.3.2 The augmented multiplicative coalescent
We will now augment the above construction and introduce a measure on a larger space that can be used to describe the joint asymptotic behavior of the component size vector and the associated surplus vector, for a broad family of random graph models.
Let \(\mathbb {N}^{\infty } = \left\{ y=(y_1, \ldots ) : y_i \in \mathbb {N}, \text{ for } \text{ all } i \ge 1\right\} \) and define
We will view \(x_i\) as the volume of the \(i\)th component and \(y_i\) the surplus of the \(i\)th component of a graph with vertex set \(\mathbb {N}\). Writing \(x = (x_i)\) and \(y = (y_i)\), we will sometimes denote \((x_i, y_i)\) as \(z=(x,y)\). We equip \(\mathbb {U}_{\downarrow }\) with the metric
The choice of this metric is discussed in Remark 4.15.
Let \(\mathbb {U}_{\downarrow }^0 = \{(x_i, y_i)_{i\ge 1}\in \mathbb {U}_{\downarrow }: \text{ if } x_k=x_m, k \le m, \text{ then } y_k \ge y_m\}\). We now introduce the augmented multiplicative coalescent (AMC). This is a continuous time Markov process with values in \((\mathbb {U}_{\downarrow }^0, \mathbf{d}_{\scriptscriptstyle \mathbb {U}})\), whose dynamics can heuristically be described as follows: The process jumps at any given time instant from state \((x,y) \in \mathbb {U}_{\downarrow }^0\) to:
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\((x^{ij}, y^{ij})\) at rate \(x_ix_j\), \(i \ne j\), where \((x^{ij}, y^{ij})\) is obtained by merging components \(i\) and \(j\) into a component with volume \(x_i+x_j\) and surplus \(y_i+y_j\) and reordering the coordinates to obtain an element in \(\mathbb {U}_{\downarrow }^0\).
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\((x, y^i)\) at rate \(x_i^2/2\), \(i \ge 1\), where \((x, y^i)\) is the state obtained by increasing the surplus in the \(i\)th component from \(y_i\) to \(y_{i}+1\) and reordering the coordinates (if needed) to obtain an element in \(\mathbb {U}_{\downarrow }^0\).
Whenever \(z=(x,y) \in \mathbb {U}_{\downarrow }^0\) is such that \(\sum _{i=1}^\infty x_i < \infty \), it is easy to construct a well defined Markov process \(\{\varvec{Z}(z,\lambda )\}_{\lambda \ge 0}\) that corresponds to the above transition mechanism, starting at time \(\lambda =0\) in the state \(z\). However when \(\sum _{i=1}^\infty x_i = \infty \), the existence of such a process requires more work. We show in Sect. 4 (see also Theorem 3.1) that in fact there is a well defined Markov process \(\{Z(z,\lambda )\}_{\lambda \ge 0}\) corresponding to the above dynamical description for any \(z \in \mathbb {U}_{\downarrow }^0\). Define, for \(\lambda \ge 0\), \(\mathcal {T}_\lambda : {\text{ BM }}(\mathbb {U}_{\downarrow }^0) \rightarrow {\text{ BM }}(\mathbb {U}_{\downarrow }^0)\) as
As for Aldous’ multiplicative coalescent, there is one particular family of distributions that plays a special role. Recall the reflected parabolic Brownian motion \(\hat{W}_\lambda (t)\) from (2.5). Let \(\mathcal {P}\) be a Poisson point process on \([0,\infty ) \times [0,\infty )\) with intensity \(\lambda _{\infty }^{\otimes 2}\) (where \(\lambda _{\infty }\) is the Lebesgue measure on \([0, \infty )\)) independent of \(\hat{W}_\lambda \). Let \((l_i,r_i)\) be the \(i\)th largest excursion of \(\hat{W}_{\lambda }\). Define
Then \(\varvec{Z}^*(\lambda ) = (\varvec{X}^*(\lambda ), \varvec{Y}^*(\lambda ))\) is a.s. a \(\mathbb {U}_{\downarrow }^0\) valued random variable, where \(\varvec{X}^* = (X^*_i)_{i\ge 1}\) and \(\varvec{Y}^* = (Y^*_i)_{i\ge 1}\). Let \(\nu _{\lambda }\) be its probability distribution. In Theorem 3.1 we will show that there exists a \(\mathbb {U}_{\downarrow }^0\) valued stochastic process \((\varvec{Z}(\lambda ))_{ -\infty < \lambda < \infty }\) such that \(\varvec{Z}(\lambda )\) has probability distribution \(\nu _{\lambda }\) for every \(\lambda \in (-\infty , \infty )\) and for all \(f \in {\text{ BM }}(\mathbb {U}_{\downarrow }^0)\), and \( \lambda _1 < \lambda _2 \), we have
The process \(\varvec{Z}\) will be referred to as the standard augmented multiplicative coalescent. We will also show that \(\{\mathcal {T}_{\lambda }\}_{\lambda \ge 0}\) is a semigroup, which is nearly Feller, in the sense made precise in the statement of Theorem 3.1. It will be seen that this process plays a similar role in characterizing the asymptotic joint distributions of the component size and surplus vector in the critical window as Aldous’ standard multiplicative coalescent does in the study of asymptotics of the component size vector.
3 Results
Our first result establishes the existence of the standard augmented coalescent process. Let \(\mathbb {U}_{\downarrow }^1 = \{z=(x,y) \in \mathbb {U}_{\downarrow }^0: \sum _i x_i = \infty \}\).
Theorem 3.1
There is a collection of maps \(\{\mathcal {T}_t\}_{t\ge 0}\), \(\mathcal {T}_t: {\text{ BM }}(\mathbb {U}_{\downarrow }^0) \rightarrow {\text{ BM }}(\mathbb {U}_{\downarrow }^0)\) and a \(\mathbb {U}_{\downarrow }^0\) valued stochastic process \(\{{\varvec{Z}}(\lambda )\}_{-\infty < \lambda < \infty }= \{(\varvec{X}(\lambda ),\varvec{Y}(\lambda ))\}_{-\infty < \lambda < \infty }\) such that the following hold.
-
(i)
\(\{\mathcal {T}_t\}\) is a semigroup: \(\mathcal {T}_t \circ \mathcal {T}_s = \mathcal {T}_{t+s}\), \(s, t \ge 0\).
-
(ii)
\(\{\mathcal {T}_t\}\) is nearly Feller: For all \(t > 0\), \(f \in {\text{ BM }}(\mathbb {U}_{\downarrow }^0)\) and \(\{z_n\} \subset \mathbb {U}_{\downarrow }^0\), such that \(f\) is continuous at all points in \(\mathbb {U}_{\downarrow }^1\) and \(z_n \rightarrow z\) for some \(z \in \mathbb {U}_{\downarrow }^1\), we have \(\mathcal {T}_{t} f(z_n) \rightarrow \mathcal {T}_tf(z)\).
-
(iii)
The marginal distribution of \(\varvec{Z}(\lambda )\) is characterized through the parabolic reflected Brownian motion \(\hat{W}_{\lambda }\): For each \(\lambda \in \mathcal {R}\), \(\varvec{Z}(\lambda )\) has the probability distribution \(\nu _{\lambda }\).
-
(iv)
The stochastic process \(\{{\varvec{Z}}(\lambda )\}\) satisfies the Markov property with semigroup \(\{\mathcal {T}_t\}\): For all \(f \in {\text{ BM }}(\mathbb {U}_{\downarrow }^0)\), and \( \lambda _1 < \lambda _2 \), we have
$$\begin{aligned} \mathbb {E}[ f(\varvec{Z}(\lambda _2))| \{\varvec{Z}(\lambda )\}_{\lambda \le \lambda _1}] =(\mathcal {T}_{\lambda _2-\lambda _1}f)( \varvec{Z}(\lambda _1) ). \end{aligned}$$ -
(v)
If \(f \in {\text{ BM }}(\mathbb {U}_{\downarrow }^0)\) is such that \(f(x,y)=g(x)\) for some \(g \in {\text{ BM }}(l^2_{\downarrow })\), then
$$\begin{aligned} (\mathcal {T}_t f)(z) = (T_t g)(x), \;\; \forall z=(x,y) \in \mathbb {U}_{\downarrow }^0. \end{aligned}$$Furthermore, \(\{\varvec{X}(\lambda ) \}_{-\infty < \lambda < \infty }\) is Aldous’s standard multiplicative coalescent.
A precise definition of \(\mathcal {T}_t\) can be found in Sect. 4. Theorem 3.1 will be proved in Sect. 5.
Throughout this work we fix \(K \in \mathbb {N}_0\), \(F \in \Omega _K^4\) and consider a \(F\) -BSR as introduced in Sect. 2.2.
The result below considers the asymptotics of the ‘susceptibility functions’. For any given time \(t\) and fixed \(k \ge 1\) define the \(k\)-susceptibility function
Define the scaled susceptibility functions by, for \(k \ge 1\),
Then Theorem 1.1 of [27] shows that for any bounded-size rule, there exists a monotonically increasing function \(s_2 : [0,t_c) \rightarrow [0, \infty )\) satisfying \(s_2(0)=1\) and \(\lim _{t \uparrow t_c}s_2(t)=\infty \), such that
Part (iii) of the following result gives a similar result for \({\bar{s}}_3\). Part (ii) in fact gives convergence of \({\bar{s}}_2, {\bar{s}}_3\) in a stronger sense. Part (i) of the theorem gives precise asymptotics of \(s_2(t)\) and \(s_3(t)\) as \(t \uparrow t_c\).
Theorem 3.2
There exist monotonically increasing functions \(s_k : [0,t_c) \rightarrow [0,\infty )\), \(k=2,3\), such that \(s_2(0)=s_3(0)=1\) and \(\lim _{t \uparrow t_c} s_2(t) = \lim _{t \uparrow t_c} s_3(t) = \infty \), having the following properties.
-
(i)
There exist \(\alpha , \beta \in (0, \infty )\) such that
$$\begin{aligned} s_2(t) = (1+O(t_c-t))\frac{\alpha }{t_c-t}, \quad s_3(t) = \beta [s_2(t)]^3(1+O(t_c-t)), \text{ as } t \uparrow t_c.\nonumber \\ \end{aligned}$$(3.2) -
(ii)
For every \(\gamma \in (1/6, 1/5)\),
$$\begin{aligned}&\sup _{t\in [0, t_n]} \left| \frac{n^{1/3}}{{\bar{s}}_2(t)} - \frac{n^{1/3}}{s_2(t)}\right| \mathop {\longrightarrow }\limits ^{\scriptscriptstyle {\mathbb P}}0 \end{aligned}$$(3.3)$$\begin{aligned}&\sup _{t\in [0, t_n]} \left| \frac{{\bar{s}}_3(t)}{({\bar{s}}_2(t))^3} - \frac{ s_3(t)}{(s_2(t))^3} \right| \mathop {\longrightarrow }\limits ^{\scriptscriptstyle {\mathbb P}}0 , \end{aligned}$$(3.4)where \(t_n = t_c - n^{-\gamma }\).
-
(iii)
For all \(t \in [0, t_c)\), \({\bar{s}}_2(t) \mathop {\longrightarrow }\limits ^{\scriptscriptstyle {\mathbb P}}s_2(t)\), \({\bar{s}}_3(t) \mathop {\longrightarrow }\limits ^{\scriptscriptstyle {\mathbb P}}s_3(t)\) as \(n \rightarrow \infty \).
We now state the main result which gives the asymptotic behavior in the critical scaling window as well as merging dynamics for all bounded-size rules.
Theorem 3.3
(Bounded-size rules: Convergence at criticality) Let \(\alpha , \beta \in (0, \infty )\) be as in Theorem 3.2. For \(\lambda \in {\mathbb {R}}\) define
Then \({\bar{\varvec{Z}}}^{\scriptscriptstyle (n)} = ({\bar{\varvec{C}}}^{\scriptscriptstyle (n)}, {\bar{\varvec{Y}}}^{\scriptscriptstyle (n)})\) is a stochastic process with sample paths in \(\mathcal {D}((-\infty ,\infty ):\mathbb {U}_{\downarrow })\) and for any set of times \(-\infty < \lambda _1< \lambda _2< \cdots < \lambda _m < \infty \)
as \(n\rightarrow \infty \), where \(\varvec{Z}\) is as in Theorem 3.1.
3.1 Background
We now make some comments on the problem background and future directions.
-
(a)
Critical random graphs: Starting with the early work of Erdős–Rényi [15, 16], there is now a large literature on understanding phase transitions in random graph models, see e.g. [12, 13, 20] and the references therein. Proving and identifying phase transitions in dynamic random graph models such as the bounded-size rule requires a relatively new set of ideas and is much more recent [27]. The study of Erdős–Rényi random graph in the critical regime was carried out in [4, 18]. In particular the paper, [4] introduced the standard multiplicative coalescent to understand the merging dynamics of the Erdős–Rényi random graph at criticality. The barely subcritical and supercritical regimes of the Bohman–Frieze process were studied respectively in [22] and [19], with the latter identifying the scaling exponents for the susceptibility functions for the special case of the Bohman–Frieze (BF) process by using the special form of the differential equations for the BF process. The current work extends this result to all bounded-size rules (Theorem 3.2) by viewing such processes as random graph processes with immigration and attachment (see Sect. 6.5).
-
(b)
Unbounded-size rules: One of the reasons for renewed interest in such models is the recent study of the product rule [1], where as before one chooses two edges at random and then uses the edge that minimizes the product of the components at the end points of the chosen edges. This is an example of an unbounded-size rule and simulations in [1] suggest different behavior at criticality as compared to the usual Erdős–Rényi or BF random graph processes. There has been recent progress in rigorously understanding the continuity at the critical point [24] as well the subcritical regime [25]. Such unbounded rules can be regarded as formal limits of \(K\)-bounded-size rules analyzed in the current work, as \(K \rightarrow \infty \). It would be of great interest to identify and understand the critical scaling window of such processes.
-
(c)
Related open questions: In the context of bounded-size rules our results suggest other related questions. In particular, there has been recent progress in understanding structural properties of the components of the Erdős–Rényi random graph at criticality, in particular see [2, 3] which use information about the surplus and component sizes in [4] to prove that the components viewed as metric spaces, converge to random fractals closely related to the continuum random tree [5] with shortcuts due to surplus edges. Our results strongly suggest the components in any bounded-size rule at criticality belong to the same universality class. Proving this will require substantially new ideas.
3.2 Organization of the paper
The two main results in this paper are Theorems 3.1 and 3.3. In Sect. 4 we introduce the semigroup \(\{\mathcal {T}_t\}_{t\ge 0}\) and, as a first step towards Theorem 3.1, establish in Theorem 4.1 the existence of a \(\mathbb {U}_{\downarrow }^0\) valued Markov process associated with this semigroup, starting from an arbitrary initial value. Then in Sect. 5 we complete the proof of Theorem 3.1. We then proceed to the analysis of bounded-size rules in Sect. 6 where we study the differential equation systems associated with the BSR process and prove Theorems 3.2. Finally in Sect. 7 we complete the proof of Theorem 3.3.
4 The augmented multiplicative coalescent
We begin by making precise the formal dynamics of the augmented multiplicative coalescent process given in Sect. 2.3.2. Fix \((x,y)\in \mathbb {U}_{\downarrow }^0\). Let \(\{\xi _{i,j}\}_{i,j \in \mathbb {N}}\) be a collection of i.i.d. rate one Poisson processes. Let \({\mathbf{G}}(z,t)\), where \(z=(x,y)\), be the random graph on vertex set \(\mathbb {N}\) given as follows:
-
(I)
For \(i \in \mathbb {N}\), put \(y_i\) initial self-loops to the vertex \(i\).
-
(II)
For \(i<j \in \mathbb {N}\), put \(\xi _{i,j}([0,t x_i x_j/2]) + \xi _{j,i}([0,t x_i x_j/2])\) edges between vertices \(i\) and \(j\). Also, for \(i \in \mathbb {N}\), put additional \(\xi _{i,i}([0,t x_i^2/2])\) self-loops to the vertex \(i\).
Note that the total number of self-loops at a vertex \(i\) at time instant \(t\) is \(y_i+\xi _{i,i}([0,t x_i^2/2])\). The self-loops coming from (I) and (II) will later be termed as “type I” and “type II” surplus.
Let \(\mathcal {F}^x_t = \sigma \{ \xi _{i,j}([0,sx_ix_j/2]): 0 \le s \le t, \; i,j \in \mathbb {N}\}\), \(t\ge 0\).
Recall the volume of a component \(\mathcal {C}\) is defined to be \({\mathbf{vol}}(\mathcal {C}) = \sum _{i \in \mathcal {C}} x_i\). The surplus of a finite connected graph was defined in (1.1). For infinite graphs the definition requires some care.
We define the surplus for a connected graph \({\mathbf{G}}\) with vertex set a subset of \(\mathbb {N}\) as
where \({\mathbf{G}}^{\scriptscriptstyle [k]}\) is the induced subgraph that has the vertex set \([k]\) (the subgraph with vertex set \([k]\) and all edges between vertices in \([k]\) that are present in \({\mathbf{G}}\)). It is easy to check that this definition of surplus does not depend on the labeling of the vertices. Further note that the surplus of a connected graph might be infinite with this definition.
Thus letting \({\tilde{\mathcal {C}}}_i(t)\) be the \(i\)th largest component (in volume) in \({\mathbf{G}}(z,t)\), define \(X_i(z,t) : = {\mathbf{vol}}({\tilde{\mathcal {C}}}_i(t))\) and \(Y_i(z,t):= {\mathbf{spls}}({\tilde{\mathcal {C}}}_i(t))\) to be the volume and the surplus of the \(i\)th largest component at time \(t\). In case two components have the same volume, the ordering of \(({\tilde{ \mathcal {C}}}_i(t): i \ge 1)\) is taken to be such that \(Y_m(z,t) \ge Y_k(z,t)\) whenever \(m \le k\) and \(X_m(z,t) = X_k(z,t)\).
Let \(\varvec{X}^z(t) := (X_i(z,t): i \ge 1 )\) and \(\varvec{Y}^z(t) := (Y_i(z,t): i \ge 1)\). The paper [4] shows that \(\varvec{X}^z(t) \in l^2_{\downarrow }\) a.s. for all \(t\ge 0\). The following result shows that \(\varvec{Z}^z(t) = (\varvec{X}^z(t), \varvec{Y}^z(t)) \in \mathbb {U}_{\downarrow }^0\) a.s., for all \(t\).
Theorem 4.1
Fix \( z=(x,y)\in \mathbb {U}_{\downarrow }^0\) and let \((\varvec{X}^z(t),\varvec{Y}^z(t))_{t \ge 0}\) be the stochastic process described above, then for any fixed \(t \ge 0\), \((\varvec{X}^z(t),\varvec{Y}^z(t)) \in \mathbb {U}_{\downarrow }^0\).
The above theorem will be proved in Sect. 4.1. For \(t\ge 0\), define \(\mathcal {T}_t: {\text{ BM }}(\mathbb {U}_{\downarrow }^0) \rightarrow {\text{ BM }}(\mathbb {U}_{\downarrow }^0)\) as
The following result shows that \(\{\mathcal {T}_t\}\) is a semigroup that is (nearly) Feller.
Theorem 4.2
For \(t, s \ge 0\), \(\mathcal {T}_t\circ \mathcal {T}_s = \mathcal {T}_{t+s}\). For all \(t > 0\), \(f \in {\text{ BM }}(\mathbb {U}_{\downarrow }^0)\) and \(\{z_n\} \subset \mathbb {U}_{\downarrow }^0\), such that \(f\) is continuous at all points in \(\mathbb {U}_{\downarrow }^1\) and \(z_n \rightarrow z\) for some \(z \in \mathbb {U}_{\downarrow }^1\), we have \(\mathcal {T}_{t} f(z_n) \rightarrow \mathcal {T}_tf(z)\).
The above theorem will be proved in Sect. 4.2. Throughout we will assume, without loss of generality, that for all \(z \in \mathbb {U}_{\downarrow }^0\), \(\varvec{Z}^z\) is constructed using the same set of Poisson processes \(\{\xi _{i,j}\}\). This coupling of \(\varvec{Z}^z\) for different values of \(z\) will not be noted explicitly in the statement of various results.
We begin with the following elementary lemma.
Lemma 4.3
Let \(\{\mathcal {F}_m\}_{m \in \mathbb {N}_0}\) be a filtration given on some probability space.
-
(i)
Let \(\{Z_m\}_{m \ge 0}\) be a \(\{\mathcal {F}_m\}\) adapted sequence of nondecreasing random variables such that \(Z_0=0\). Let \(\lim _{m \rightarrow \infty }Z_m=Z_\infty \). Suppose there exists a nonnegative random variable \(U\) such that \(U<\infty \) a.s. and \( \sum _{m=1}^\infty \mathbb {E}[Z_m-Z_{m-1} | \mathcal {F}_{m-1} ] \le U.\) Then for any \(\epsilon \in (0,1)\),
$$\begin{aligned} \mathbb {P}\{ Z_\infty > \epsilon \} \le \frac{1+\epsilon }{\epsilon } \mathbb {E}[U \wedge 1]. \end{aligned}$$ -
(ii)
Let \(\{A_m\}\) be a sequence of events such that \(A_m \in \mathcal {F}_m\). Suppose there exists a random variable \(U<\infty \) a.s. such that \(\sum _{m=1}^\infty \mathbb {E}[ {\mathbb {1}}_{A_m} | \mathcal {F}_{m-1}] \le U\). Then \(\mathbb {P}\{ A_m \text{ i.o. } \}=0\). Furthermore,
$$\begin{aligned} \mathbb {P}\{ \cup _{m=1}^\infty A_m \} \le 2 \mathbb {E}[U \wedge 1]. \end{aligned}$$
Proof
(i) Define \(B_0 =0\) and \(B_m := \sum _{i=1}^m \mathbb {E}[Z_i-Z_{i-1}| \mathcal {F}_{i-1}]\) for \(m=1,2,\ldots \) Note that \(B_m\) is nondecreasing and \(\mathcal {F}_{m-1}\)-measurable. Define \(\tau =\inf \{ l: B_{l+1} > 1 \}\) where the infimum over an empty set is taken to be \(\infty \). Since \(B_m\) is predictable, \(\tau \) is a stopping time and, for all \(m\), \(B_{m\wedge \tau } \le 1\). Let \(B_{\infty } = \lim _{m\rightarrow \infty } B_m\). Since \(Z_{m \wedge \tau }-B_{m\wedge \tau }\) is a martingale, by the optimal stopping theorem and monotone convergence,
Thus
(ii) The first statement is immediate from the Borel–Cantelli lemma (cf. [14, Theorem 5.3.2]). For the second statement note that for any \(\epsilon \in (0, 1)\), we have \( { \cup _{m=1}^\infty A_m } = \{\sum _{m=1}^\infty {\mathbb {1}}_{A_m} > \epsilon \} \). Now applying part (i) to \(Z_m = \sum _{k=1}^m {\mathbb {1}}_{A_k}\) and taking \(\epsilon \rightarrow 1\) yields the desired result. \(\square \)
Next, we present a result from [4] that will be used here. We begin with some notation. For \(x \in l^2_{\downarrow }\), we write \(x^{\scriptscriptstyle [k]}=(x_1,\ldots ,x_k, 0, 0, \ldots )\) for the \(k\)-truncated version of \(x\). Similarly, for a sequence \(x^{\scriptscriptstyle (n)}=(x_1^{\scriptscriptstyle (n)},x_2^{\scriptscriptstyle (n)},\ldots )\) of elements in \(l^2_{\downarrow }\), \(x^{\scriptscriptstyle (n)[k]}\) is the \(k\)-truncation of \(x^{\scriptscriptstyle (n)}\). For \(z = (x,y), z^{\scriptscriptstyle (n)} = (x^{\scriptscriptstyle (n)}, y^{\scriptscriptstyle (n)}) \in \mathbb {U}_{\downarrow }^0\) \(z^{\scriptscriptstyle [k]}, y^{\scriptscriptstyle [k]}, z^{\scriptscriptstyle (n)[k]}, y^{\scriptscriptstyle (n)[k]}\) are defined similarly.
Recall the construction of \({\mathbf{G}}(z,t)\) described in items (I) and (II) at the beginning of the section. We will distinguish the surplus created in \({\tilde{\mathcal {C}}}_i(t)\) by the action in item (I) and that in item (II). The former will be referred to as the type I surplus and denoted by \(\tilde{Y}_i(z,t)\) while the latter will be referred to as the type II surplus and denoted by \(\hat{Y}_i(z,t) \equiv \hat{Y}_i(x,t)\). More precisely,
Also define
and
The following properties of \(S\) and \(\varvec{X}\) have been established in [4, Proposition 5, Corollary 18, Lemma 22].
Theorem 4.4
-
(i)
For every \(x \in l^2_{\downarrow }\) and \(t\ge 0\), we have \(S(x,t)<\infty \) a.s. and \(S(x^{\scriptscriptstyle [k]}, t) \uparrow S(x,t)\) as \(k \rightarrow \infty \).
-
(ii)
If \(x^{\scriptscriptstyle (n)} \rightarrow x\) in \(l^2_{\downarrow }\), then \(\varvec{X}(x^{\scriptscriptstyle (n)},t) \mathop {\longrightarrow }\limits ^{\scriptscriptstyle {\mathbb P}}\varvec{X}(x,t)\) in \(l^2_{\downarrow }\), as \(n \rightarrow \infty \). In particular, \(\{S(x^{\scriptscriptstyle (n)},t)\}_{n \ge 1}\) is tight.
4.1 Existence of the augmented MC
This section proves Theorem 4.1. We begin by considering the type I surplus.
Proposition 4.5
For any \(t \ge 0\) and \(z \in \mathbb {U}_{\downarrow }^0\), \( \tilde{R}(z,t) = \sum _{i=1}^\infty X_i(z,t) \tilde{Y}_i(z,t) < \infty \) a.s.
Proof of Proposition 4.5 is given below Lemma 4.7. The basic idea is to bound the truncated version \(\tilde{R}^{\scriptscriptstyle [k]}=\tilde{R}(z^{\scriptscriptstyle [k]},t)\) using a martingale argument, and then let \(k \rightarrow \infty \). The truncation error is controlled using Lemma 4.6 below and a suitable supermartingale is constructed in Lemma 4.7.
Lemma 4.6
For every \(z \in \mathbb {U}_{\downarrow }^0\) and \(t \ge 0\), as \(k\rightarrow \infty \), \(\tilde{R}(z^{\scriptscriptstyle [k]},t) \rightarrow \tilde{R}(z,t) \le \infty \) a.s.
Proof
Fix \(t\ge 0\). Denote by \(E_{ij}\) [resp. \(E_{ij}^{\scriptscriptstyle [k]}\)] the event that there exists a path from \(i\) to \(j\) in \({\mathbf{G}}(z,t)\) [resp. \({\mathbf{G}}(z^{\scriptscriptstyle [k]},t)\)], with the convention that \(\mathbb {P}\{E_{ii}\} = \mathbb {P}\{ E_{ii}^{\scriptscriptstyle [k]} \}= 1\). Let
Then
Since \(E_{ij}^{\scriptscriptstyle [k]} \uparrow E_{ij}\), we have \( f_i^{\scriptscriptstyle [k]} \uparrow f_i\). The result now follows from an application of monotone convergence theorem. \(\square \)
Lemma 4.7
Suppose that \(z = (x,y) = z^{\scriptscriptstyle [k]}\) for some \(k \ge 1\) and that \(\sum _j y_j \ne 0\). Then
is a supermartingale with respect to the filtration \(\mathcal {F}_t^x = \sigma \{ \xi _{i,j}([0,sx_ix_j/2]); 0 \le s \le t, \; i,j \in \mathbb {N}\}\).
Proof
From the construction of \(\varvec{Z}(z, \cdot )\) we see that \( \tilde{R}(z,t)\) is a pure jump, nondecreasing process that at any time instant \(t\), jumps at rate \(X_i(z,t-)X_j(z,t-)\), \(1 \le i < j \le k\), with jump sizes \(B_{ij}(t-) = X_i(z,t-)\tilde{Y}_j(z,t-)+X_j(z,t-) \tilde{Y}_i(z,t-)\). Consequently \(\log \tilde{R}(z,t)\) jumps at the same rate, with corresponding jump size \(\log ( 1 + \frac{B_{ij}(t-)}{\tilde{R}(z,t-)})\). From this and elementary properties of Poisson processes it follows that
where \(M\) is a \(\mathcal {F}_t^x \) martingale. Consequently, for \(0 \le s < t < \infty \)
Next note that, for \(u \ge 0\)
Using this observation in (4.1) we now have
The result follows. \(\square \)
Proof of Proposition 4.5
Fix \(z = (x,y) \in \mathbb {U}_{\downarrow }^0\). The result is trivially true if \(\sum _i y_i = 0\). Assume now that \(\sum _i y_i \ne 0\). For \(k \ge 1\) and \(a \in (0, \infty )\), define \(T_{a}^{\scriptscriptstyle [k]} =\inf \{ s \ge 0: S(z^{\scriptscriptstyle [k]},s) \ge a \}.\) Fix \(k \ge 1\) and assume without loss of generality that \(\sum _{i=1}^k y_i > 0\). Write \(R^{\scriptscriptstyle [k]}(t)=R(z^{\scriptscriptstyle [k]},t)\), and \(A^{\scriptscriptstyle [k]}(t)=A(z^{\scriptscriptstyle [k]},t)\) where \(A\) is as in Lemma 4.7. From the supermartingale property \(\mathbb {E}[A^{\scriptscriptstyle [k]} (T_{a}^{\scriptscriptstyle [k]} \wedge t)] \le \mathbb {E}[A^{\scriptscriptstyle [k]}(0)]= \log \tilde{R}^{\scriptscriptstyle [k]}(0)\). Therefore
Thus
By Lemma 4.6, \(\tilde{R}^{\scriptscriptstyle [k]}(t) \rightarrow \tilde{R}(z,t)\), and by Theorem 4.4 (i), \(S(z^{\scriptscriptstyle [k]},t) \rightarrow S(z,t)\) when \(k \rightarrow \infty \). Therefore letting \(k \rightarrow \infty \) on both sides of the above inequality, we have
The result now follows on first letting \(m \rightarrow \infty \) and then \(a \rightarrow \infty \) in the above inequality. \(\square \)
The following result is an immediate consequence of the estimate in (4.2) and Theorem 4.4(ii).
Corollary 4.8
If \(z^{\scriptscriptstyle (n)} \rightarrow z\) in \(\mathbb {U}_{\downarrow }^0\), then for every \(t\ge 0\), \(\{\tilde{R}(z^{\scriptscriptstyle (n)},t)\}_{ n \ge 1}\) is tight.
Next we consider the type II surplus. Let, for \(x \in l^2_{\downarrow }\)
The \(\sigma \)-field \(\mathcal {G}_t^x\) records whether or not \(i\) and \(j\) are in the same component at time \(s\), for all \(i,j\) and for all \(s \le t\). In particular, components \(\{{\tilde{\mathcal {C}}}_i(s), i \ge 1, s \le t\}\) can be determined from the information in \(\mathcal {G}_t^x\) and consequently, \(\varvec{X}(x,t)\) is \(\mathcal {G}_t^x \) measurable.
Lemma 4.9
-
(i)
Fix \(x \in l^2_{\downarrow }\) and \(t \ge 0\). Then \(\hat{R}(x,t) < \infty \) a.s.
-
(ii)
Let \(x^{\scriptscriptstyle (n)} \rightarrow x\) in \(l^2_{\downarrow }\). Then the sequence \(\{ \hat{R}(x^{\scriptscriptstyle (n)},t) \}_{n \ge 1}\) is tight.
Proof
Note that (i) is an immediate consequence of (ii). Consider now (ii). For fixed \(x \in l^2_{\downarrow }\) and \(t \ge 0\), let \(\hat{\mu }_i(x,t)\) denote the conditional law of \(\hat{Y}_i(x,t)\), conditioned on \(\mathcal {G}_t^x\). Then, for a.e. \(\omega \), \(\hat{\mu }_i(x,t)\) is Poisson distribution with parameter
where the last inequality is a consequence of the inequality \(\sum _{j:{\tilde{\mathcal {C}}}_j(s) \subset {\tilde{\mathcal {C}}}_i(t) }(X_j(x,s))^2 \le (X_i(x,t))^2\). Therefore \(\hat{\mu }_i(x,t)\le _d \hat{\nu }_i(x,t)\), a.s., where \(\hat{\nu }_i(x,t)\) is a random probability measure on \(\mathbb {N}\) such that for a.e. \(\omega \), \(\hat{\nu }_i(x,t)\) is Poisson distribution with parameter \(\frac{t}{2} (X_i(x,t, \omega ))^2 \).
A similar argument shows that the conditional distribution of \(\sum _{i=1}^{\infty } \hat{Y}_i(x,t)\), given \(\mathcal {G}_t^x\) is a.s. stochastically dominated by a random measure on \(\mathbb {N}\) that, for a.e. \(\omega \) has a Poisson distribution with parameter \(\sum _{i=1}^{\infty }\frac{t}{2} (X_i(x,t, \omega ))^2 = \frac{t}{2} S(x,t)\). Also, if \(x^{\scriptscriptstyle (n)}\) is a sequence converging to \(x\) in \(l^2_{\downarrow }\), we have that for each \(n\), the conditional distribution of \(\sum _{i=1}^{\infty } \hat{Y}_i(x^{\scriptscriptstyle (n)},t)\), given \(\mathcal {G}_t^{x^{\scriptscriptstyle (n)}}\) is a.s. stochastically dominated by a Poisson random variable with parameter \(\frac{t}{2} S(x^{\scriptscriptstyle (n)},t)\). From Theorem 4.4(ii), \(\{S(x^{\scriptscriptstyle (n)},t)\}_{n\ge 1}\) is tight. Combining these facts we have that \(\{\sum _{i=1}^{\infty } \hat{Y}_i(x^{\scriptscriptstyle (n)},t)\}_{n\ge 1}\) is a tight family. Finally, note that \(\hat{R}(x^{\scriptscriptstyle (n)},t) \le X_1(x^{\scriptscriptstyle (n)},t)\left( \sum _{i=1}^{\infty } \hat{Y}_i(x^{\scriptscriptstyle (n)},t) \right) \). The tightness of \(\{ \hat{R}(x^{\scriptscriptstyle (n)},t) \}_{n \ge 1}\) now follows on combining the above established tightness of \(\{\sum _{i=1}^{\infty } \hat{Y}_i(x^{\scriptscriptstyle (n)},t)\}_{n\ge 1}\) and the tightness of \(\{X_1(x^{\scriptscriptstyle (n)},t)\}_{n\ge 1}\), where the latter is once again a consequence of Theorem 4.4(ii).\(\square \)
We now complete the proof of Theorem 4.1.
Proof of Theorem 4.1
Fix \(z=(x,y) \in \mathbb {U}_{\downarrow }^0\) and \(t \ge 0\). From Lemma 4.9 (i) \(\hat{R}(x,t) < \infty \) a.s. Also, from Proposition 4.5, \(\tilde{R}(z,t) < \infty \) a.s. The result now follows on recalling that \(R(z,t) = \hat{R}(x,t) + \tilde{R}(z,t)\). \(\square \)
We also record the following consequence of Lemma 4.9 and Corollary 4.8 for future use.
Corollary 4.10
If \(z^{\scriptscriptstyle (n)} \rightarrow z\) in \(\mathbb {U}_{\downarrow }^0\), then \( \{R(z^{\scriptscriptstyle (n)},t)\}_{n\ge 1} \) is tight.
4.2 Feller property of the augmented MC
In this section, we will prove Theorem 4.2. In fact we will show that if \( z^{\scriptscriptstyle (n)}= (x^{\scriptscriptstyle (n)},y^{\scriptscriptstyle (n)})\) converges to \(z=(x,y)\) in \(\mathbb {U}_{\downarrow }^0\), and \(z \in \mathbb {U}_{\downarrow }^1\), then
We start with the following elementary lemma.
Lemma 4.11
Suppose \((x,y), (x^{\scriptscriptstyle (n)}, y^{\scriptscriptstyle (n)}) \in \mathbb {U}_{\downarrow }\) for \(n \ge 1\). Then
if and only if the following three conditions hold:
-
(i)
\(\lim _{n \rightarrow \infty }\sum _{i=1}^\infty (x_i^{\scriptscriptstyle (n)}-x_i)^2 = 0\).
-
(ii)
\(y_i^{\scriptscriptstyle (n)}=y_i\) for \(n\) sufficiently large, for all \(i \ge 1\).
-
(iii)
\(\lim _{n \rightarrow \infty }\sum _{i=1}^\infty x_i^{\scriptscriptstyle (n)}y_i^{\scriptscriptstyle (n)} = \sum _{i=1}^\infty x_iy_i\).
Proof
The “only if” part is immediate. To see the “if” part, note that the first two conditions imply \(\lim _{n\rightarrow \infty } x_i^{\scriptscriptstyle (n)} y_i^{\scriptscriptstyle (n)}= x_i y_i\) for all \(i \ge 1\). By the third condition and Scheffe’s lemma, we now have \(\lim _{n\rightarrow \infty } \sum _i | x_i^{\scriptscriptstyle (n)} y_i^{\scriptscriptstyle (n)} - x_i y_i|=0\). The result follows.\(\square \)
The key ingredient in the proof is the following lemma the proof of which is given after Lemma 4.14.
Lemma 4.12
Let \( z^{\scriptscriptstyle (n)}= (x^{\scriptscriptstyle (n)},y^{\scriptscriptstyle (n)})\) converge to \(z=(x,y)\) in \(\mathbb {U}_{\downarrow }^0\). Suppose that \(z \in \mathbb {U}_{\downarrow }^1\). Then
-
(i)
\(Y_i(z^{\scriptscriptstyle (n)},t) \mathop {\longrightarrow }\limits ^{\scriptscriptstyle {\mathbb P}}Y_i(z,t)\) for all \(i \ge 1\).
-
(ii)
\(\sum _{i=1}^\infty X_i(z^{\scriptscriptstyle (n)},t)Y_i(z^{\scriptscriptstyle (n)},t) \mathop {\longrightarrow }\limits ^{\scriptscriptstyle {\mathbb P}}\sum _{i=1}^\infty X_i(z,t)Y_i(z,t)\).
Proof of Theorem 4.2 can now be completed as follows.
Proof of Theorem 4.2
The first part of the theorem is immediate from the construction given at the beginning of Sect. 4 and elementary properties of Poisson processes. For the second part, consider \( z^{\scriptscriptstyle (n)}= (x^{\scriptscriptstyle (n)},y^{\scriptscriptstyle (n)})\), \(z=(x,y)\) as in the statement of the theorem. It suffices to prove (4.3). From Theorem 4.4(ii), \(\varvec{X}(z^{\scriptscriptstyle (n)},t) \rightarrow \varvec{X}(z,t)\) in probability, in \(l^2_{\downarrow }\). The result now follows on combining this convergence with the convergence in Lemma 4.12 (on noting that \((\varvec{X}(z,t),\varvec{Y}(z,t)) \in \mathbb {U}_{\downarrow }^1\) a.s.) and applying Lemma 4.11.\(\square \)
Rest of this section is devoted to the proof of Lemma 4.12. The key idea of the proof is as follows. Consider the induced subgraphs on the first \(k\) vertices \({\mathbf{G}}^{\scriptscriptstyle [k]}={\mathbf{G}}(z^{\scriptscriptstyle [k]},t)\) and \({\mathbf{G}}^{\scriptscriptstyle (n)[k]}={\mathbf{G}}(z^{\scriptscriptstyle (n)[k]},t)\). Since there are only finite number of vertices in \({\mathbf{G}}^{\scriptscriptstyle [k]}\), when \(n \rightarrow \infty \), \({\mathbf{G}}^{\scriptscriptstyle (n)[k]}\) will eventually be identical to \({\mathbf{G}}^{[k]}\) almost surely. The main step in the proof is to control the difference between \({\mathbf{G}}^{\scriptscriptstyle (n)[k]}\) and \({\mathbf{G}}^{\scriptscriptstyle (n)}\) when \(k\) is large, uniformly for all \(n\). For this we first analyze the difference between \({\mathbf{G}}^{\scriptscriptstyle (n)[k]}\) and \({\mathbf{G}}^{\scriptscriptstyle (n)[k+1]}\) in the lemma below.
Consider the set of vertices \([k+1]=\{1,2,\ldots ,k, k+1\}\), and for every \(i \in [k+1]\), let vertex \(i\) have label \((x_i, y_i)\) representing its size and surplus, respectively. Suppose \(x_1 \ge x_2 \ge \cdots \ge x_{k+1}\). Fix \(t > 0\). Define a random graph \({\mathbf{G}}^*\) on the above vertex set as follows. For \(i\le k\), the number of edges, \(N_i\), between \(i\) and \(k+1\) is distributed as Poisson \((tx_i x_{k+1})\). In addition, there are \(N_0\) = Poisson \((t x^2_{k+1}/2)\) self-loops to the vertex \(k+1\). All the Poisson random variables are taken to be mutually independent.
Denote \(X_i\) and \(Y_i\) for the component volumes and surplus of the resulting star-like graph if \(i\) is the smallest labeled vertex in its component; otherwise let \(X_i=Y_i=0\). A precise definition of \((X_i, Y_i)\) is as follows. Write \(i \sim k+1\) if there is an edge between \(i\) and \(k+1\) in \({\mathbf{G}}^*\). By convention \((k+1) \sim (k+1)\). Let \(\mathcal {J}_k = \{i \in [k+1]: i \sim k+1\}\), and \(i_0 = \min \{i: i \in \mathcal {J}_k\}\). Then
Define \(R_k=\sum _{i=1}^k x_i y_i\), \(S_k=\sum _{i=1}^k x_i^2\), \(R_{k+1}=\sum _{i=1}^{k+1} X_iY_i\). Then we have the following result.
Lemma 4.13
-
(i)
\(\mathbb {P}\{ Y_i \ne y_i \} \le tx_{k+1}y_{k+1} x_1 + t x_{k+1}^2 \left( 1 + it x_1^2 + tS_k +tR_k x_1 \right) \).
-
(ii)
\( \mathbb {E}[R_{k+1}-R_k] \le x_{k+1}y_{k+1} ( 1 + t S_k) + x_{k+1}^2 (t R_k + t^2 S_k R_k + t^2 S_k x_1) + tx_{k+1}^3 (1 + 2tS_k +t^2S_k^2). \)
Proof
(i) It is easy to see that, for \(i = 1, \ldots k\),
Using the observation that for a Poisson\((\lambda )\) random variable \(Z\), \(\mathbb {P}\{Z \ge 1\} < \lambda \) and \(\mathbb {P}\{ Z \ge 2\} < \lambda ^2\), we now have that
The proof is now completed on collecting all the terms and using the fact that \(x_i \le x_1\) for every \(i\).
(ii) Note that
Then
The result now follows on taking expectations in the above equation and using the fact that \(\mathbb {E}[(N_j-1)^+] < (tx_jx_{k+1})^2 \).\(\square \)
Recall that, by construction, \(X_i(z,t) \ge X_{i+1}(z,t)\) for all \(z \in \mathbb {U}_{\downarrow }\), \(t\ge 0\) and \(i \in \mathbb {N}\). The following lemma which is a key ingredient in the proof of Lemma 4.12 says that if \(z \in \mathbb {U}_{\downarrow }^1\), ties do not occur, a.s.
Lemma 4.14
Let \(z \in \mathbb {U}_{\downarrow }^1\). Then for every \(t > 0\) and \(i \in \mathbb {N}\), \(X_i(z,t) > X_{i+1}(z,t)\) a.s.
Proof
Fix \(t > 0\). Consider the graph \({\mathbf{G}}(z,t)\) and write \(\mathcal {C}_{x_i} \equiv \mathcal {C}_{x_i}(t)\) for the component of vertex \((x_i,y_i)\) at time \(t\). It suffices to show for all \(i\ne j\)
The key property we shall use is that for \(z=(x,y) \in \mathbb {U}_{\downarrow }^1\), \(\sum _{i=1}^\infty x_i =\infty \). Now fix \(i\ge 1\). It is enough to show that \(|\mathcal {C}_{x_i}|\) has no atom i.e for all \((x,y)\in \mathbb {U}_{\downarrow }^1\)
To see this, first note that since \(|\mathcal {C}_{x_i}|< \infty \) a.s., conditional on \(\mathcal {C}_{x_i}\) the vector \(z^* = ((x_k,y_k): x_k\notin \mathcal {C}_{x_i}) \in \mathbb {U}_{\downarrow }^1\) almost surely. Thus on the event \(x_j\notin \mathcal {C}_{x_i}\), conditional on \(\mathcal {C}_{x_i}\), using (4.5) with \(a= |\mathcal {C}_{x_i}|\) implies that \(\mathbb {P}(|\mathcal {C}_{x_j}| = |\mathcal {C}_{x_i}|\mid \mathcal {C}_{x_i}) = 0\) and this completes the proof. Thus it is enough to prove (4.5). For the rest of the argument, to ease notation let \(i=1\). Let us first show the simpler assertion that the volume of direct neighbors of \(x_1\) has a continuous distribution. More precisely, let \(N_{i,j}(t):=\xi _{i,j}([0,tx_ix_j/2])+\xi _{j,i}([0,tx_ix_j/2])\), \(1 \le i < j\), denote the number of edges between any two vertices \(x_i\) and \(x_j\) by time \(t\). Then the volume of direct neighbors of the vertex \(x_1\) is \(L := \sum _{i = 2 }^\infty x_i {\mathbb {1}}_{\{N_{1,i}(t) \ge 1\}} \) and we will first show that \(L\) has no atom, namely
For any random variable \(X\) define the maximum atom size of \(X\) by
For two independent random variables \(X_1\) and \(X_2\) we have \(\mathbf{atom}(X_1 + X_2) \le \min \{ \mathbf{atom}(X_1), \mathbf{atom}(X_2) \}\). For \(m \ge 2\), define \(L_m = \sum _{i=m}^\infty x_i {\mathbb {1}}_{\{N_{1,i}(t) \ge 1\}}\). Since \(L_m\) and \(L-L_m\) are independent, we have \( \mathbf{atom}(L) \le \mathbf{atom}(L_m).\) Define the event
and write
Then \(L^*_m(t)\) is a pure jump Levy process with Levy measure \(\nu (du) = \sum _{i=m}^\infty x_1 x_i \delta _{x_i}(du)\). By [17], such a Levy process has continuous marginal distribution since the Levy measure is infinite (\( \nu (0,\infty ) = (\sum _{i=m}^\infty x_i) x_1 = \infty \)) . Thus \(L^*_m(t)\) has no atom. Next, for any \(a \in \mathbb {R}\),
Thus \(\mathbf{atom}(L) \le \mathbf{atom}(L_m) \le \frac{t^2 x_1^2}{2} \sum _{i=m}^\infty x_i^2\). Since \(m\) is arbitrary, we have \(\mathbf{atom}(L)=0\). Thus \(L\) is a continuous variable, and (4.6) is proved.
Let us now strengthen this to prove (4.5). Let \(\tilde{\mathbf{G}}\) be the subgraph of \({\mathbf{G}}(z,t)\) obtained by deleting the vertex \(x_1\) and all related edges. Let \(\tilde{X}_i\) be the volume of the \(i\)th largest component of \(\tilde{\mathbf{G}}\). Note that \(\sum _{i=1}^\infty \tilde{X}_i = \sum _{i=2}^\infty x_i = \infty \) a.s. Conditional on \((\tilde{X}_i)_{i\ge 1}\), let \(\tilde{N}_{1,i}\) have Poisson distribution with parameter \(t x_1 \tilde{X}_i\). Then
where the second term has the same form as the random variable \(L\). Using (4.6) completes the proof. \(\square \)
We now proceed to the proof of Lemma 4.12.
Proof of Lemma 4.12
Fix \(t > 0\) and \(z^{\scriptscriptstyle (n)}, z\) as in the statement of the lemma. Denote \( Y^{\scriptscriptstyle [k]} =Y(z^{\scriptscriptstyle [k]},t), \;\; Y^{\scriptscriptstyle (n)[k]}= Y(z^{\scriptscriptstyle (n)[k]},t)\). Similarly, denote \(\mathcal {C}^{\scriptscriptstyle [k]}_i\) and \(\mathcal {C}_i^{\scriptscriptstyle (n)[k]}\) for the corresponding \(i\)th largest component; and \(X_i^{\scriptscriptstyle [k]}\) and \(X_i^{\scriptscriptstyle (n)[k]}\) for their respective sizes. Also, write \(X^{\scriptscriptstyle (n)} = X(x^{\scriptscriptstyle (n)},t)\) and define \(Y^{\scriptscriptstyle (n)}, R^{\scriptscriptstyle (n)}, S^{\scriptscriptstyle (n)}\) similarly.
For \(i \in \mathbb {N}\), define the event \(E_i^{\scriptscriptstyle (n)[k]}\) as,
and define \(E_i^{\scriptscriptstyle [k]}\) similarly. Then
Note that
Thus the probability of the event \(\{ Y_i^{\scriptscriptstyle (n)[m+1]} \ne Y_i^{\scriptscriptstyle (n)[m]}, E_i^{\scriptscriptstyle (n)[k]} \}\), for \(m \ge k\), can be estimated using Lemma 4.13 (i). More precisely, let \(\mathcal {F}^{\scriptscriptstyle [m]}=\sigma \{ \xi _{i,j}; {i , j \le m} \}\) for \( m \ge 1\). Then by Lemma 4.13 (i),
where \(S^{\scriptscriptstyle (n)[m]} = \sum _i (X_i^{\scriptscriptstyle (n)[m]})^2\) and \(R^{\scriptscriptstyle (n)[m]} = \sum _i (X_i^{\scriptscriptstyle (n)[m]}Y_i^{\scriptscriptstyle (n)[m]})\).
Note that \(X_1^{\scriptscriptstyle (n)[k]} \le X_1^{\scriptscriptstyle (n)}\), \(R^{\scriptscriptstyle (n)[k]} \le R^{\scriptscriptstyle (n)}\) and \(S^{\scriptscriptstyle (n)[k]} \le S^{\scriptscriptstyle (n)}\). Thus we have
Denote the right hand side of the above inequality as \(U^{\scriptscriptstyle (n)[k]}\). Then by Lemma 4.3(ii), we have
and therefore
Next note that \(X^{\scriptscriptstyle (n)}_1\), \(S^{\scriptscriptstyle (n)}\) and \(R^{\scriptscriptstyle (n)}\) are all tight sequences by Corollary 4.10 and Theorem 4.4(ii). Thus \((1+ it (X_1^{\scriptscriptstyle (n)})^2 + tS^{\scriptscriptstyle (n)} + tR^{\scriptscriptstyle (n)} X_1^{\scriptscriptstyle (n)})\) is also tight. Also, since \(z^{\scriptscriptstyle (n)} \rightarrow z\),
Combining the above observations we have that \(\limsup _{k \rightarrow \infty } \limsup _{ n \rightarrow \infty } \mathbb {P}\{U^{\scriptscriptstyle (n)[k]}>\epsilon \}=0\) for all \(\epsilon > 0\). From the inequality
we now see that
Next, from a straightforward extension of Proposition 5 of Aldous [4] we have that \((\varvec{X}^{\scriptscriptstyle (n)}, X_1^{\scriptscriptstyle (n)[k]}, \ldots , X_i^{\scriptscriptstyle (n)[k]} ) \mathop {\longrightarrow }\limits ^{d}(\varvec{X}(t), X_1^{\scriptscriptstyle [k]}, \ldots , X_i^{\scriptscriptstyle [k]} )\) in \(l^2_{\downarrow }\times \mathbb {R}^i\) when \(n \rightarrow \infty \), for each fixed \(i\) and \(k\). Combining this with Lemma 4.14 we now see that for fixed \(i\)
Also, for each fixed \(k\)
Observing that \(\lim _{k \rightarrow \infty } Y_i^{\scriptscriptstyle [k]}=Y_i(t)\) and the last term in (4.9) does not depend on \(n\), we have that
Part (i) of the lemma now follows on combining the above observations and taking limit as \(n \rightarrow \infty \) and then \(k \rightarrow \infty \) in (4.9).
We now prove part (ii) of the lemma.
Note that
With a similar argument as in Lemma 4.6, we have \(R^{[k]} \rightarrow R(z,t)\) as \(k \rightarrow \infty \). Thus sending \(k \rightarrow \infty \) in the above display we have
To complete the proof, it suffices to show that
Note that
The second term on the right side above goes to zero for each fixed \(k\), as \(n \rightarrow \infty \). For the first term, note that by Lemma 4.13(ii), for all \(m \ge k\)
where \(U_1^{\scriptscriptstyle (n)}= 1 + t S^{\scriptscriptstyle (n)}\), \( U_2^{\scriptscriptstyle (n)}= t R^{\scriptscriptstyle (n)} + t^2 S^{\scriptscriptstyle (n)}R^{\scriptscriptstyle (n)}+t^2 S^{\scriptscriptstyle (n)} X_1^{\scriptscriptstyle (n)}\) and \(U_3^{\scriptscriptstyle (n)}= t(1 + 2t S^{\scriptscriptstyle (n)} + t^2 (S^{\scriptscriptstyle (n)})^2)\). Thus by Lemma 4.3 (i),
where \(U^{\scriptscriptstyle (n)[k]}=( \sum _{m=k+1}^\infty x^{\scriptscriptstyle (n)}_m y^{\scriptscriptstyle (n)}_m ) U_1^{\scriptscriptstyle (n)} + (\sum _{m=k+1}^\infty (x^{\scriptscriptstyle (n)}_m)^2) U_2^{\scriptscriptstyle (n)} + (\sum _{m=k+1}^\infty (x^{\scriptscriptstyle (n)}_{m+1})^3) U_3^{\scriptscriptstyle (n)} \). Note that \(U_1^{\scriptscriptstyle (n)}\), \(U_2^{\scriptscriptstyle (n)}\) and \(U_3^{\scriptscriptstyle (n)}\) are all tight sequences and \(z^{\scriptscriptstyle (n)} \rightarrow z\). An argument similar to the one used to prove (4.10) now shows that, for all \(\epsilon > 0\),
The statement in (4.12) now follows on using the above convergence in (4.13) and combining it with the observation below (4.13). This completes the proof of part (ii). \(\square \)
Remark 4.15
Lemma 4.12 is at the heart of the (near) Feller property in Theorem 4.2 which is crucial for the proof of the joint convergence in (3.5). The proof of the lemma reveals the reason for considering the metric \(\mathbf{d}_{\scriptscriptstyle \mathbb {U}} \) on \(\mathbb {U}_{\downarrow }\).
One natural metric on \(\mathbb {U}_{\downarrow }\), denoted by \(\mathbf{d}_1\), is the one obtained by replacing the second term in (2.6) with
This metric corresponds to the topology on \(\mathbb {U}_{\downarrow }\) inherited from \(\ell ^2 \times \mathbb {N}^{\infty }\) taking the topology generated by the inner product \(\langle \cdot , \cdot \rangle \) on \(\ell ^2\) and the product topology on \(\mathbb {N}^{\infty }\); and then considering the product topology on \(\ell ^2 \times \mathbb {N}^{\infty }\).
Another metric (which we denote by \(\mathbf{d}_2\)) that can be considered on \(\mathbb {U}_{\downarrow }\) corresponds to replacing the second term in (2.6) with \(\mathbf{d}_{vt}(\mu _z, \mu _{z'})\), where \(\mu _z = \sum _{i=1}^{\infty } \delta _{z_i}\), \(\mu _{z'} = \sum _{i=1}^{\infty } \delta _{z'_i}\) and \(\mathbf{d}_{vt}\) is the metric corresponding to the vague topology on the space of \(\mathbb {N}\cup \{\infty \}\) valued locally finite measures on \((0,\infty ) \times \mathbb {N}\).
The proof of Lemma 4.12 hinges upon the convergence of \(\sum _{m=1}^\infty x^{\scriptscriptstyle (n)}_m y^{\scriptscriptstyle (n)}_m \) to \(\sum _{m=1}^\infty x_m y_m \), as \(n\rightarrow \infty \), even for the proof of convergence of \(Y_i(z^{\scriptscriptstyle (n)},t) \mathop {\longrightarrow }\limits ^{\scriptscriptstyle {\mathbb P}}Y_i(z,t)\). Since \(\mathbf{d}_1\) and \(\mathbf{d}_2\) give no control over sums of the form \(\sum _{m=1}^\infty x_m y_m \), this suggests that the convergence in \(\mathbf{d}_1\) or \(\mathbf{d}_2\) is “too weak” to yield the desired Feller property.
5 The standard augmented multiplicative coalescent
In this section we prove Theorem 3.1. The Proposition 4 of [4] proves a very useful result on convergence of component size vectors of a general family of non-uniform random graph models to the ordered excursion lengths of \(\hat{W} _{\lambda }\). We begin in this section by extending this result to the joint convergence of component size and component surplus vectors in \(\mathbb {U}_{\downarrow }\), under a slight strengthening of the conditions assumed in [4].
Recall the excursion lengths and mark count process \(\varvec{Z}^*(\lambda ) = (\varvec{X}^*(\lambda ), \varvec{Y}^*(\lambda ))\) defined in Sect. 2.3.2. Our first result below shows that, for fixed \(\lambda \in \mathbb {R}\), \(\varvec{Z}^*(\lambda )\) arises as a limit of \(\varvec{Z}(z^{\scriptscriptstyle (n)}, q^{\scriptscriptstyle (n)})\) in \(\mathbb {U}_{\downarrow }\) for all sequences \(\{z^{\scriptscriptstyle (n)}\} \subset \mathbb {U}_{\downarrow }\) and \(q^{\scriptscriptstyle (n)}=q^{\scriptscriptstyle (n)}_\lambda \subset (0, \infty )\) that satisfy certain regularity conditions.
For \(n \ge 1\), let \(z^{\scriptscriptstyle (n)} = (x^{\scriptscriptstyle (n)}, y^{\scriptscriptstyle (n)}) \in \mathbb {U}_{\downarrow }^0\).
Writing \(z_i^{\scriptscriptstyle (n)} = (x_i^{\scriptscriptstyle (n)}, y_i^{\scriptscriptstyle (n)})\), \(i \ge 1\), define
Note that \(x^{* \scriptscriptstyle (n)}=x_1^{\scriptscriptstyle (n)}\) since the sequence is ordered. Let \(\{q^{\scriptscriptstyle (n)}\}\) be a nonnegative sequence. We will suppress \((n)\) from the notation unless needed.
Theorem 5.1
Let \(z^{\scriptscriptstyle (n)} = (z_1^{\scriptscriptstyle (n)}, \ldots )\in \mathbb {U}_{\downarrow }^0\) be such that \(x^{\scriptscriptstyle (n)}_i = 0\) for all \(i >n\) and \(y^{\scriptscriptstyle (n)}_i = 0\) for all \(i \ge 1\). Suppose that, as \(n\rightarrow \infty \),
and, for some \(\varsigma \in (0, \infty )\),
Then \(\varvec{Z}^{\scriptscriptstyle (n)} = \varvec{Z}(z^{\scriptscriptstyle (n)},q^{\scriptscriptstyle (n)})\) converges in distribution in \(\mathbb {U}_{\downarrow }\) to \(\varvec{Z}^*(\lambda )\).
Remark
The convergence assumption in (5.1) is the same as that in Proposition 4 of [4]. The additional assumption in (5.2) is not very stringent as will be seen in Sect. 7 when this result is applied to a general family of bounded-size rules.
Given Theorem 5.1, the proof of Theorem 3.1 can now be completed as follows.
Proof of Theorem 3.1
The first two parts of the theorem were shown in Theorem 4.2. Also, part (v) of the theorem is immediate from the definition of \(\{T_t\}\) in Sect. 2.3.1. Recall the definition of \(\nu _{\lambda }\) from Sect. 2.3.2. In order to prove parts (iii)-(iv) it suffices to show that
Indeed, using the semigroup property of \((\mathcal {T}_{\lambda })\) and the above relation, it is straightforward to define a consistent family of finite dimensional distributions \(\mu _{\lambda _1, \ldots \lambda _k}\) on \((\mathbb {U}_{\downarrow })^{\otimes k}\), \(-\infty < \lambda _1 < \lambda _2, \ldots \lambda _k<\infty \), \(k \ge 1\), such that \(\mu _{\lambda } = \nu _{\lambda }\) for every \(\lambda \in \mathbb {R}\). The desired result then follows from Kolmogorov’s consistency theorem.
We now prove (5.3). Let
We set \(z^{\scriptscriptstyle (n)}_i =0\) for \(i > n\). Note that with this choice of \(x^{\scriptscriptstyle (n)}\), \(s_1=n^{1/3}, s_2 = n^{-1/3}, s_3 = n^{-1}\) and so clearly (5.1) and (5.2) (with any \(\varsigma > 1\)) are satisfied with \(q = q_{\lambda _j}\), \(\lambda = \lambda _j\), \(j=1,2\). Thus, denoting the distribution of \(\varvec{Z}(z^{\scriptscriptstyle (n)}, q^{\scriptscriptstyle (n)}_{\lambda _j})\) by \(\nu _{\lambda _j}^{\scriptscriptstyle (n)}\), we have by Theorem 5.1 that
Also, from the construction of \(\varvec{Z}(z,t)\) in Sect. 4, it is clear that \(\nu _{\lambda _1}^{\scriptscriptstyle (n)}\mathcal {T}_{\lambda _{2}-\lambda _1} = \nu _{\lambda _2}^{\scriptscriptstyle (n)}\). The result now follows on combining the convergence in (5.4) with Theorem 4.2 and observing that \(\varvec{Z}^*(\lambda ) \in \mathbb {U}_{\downarrow }^1\) a.s. for every \(\lambda \in \mathbb {R}\). \(\square \)
Rest of this section is devoted to the proof of Theorem 5.1 and is organized as follows. Recall the random graph process \({\mathbf{G}}(z,q)\), for \(z \in \mathbb {U}_{\downarrow }\), \(q\ge 0\), defined at the beginning of Sect. 4. In Sect. 5.1 we will give an equivalent in law construction of \({\mathbf{G}}(z,q)\), from [4], that defines the random graph simultaneously with a certain breadth-first-exploration random walk. The excursions of the reflected version of this walk encode the component sizes of the random graph while the area under the excursions gives the parameter of the Poisson distribution describing the (conditional) law of the surplus associated with the corresponding component. Using this construction, in Theorem 5.2, we will first prove a weaker result than Theorem 5.1 which proves the convergence in distribution of \(\varvec{Z}^{\scriptscriptstyle (n)}\) to \(\varvec{Z}^*(\lambda )\) in \(l^2_{\downarrow }\times \mathbb {N}^{\infty }\), where we consider the product topology on \(\mathbb {N}^{\infty }\). This result is proved in Sect. 5.2. In Sect. 5.3 we will give the proof of Theorem 5.1 using Theorem 5.2 and an auxiliary tightness lemma (Lemma 5.4). Finally, proof of Lemma 5.4 is given in Sect. 5.4.
5.1 Breadth first exploration walk
In this section, following [4], we will give an equivalent in law construction of \({\mathbf{G}}(z,q)\) that defines the random graph simultaneously with a certain breadth-first-exploration random walk. Given \(q \in (0, \infty )\) and \(z \in \mathbb {U}_{\downarrow }^0\) such that \(x_i=0\) for all \(i > n\) and \(y_i=0\) for all \(i\), we will construct a random graph \({\bar{{\mathbf{G}}}}(z,q)\) that is equivalent in law to \({\mathbf{G}}(z,q)\), in two stages, as follows. We begin with a graph on \([n]\) with no edges. Let \(\{\eta _{i,j}\}_{i,j \in \mathbb {N}}\) be independent Poisson point processes on \([0,\infty )\) such that \(\eta _{ij}\) for \(i\ne j\) has intensity \(qx_j\); and for \(i=j\) has intensity \(qx_i/2\).
Stage I: The breadth-first-search forest and associated random walk: Choose a vertex \(v(1) \in [n]\) with \(\mathbb {P}(v(1) = i)\propto x_i\). Let
Form an edge between \(v(1)\) and each \(j \in \mathbb {I}_1\). Let \(c(1) = |\mathbb {I}_1|\). Let \(m_{v(1),j}\) be the first point in \(\eta _{v(1),j}\) for each \(j \in \mathbb {I}_1\). Order the vertices in \(\mathbb {I}_1\) according to increasing values of \(m_{v(1),j}\) and label these as \(v(2), \ldots v(c(1) + 1)\). Let
Having defined \(\mathcal {V}_{i'}\), \(\mathcal {N}_{i'}\), \(l_{i'}\), \(d_{i'}\) and the edges up to step \(i'\), with \(\mathcal {V}_{i'} = \{v(1), \ldots v(i')\}\), \(\mathcal {N}_{i'} = \{ v(i'+1), v(i'+2), \ldots v(d_{i'} +1)\}\) for \(1 \le i' \le i-1\), define, if \(\mathcal {N}_{i-1} \ne \emptyset \)
and form an edge between \(v(i)\) and each \(j \in \mathbb {I}_i\). Let \(c(i) = |\mathbb {I}_i|\) and let \(m_{v(i),j}\) be the first point in \(\eta _{v(i),j}\) for each \(j \in \mathbb {I}_i\). Order the vertices in \(\mathbb {I}_i\) according to increasing values of \(m_{v(i),j}\) and label these as \(v(d_{i-1}+2), \ldots v(d_i + 1)\), where \(d_i = d_{i-1} + c(i)\). Set
In case \(\mathcal {N}_{i-1} = \emptyset \), we choose \(v(i)\in [n]{\setminus } \mathcal {V}_{i-1}\) with probability proportional to \(x_j\), \(j \in [n]{\setminus } \mathcal {V}_{i-1}\) and define \(\mathbb {I}_i, c(i), d_i, l_i, \mathcal {V}_i, \mathcal {N}_i\) and the edges at step \(i\) exactly as above.
This procedure terminates after exactly \(n\) steps at which point we obtain a forest-like graph with no surplus edges. We will include surplus to this graph in stage II below.
Associate with the above construction an (interpolated) random walk process \(H^{\scriptscriptstyle (n)}(\cdot )\) defined as follows. \(H^{\scriptscriptstyle (n)}(0) = 0\) and
where by convention \(l_0 =0\) and \(\mathcal {N}_0 = \emptyset \). This defines \(H^{\scriptscriptstyle (n)}(t)\) for all \(t \in [0, l_n)\). Define \(H^{\scriptscriptstyle (n)}(t)=H^{\scriptscriptstyle (n)}(l_n-)\) for all \(t \ge l_n\).
Stage II: Construction of surplus edges: For each \(i=1, \ldots , n\), we construct surplus edges on the graph obtained in Stage I and a point process \(\mathcal {P}_x\) on \([0, l_n]\), simultaneously, as follows.
-
(i)
For each \(v \in \mathbb {I}_i\) and \(\tau \in \eta _{v(i),v}\cap [0, x_{v(i)}] {\setminus } \{ m_{v(i),v} \}\), construct an edge between \(v(i)\) and \(v\). This corresponds to multi-edges between the two vertices \(v(i)\) and \(v\).
-
(ii)
For each \(\tau \in \eta _{v(i), v(i)} \cap [0, x_{v(i)}]\), construct an edge between \(v(i)\) and itself. This corresponds to self-loops at the vertex \(v(i)\).
-
(iii)
For each \(v(j) \in \mathcal {N}_{i-1} {\setminus } \{v(i)\}\) and \(\tau \in \eta _{v(i), v(j)} \cap [0, x_{v(i)}]\), construct an edge between \(v(i)\) and \(v(j)\). This corresponds to additional edges between two vertices, \(v(i)\) and \(v(j)\), that were indirectly connected in stage I. For each of the above cases, we also construct points for the point process \(\mathcal {P}_x\) at time \(l_{i-1}+\tau \in [0,l_n]\).
This completes the construction of the graph \({\bar{{\mathbf{G}}}}(z,q)\) and the random walk \(H^{\scriptscriptstyle (n)}(\cdot )\). This graph has the same law as \({\mathbf{G}}(z,q)\), so the associated component sizes and surplus vector denoted by \(({\bar{\varvec{X}}}(z,q), {\bar{\varvec{Y}}}(z,q))\) has the same law as that of \(( \varvec{X}(z,q), \varvec{Y}(z,q))\). Furthermore, conditioned on \(H^{\scriptscriptstyle (n)}\), \(\mathcal {P}_x\) is Poisson point process on \([0, l_n]\) whose intensity we denote by \(r_x(t)\).
Using the above construction we will show in next section, as a first step, a weaker result than Theorem 5.1.
5.2 Convergence in \(l^2_{\downarrow }\times \mathbb {N}^{\infty }\)
The following is the main result of this section.
Theorem 5.2
Let \(z^{\scriptscriptstyle (n)} \in \mathbb {U}_{\downarrow }^0\) and \(q^{\scriptscriptstyle (n)} \in (0,\infty )\) be sequences that satisfy the conditions in Theorem 5.1. Then
in the space \(l^2_{\downarrow }\times \mathbb {N}^\infty \) as \(n\rightarrow \infty \), where we consider the product topology on \(\mathbb {N}^\infty \).
The key ingredient in the proof is the following result. With \(z^{\scriptscriptstyle (n)}\) and \(q^{\scriptscriptstyle (n)}\) as in the above theorem, define \({\bar{\varvec{X}}}^{\scriptscriptstyle (n)} = {\bar{\varvec{X}}}(z^{\scriptscriptstyle (n)}, q^{\scriptscriptstyle (n)})\), \({\bar{\varvec{Y}}}^{\scriptscriptstyle (n)} = {\bar{\varvec{Y}}}(z^{\scriptscriptstyle (n)}, q^{\scriptscriptstyle (n)})\) and \(r^{\scriptscriptstyle (n)}(t) = r_{x^{\scriptscriptstyle (n)}}(t)\mathbb {1}_{[0, l_n]}(t)\), \(t \ge 0\). Denote the random walk process from Sect. 5.1 constructed using \((x^{\scriptscriptstyle (n)}, q^{\scriptscriptstyle (n)})\) (rather than \((x,q)\)), once more, by \(H^{\scriptscriptstyle (n)}(\cdot )\).
Define the rescaled process \(\bar{H}^{\scriptscriptstyle (n)}(\cdot )\) and its reflected version \(\hat{H}^{\scriptscriptstyle (n)} (\cdot )\) as follows
Lemma 5.3
-
(i)
As \(n \rightarrow \infty \), the process \(\bar{H}^{\scriptscriptstyle (n)} \mathop {\longrightarrow }\limits ^{d}W_\lambda \) in \(\mathcal {D}([0,\infty ): \mathbb {R})\).
(ii) For \(n \ge 1\),
Given Lemma 5.3, the proof of Theorem 5.2 can be completed as follows.
Proof of Theorem 5.2
The paper [4] shows that the vector \({\bar{\varvec{X}}}^{\scriptscriptstyle (n)}\) can be represented as the ordered sequence of excursion lengths of the process \(\hat{H}^{\scriptscriptstyle (n)} \). Also, weak convergence of \(\bar{H}^{\scriptscriptstyle (n)}\) to \(W_\lambda \) in Lemma 5.3 (i) implies the convergence of \(\hat{H}^{\scriptscriptstyle (n)} \) to \(\hat{W}_{\lambda }\). Using these facts, Proposition 4 of [4] shows that \({\bar{\varvec{X}}}^{\scriptscriptstyle (n)}\) converges in distribution to the ordered excursion length sequence of \(\hat{W}_{\lambda }\), namely \(\varvec{X}^*(\lambda )\), in \(l^2_{\downarrow }\). Also, conditional on \(\hat{H}^{\scriptscriptstyle (n)}\), \(\mathcal {P}_x\) is a Poisson point process on \([0, \infty )\) with rate \(r^{\scriptscriptstyle (n)}(t)\) and for \(i \ge 1\), \({\bar{Y}}_i^{\scriptscriptstyle (n)}\) has a Poisson distribution with parameter \(\int _{[a_i^{\scriptscriptstyle (n)}, b_i^{\scriptscriptstyle (n)}]} r^{\scriptscriptstyle (n)}(s) ds\), where \(a_i^{\scriptscriptstyle (n)}, b_i^{\scriptscriptstyle (n)}\) are the left and right endpoints of the \(i\)th ordered excursion of \(\hat{H}^n\). From conditions in (5.1) it follows that \(x^* \rightarrow 0\) and \(s_2 \rightarrow 0\), further more we have \( q x^* \rightarrow 0\) and \( q \sqrt{s_3/s_2} \rightarrow 1\). Lemma 5.3 (ii) then shows that \(\int _{[a_i^{\scriptscriptstyle (n)}, b_i^{\scriptscriptstyle (n)}]} r^{\scriptscriptstyle (n)}(s) ds\) converges in distribution to \(\int _{[a_i, b_i]} \hat{W}_{\lambda }(s) ds\), where \(a_i, b_i\) are the left and right endpoints of the \(i\)th ordered excursion of \(\hat{W}_{\lambda }\). In fact we have the joint convergence of \(\left( \hat{H}^{\scriptscriptstyle (n)}, \left( \int _{[a_i^{\scriptscriptstyle (n)}, b_i^{\scriptscriptstyle (n)}]} r^{\scriptscriptstyle (n)}(s) ds\right) _{i\ge 1}\right) \) to \(\left( \hat{W}_{\lambda }, \left( \int _{[a_i, b_i]} \hat{W}_{\lambda }(s) ds\right) _{i\ge 1}\right) \). This proves the convergence of \(({\bar{\varvec{X}}}^{\scriptscriptstyle (n)},{\bar{\varvec{Y}}}^{\scriptscriptstyle (n)})\) to \((\varvec{X}^*(\lambda ), \varvec{Y}^*(\lambda ))\) in \(l^2_{\downarrow }\times \mathbb {N}^\infty \). The result follows since \(({\bar{\varvec{X}}}^{\scriptscriptstyle (n)}, {\bar{\varvec{Y}}}^{\scriptscriptstyle (n)})\) has the same law as \(( \varvec{X}^{\scriptscriptstyle (n)}, \varvec{Y}^{\scriptscriptstyle (n)})\).\(\square \)
Proof of Lemma 5.3
Part (i) was proved in Proposition 4 of [4]. Consider now (ii).
For \(j=1,2,\ldots \), define \(\delta _{v(j)} = 1_{\{\mathcal {N}_{j-1}=\emptyset \}}\), i.e. \(\delta _{v(j)}\) is \(1\) if \(v(j)\) is the first vertex that is explored in its component during the breadth-first-search, and is \(0\) otherwise. It is easy to verify that \(H^{\scriptscriptstyle (n)}\) satisfies
The above equation implies that for all \(k \le i\), \(H^{\scriptscriptstyle (n)}(l_{k}) \ge - \sum _{j=1}^{i} \delta _{v(j)} x_{v(j)}\). In addition, taking \(k_0 = \sup \{j \le i: \delta _{v(j)}=1\} \) we have \(H^{\scriptscriptstyle (n)}(l_{k_0}) = - \sum _{j=1}^{i} \delta _{v(j)} x_{v(j)}\). In particular, this implies that \(\inf _{j \le i} H^{\scriptscriptstyle (n)}(l_{j}) = - \sum _{j=1}^{i} \delta _{v(j)} x_{v(j)}\). Also, from (5.5) we have that for \(t \in (l_{i-1},l_i]\), \( H^{\scriptscriptstyle (n)}(t) \ge H^{\scriptscriptstyle (n)}(l_{i-1}) - x^*\). Consequently
Let \(\mathcal {N}_{i-1}=\{v(i), v(i+1), \ldots , v(i+l)\}\). From the above expression for \(H^{\scriptscriptstyle (n)}(l_{i})\), we have that for \(t \in (l_{i-1},l_i]\)
Also, accounting for the three sources of surplus described in Stage II of the construction, one has the following formula for \(r^{\scriptscriptstyle (n)}(t)\) at time \(t \in (l_{i-1},l_i]\):
The three terms in the above expression correspond to self-loops; edges between vertices that in stage I were only connected indirectly; and additional edges between two vertices that were directly connected in stage I. Combining the above expression with (5.10) and (5.9), we have
The result follows. \(\square \)
5.3 Proof of Theorem 5.1
In this section we complete the proof of Theorem 5.1. The key step in the proof is the following lemma whose proof is given in Sect. 5.4.
Lemma 5.4
Let \(z^{\scriptscriptstyle (n)} \in \mathbb {U}_{\downarrow }^0\) and \(q^{\scriptscriptstyle (n)} \in (0,\infty )\) be as in Theorem 5.1. Let \(\hat{H}^{\scriptscriptstyle (n)}\) be as introduced in (5.7). Then \( \{\sup _{t \ge 0} \hat{H}^{\scriptscriptstyle (n)}(t) \}_{n \ge 1}\) is a tight family of \(\mathbb {R}_+\) valued random variables.
Remark 5.5
In fact one can establish a stronger statement, namely \( \sup _{u \ge t} \sup _{n \ge 1} \hat{H}_u^{\scriptscriptstyle (n)} \rightarrow 0\) in probability as \(t \rightarrow \infty \). Also, although not used in this work, using very similar techniques as in the proof of Lemma 5.4, it can be shown that \( \sup _{u \ge t} \hat{W}_\lambda (u) \) converges a.s. to \(0\), as \(t \rightarrow \infty \).
Proof of Theorem 5.1
Since \((\varvec{X}^{\scriptscriptstyle (n)}, \varvec{Y}^{\scriptscriptstyle (n)})\) has the same distributions as \(({\bar{\varvec{X}}}^{\scriptscriptstyle (n)}, {\bar{\varvec{Y}}}^{\scriptscriptstyle (n)})\), we can equivalently consider the convergence of the latter sequence. From Theorem 5.2 we have that \(({\bar{\varvec{X}}}^{\scriptscriptstyle (n)}, {\bar{\varvec{Y}}}^{\scriptscriptstyle (n)})\) converges to \((\varvec{X}^*(\lambda ), \varvec{Y}^*(\lambda ))\), in distribution, in \(l^2_{\downarrow }\times \mathbb {N}^{\infty }\) (with product topology on \(\mathbb {N}^{\infty }\)). By appealing to Skorohod representation theorem, we can assume without loss of generality that the convergence is almost sure. By the definition of \(\mathbf{d}_{\mathbb {U}_{\downarrow }}\) it now suffices to argue that
Fix \(\epsilon > 0\). Then, for any \(k \in \mathbb {N}\),
From the convergence of \(({\bar{\varvec{X}}}^{\scriptscriptstyle (n)}, {\bar{\varvec{Y}}}^{\scriptscriptstyle (n)})\) to \((\varvec{X}^*(\lambda ), \varvec{Y}^*(\lambda ))\) in \(l^2_{\downarrow }\times \mathbb {N}^{\infty }\) we have that
Consider now the second term in (5.12). Let \(E^{\scriptscriptstyle (n)}_L = \{ \sup _{t\ge 0} r_t^{\scriptscriptstyle (n)} \le L \}\). Then
Let \(\mathcal {G}= \sigma \{\hat{H}^{\scriptscriptstyle (n)}(t): t \ge 0\}\). Since \(r_t^{\scriptscriptstyle (n)}\) is \(\mathcal {G}\) measurable for all \(t \ge 0\), \(E^{\scriptscriptstyle (n)}_L \in \mathcal {G}\). Then
where the last inequality follows on observing that, conditionally on \(\mathcal {G}\), \(\bar{Y}^{\scriptscriptstyle (n)}_i\) has a Poisson distribution with rate that is dominated by \(\bar{X}^{\scriptscriptstyle (n)}_i \cdot (\sup _{t\ge 0} r_t^{\scriptscriptstyle (n)})\). Using the convergence of \({\bar{\varvec{X}}}^{\scriptscriptstyle (n)}\) to \(\varvec{X}^*\), we now have
Let \(\delta > 0\) be arbitrary. Using Lemma 5.4 and Lemma 5.3 (ii) we can choose \(L \in (0, \infty )\) such that \(\mathbb {P}\{( E^{\scriptscriptstyle (n)}_L)^c\} \le \delta \). Finally, taking limit as \(n \rightarrow \infty \) in (5.12) we have that
The result now follows on sending \(k \rightarrow \infty \) in the above display and recalling that \(\sum _{i=1}^\infty ( X^*_i(\lambda ))^2 < \infty \) and \(\sum _{i=1}^\infty X^{*}_i(\lambda ) Y^{*}_i(\lambda ) < \infty \) a.s. and \(\delta > 0\) is arbitrary. \(\square \)
5.4 Proof of Lemma 5.4
In this section we prove Lemma 5.4. We will only treat the case \(\lambda = 0\). The general case can be treated similarly. The key step in the proof is the following proposition whose proof is given at the end of the section.
Note that \(\sup _{t \ge 0}|\bar{H}^{\scriptscriptstyle (n)}(t) - \bar{H}^{\scriptscriptstyle (n)}(t-)| \le x^* \sqrt{s_2/s_3} \rightarrow 0\) as \(n \rightarrow \infty \). Also, as \(n \rightarrow \infty \), \(qs_2 \rightarrow 1\). Thus, without loss of generality, we will assume that
Fix \(\vartheta \in (0, 1/2)\) and define \(t^{* \scriptscriptstyle (n)} = \left( \frac{s_2}{x^*}\right) ^{\vartheta }\). Denote by \(\{\mathcal {F}_t^{\scriptscriptstyle (n)}\}\) the filtration generated by \(\{\bar{H}^{\scriptscriptstyle (n)}(t)\}_{t \ge 0}\). For ease of notation, we write \(\sup _{t \in [a,b]}=\sup _{[a,b]}\). We will suppress \((n)\) in the notation, unless needed.
Proposition 5.6
There exist \(\Theta \in (0, \infty )\), events \(G^{\scriptscriptstyle (n)}\), increasing \(\mathcal {F}_t^{\scriptscriptstyle (n)}\)-stopping times \(1= \sigma ^{\scriptscriptstyle (n)}_0 < \sigma ^{\scriptscriptstyle (n)}_1 < \ldots \), and a real positive sequence \(\{\kappa _i\}\) with \(\sum _{i=1}^\infty \kappa _i < \infty \), such that the following hold:
-
(i)
For every \(i \ge 1\), \(\{\sigma _i^{\scriptscriptstyle (n)}\}_{n \ge 1}\) is tight.
-
(ii)
For every \(i \ge 1\),
$$\begin{aligned} \mathbb {P}\left( \left\{ \sup _{[{\sigma ^{\scriptscriptstyle (n)}_{i-1}},{\sigma ^{\scriptscriptstyle (n)}_i}]} \hat{H}^{\scriptscriptstyle (n)}(t) > 2\Theta + 1 \right\} \cap \left\{ \sigma ^{\scriptscriptstyle (n)}_{i-1} < t^{* \scriptscriptstyle (n)} \right\} \cap G^{\scriptscriptstyle (n)}\right) \le \kappa _i. \end{aligned}$$ -
(iii)
As \(n \rightarrow \infty \), \( \mathbb {P}\{ \sup _{[\sigma ^{*\scriptscriptstyle (n)}, \infty )} \hat{H}^{\scriptscriptstyle (n)}(t) > \Theta ; G^{\scriptscriptstyle (n)}\} \rightarrow 0\), where \(\sigma ^{*\scriptscriptstyle (n)}= \inf \{ \sigma _i^{\scriptscriptstyle (n)}: \sigma _i^{\scriptscriptstyle (n)} \ge t^{*\scriptscriptstyle (n)}\} \).
-
(iv)
As \(n \rightarrow \infty \), \(\mathbb {P}(G^{\scriptscriptstyle (n)}) \rightarrow 1\).
Given Proposition 5.6, the proof of Lemma 5.4 can be completed as follows.
Proof of Lemma 5.4
Fix \(\epsilon \in (0,1)\). Let \(\Theta \in (0, \infty )\), \(G^{\scriptscriptstyle (n)}\), \(\sigma _i^{\scriptscriptstyle (n)}\), \(\kappa _i\) be as in Proposition 5.6. Choose \(i_0 > 1\) such that \(\sum _{i\ge i_0} \kappa _i \le \epsilon \). Since \(\{\sigma _{i_0-1}^{\scriptscriptstyle (n)}\}\) is tight, there exists \(T \in (0,\infty )\) such that \(\limsup _{n \rightarrow \infty } \mathbb {P}\{ \sigma _{i_0-1}^{\scriptscriptstyle (n)} > T \} \le \epsilon \). Thus for any \(M' > 2\Theta + 1\), we have
Taking \(\limsup _{n \rightarrow \infty }\) on both sides
Since \(\left\{ \sup _{[1,T]}\hat{H}^{\scriptscriptstyle (n)}(t)\right\} _{n \ge 1}\) is tight, we have,
Since \(\epsilon > 0\) is arbitrary, the result follows. \(\square \)
We now proceed to the proof of Proposition 5.6. The following lemma is key.
Lemma 5.7
There are \(\{\mathcal {F}_t^{\scriptscriptstyle (n)}\}\) adapted processes \(\{A^{\scriptscriptstyle (n)}(t)\}\), \(\{B^{\scriptscriptstyle (n)}(t)\}\) and \(\mathcal {F}_t^{\scriptscriptstyle (n)}\)-martingale \(\{M^{\scriptscriptstyle (n)}(t)\}\) such that
-
(i)
\(A^{\scriptscriptstyle (n)}(\cdot )\) is a non-increasing function of \(t\), a.s. For all \(t \ge 0\), \(\bar{H}^{\scriptscriptstyle (n)}(t) = \int _0^t A^{\scriptscriptstyle (n)}(u) du + M^{\scriptscriptstyle (n)}(t)\).
-
(ii)
For \(t \ge 0\), \(\langle M^{\scriptscriptstyle (n)},M^{\scriptscriptstyle (n)} \rangle _t = \int _0^t B^{\scriptscriptstyle (n)}(u) du\).
-
(iii)
\(\sup _{n \ge 1} \sup _{u \ge 0} B^{\scriptscriptstyle (n)}(u) \le 2\).
-
(iv)
With \( G^{\scriptscriptstyle (n)} = \{ A(t) < -t/2 \hbox { for all } t \in [1, t^{*(n)}] \} \), \(\mathbb {P}(G^{\scriptscriptstyle (n)}) \rightarrow 1\) as \(n\rightarrow \infty \).
-
(v)
For any \(\alpha \in (0, \infty )\) and \(t > 0\),
$$\begin{aligned} \mathbb {P}\left\{ \sup _{ u \in [0,t]} |M^{\scriptscriptstyle (n)}(u)| > \alpha \right\} \le 2\exp \left\{ \alpha \right\} \cdot \exp \left\{ - \alpha \log \left( 1+ \frac{\alpha }{2t} \right) \right\} . \end{aligned}$$(5.15)
Proof
Recall the notation from Sect. 5.1. Parts (i) and (ii) are proved in [4]. Furthermore, from Lemma 11 of [4] it follows that, for \( t \in [l_{i-1}, l_i)\), writing \(Q_2(t) = \sum _{j=1}^i (x_{v(j)})^2 \), we have
Part (iii) now follows on recalling from (5.14) that \(qs_2 \le 2\). To prove (iv) it suffices to show that
To prove this we will use the estimate on Page 832, Lemma 13 of [4], which says that for any fixed \(\epsilon \in (0,1)\), and \( L \in (0,\infty )\)
Note that the first term on the right hand side determine its order when \(L \rightarrow \infty \). Taking \(L = t^*\) in the above estimate we see that, since \(\vartheta \in (0, 1/2)\), the expression on the right side above goes to \(0\) as \(n \rightarrow \infty \). This proves (5.16) and thus completes the proof of (iv). Finally, proof of (v) uses standard concentration inequalities for martingales. Indeed, recalling that the maximal jump size of \(\bar{H}\), and consequently that of \(M\), is bounded by \(1\) and \(\langle M,M \rangle _t \le 2t\), we have from Section 4.13, Theorem 5 of [23] that, for any fixed \(\alpha >0\) and \(t>0\),
where \(\phi (\lambda ) = (e^{\lambda }-1-\lambda ).\) A straightforward calculation shows
The result follows. \(\square \)
The bound (5.15) continues to hold if we replace \(M(u)\) with \(M(\tau +u)-M(\tau )\) for any finite stopping time \(\tau \). From this observation we immediately have the following corollary.
Corollary 5.8
Let \(M\) be as in Lemma 5.7. Then, for any finite stopping time \(\tau \):
-
(i)
\(\mathbb {P}\{ \sup _{ u \in [0,t]} |M(\tau +u)-M(\tau )| > \alpha \} \le 2 e^{-\alpha }\), whenever \(\alpha > 2(e^2-1)t\).
-
(ii)
\(\mathbb {P}\{ \sup _{ u \in [0,t]} |M(\tau +u)-M(\tau )| > \alpha \} \le 2 (2e/\alpha )^{\alpha } t^{\alpha }\), for all \(t >0\) and \(\alpha > 0\).
Part (i) of the corollary is useful when \(\alpha \) is large and part (ii) is useful when \(t\) is small. Finally we now give the proof of Proposition 5.6.
Proof of Proposition 5.6
From Lemma 5.3 (i) we have that \(\hat{H}^{\scriptscriptstyle (n)}\) converges in distribution to \(\hat{W}_0\) (recall we assume that \(\lambda = 0\)) as \(n \rightarrow \infty \). Let \(\{\epsilon _i\}_{i \ge 1}\) be a positive real sequence bounded by \(1\) and fix \(\Theta \in (2, \infty )\). Choice of \(\Theta \) and \(\epsilon _i\) will be specified later in the proof. Let \(\sigma ^{\scriptscriptstyle (n)}_0 < \tau ^{\scriptscriptstyle (n)}_1 \le \sigma ^{\scriptscriptstyle (n)}_1 < \tau ^{\scriptscriptstyle (n)}_2 \le \sigma ^{\scriptscriptstyle (n)}_2 < \ldots \) be a sequence of stopping times such that \(\sigma ^{\scriptscriptstyle (n)}_0=1\), and for \(i \ge 1\),
Similarly define stopping times \(1 = \bar{\sigma }_0 < \bar{\tau }_1 \le \bar{\sigma }_1 < \bar{\tau }_2 \le \bar{\sigma }_2 < \ldots \) by replacing \(\hat{H}^{\scriptscriptstyle (n)}\) in (5.17) with \(\hat{W}_0\). Due to the negative quadratic drift in the definition of \(W_0\) it follows that \(\bar{\sigma }_i < \infty \) for every \(i\) and from the weak convergence of \(\hat{H}^{\scriptscriptstyle (n)}\) to \(\hat{W}_0\) it follows that \(\sigma ^{\scriptscriptstyle (n)}_i \rightarrow \bar{\sigma }_i \) and \(\tau ^{\scriptscriptstyle (n)}_i \rightarrow \bar{\tau }_i \), in distribution, as \(n \rightarrow \infty \). Here we have used the fact that if \(\zeta \) denotes the first time \(W_0\) hits the level \(\alpha \in (0, \infty )\) then, a.s., for any \(\delta > 0\), there are infinitely many crossings of the level \(\alpha \) in \((\zeta , \zeta + \delta )\). In particular we have that \(\{\sigma ^{\scriptscriptstyle (n)}_i\}_{n \ge 1}\) is a tight sequence, and this proves part (i) of Proposition 5.6.
For the rest of the proof we suppress \((n)\) from the notation. Since the jump size of \(\hat{H}\) is bounded by \(1\), we have that \( \sup _{[{\sigma _{i-1}},{\sigma _{i-1}+ \epsilon _i}]} \hat{H}(t) \le \Theta \) implies \( \sup _{[\sigma _{i-1},\tau _i]} \hat{H}(t) \le \Theta +1 \) and thus, in this case, when \(t \in [\tau _i,\sigma _i]\), we have \(\hat{H}(t) = \hat{H}(\tau _i) + ( \bar{H}(t) - \bar{H}(\tau _i) )\le \Theta +1 +( \bar{H}(t) - \bar{H}(\tau _i))\). Let \(G \equiv G^{\scriptscriptstyle (n)}\) be as in Lemma 5.7 (iv) and let \(H_i = G \cap \{\sigma _{i-1} < t^*\}\), then writing \(\mathbb {P}(\cdot \cap H_i)\) as \(\mathbb {P}_i(\cdot )\),
Denote the two terms on the right side by \(\mathbb {T}_1\) and \(\mathbb {T}_2\) respectively. Recalling that \(\hat{H}(\sigma _{i-1})\le 2\), we have from the decomposition in Lemma 5.7 (i) and Corollary 5.8(ii) that
Here, for \(\alpha > 0\), \(C_{\alpha } = 2 (2e/\alpha )^{\alpha }\) and we have used the fact that on \(H_i\), \(A(t) \le -t/2\le 0\) for all \(t \in [\sigma _{i-1}, \sigma _{i-1}+ \epsilon _i]\).
Next, let \(\{\delta _i\}_{i\ge 1}\) be a sequence of positive reals bounded by \(1\). Setting \(d_i = \sum _{j=1}^{i-1} \epsilon _i\), we have
whenever
Fix \(\Theta > 14\). Then \(\max \{C_{\Theta }, C_{(\Theta -2)/2}\}\le 2\). We will impose additional conditions on \(\Theta \) later in the proof. Combining (5.20) and (5.21), we have
Let
Then, (5.22) holds for \(i\) large enough, and
which, since \(\Theta >14\), is summable. This proves part (ii) of the Proposition.
Now we consider part (iii). We will construct another sequence of stopping times with values in \([t^*, \infty )\), as follows. Define \(\sigma _0^* :=\inf \left\{ \sigma _i : \sigma _i \ge t^*\right\} = \inf \{t \ge t^*: \hat{H}(t) \le 1 \}\), then define \(\tau ^*_i, \sigma ^*_i\) for \(i \ge 1\) similarly as in (5.17). Similar arguments as before give a bound as (5.23) with \(d_i\) replaced by \(t^*\), \(\delta _i\) replaced by \(1/\sqrt{t^*}\), \(\epsilon _i\) replaced with \(1/t^*\) and \(\Theta \) replaced by any \(\Theta _0 > 14\). Namely,
Here we have used the fact that since \(A(t)\) is non-increasing, on \(G^{\scriptscriptstyle (n)}\), \(A(t) \le -t^*/2\) for all \(t \ge t^*\).
Recall that, by construction, \(\hat{H}(t) = 0\) when \(t \ge s_1\). So there exist \(i_0\) such that \(\tau ^*_{i_0}= \infty \), in fact since \(\sigma ^*_i \ge \sigma ^*_{i-1}+ \epsilon \), we have that \(i_0 \le s_1/\epsilon \). Thus, we have from the above display that
Taking \(\Theta > 29\), we have on setting \(\Theta _0 = \frac{\Theta -1}{2}\) in the above display
From (5.2) we have that \(s_1 \cdot (\frac{1}{t^*})^{\varsigma /\vartheta } \rightarrow 0\). So if \(\Theta \ge 4 (\frac{\varsigma }{\vartheta } +2) + 1\), the above expression approaches \(0\) as \(n \rightarrow \infty \). The result now follows on taking \(\Theta = \max \{29, 4 (\frac{\varsigma }{\vartheta } +2) + 1\}\).\(\square \)
6 Bounded-size rules at time \(t_c- n^{-\gamma }\)
Throughout Sects. 6 and 7 we take \(T = 2t_c\) which is a convenient upper bound for the time parameters of interest. In this section we prove Theorem 3.2.
We begin with some notation associated with BSR processes, which closely follows [27]. Recall from Sect. 2.2 the set \(\Omega _K\) and the random graph process \(\mathbf{{BSR}}^{\scriptscriptstyle (n)}(t)\) associated with a given \(K\)-BSR \(F \subset \Omega _K^4\). Frequently we will suppress \(n\) in the notation. Also recall the definition of \(c_t(v)\) from Sect. 2.2.
For \(i \in \Omega _K\), define
Denote by \(\mathbf{{BSR}}^*(t)\) the subgraph of \(\mathbf{{BSR}}(t)\) consisting of all components of size greater than \(K\), and define, for \(k=1,2,3\)
where \(\{\mathcal {C}\subset \mathbf{{BSR}}^*(t)\}\) denotes the collection of all components in \(\mathbf{{BSR}}^*(t)\). For notational convenience in long formulae, we sometimes write \(\mathbf{{BSR}}(t) = \mathbf{{BSR}}_t\) and similarly \(\mathbf{{BSR}}^*(t) = \mathbf{{BSR}}_t^*\). Similar notation will be used throughout the paper.
Clearly
Also note that \(\mathcal {S}_1(t) = n\) and \(\mathcal {S}_{1,\varpi }(t) = X_\varpi (t)\).
Recall the Poisson processes \(\mathcal {P}_{\varvec{v}}\) introduced in Sect. 2.2. Let \(\mathcal {F}_t \!=\! \sigma \{\mathcal {P}_{\varvec{v}}(s): s \!\le \! t, \varvec{v} \!\in \! [n]^4\}\). For \(T_0 \!\in \! [0,T]\) and a \(\{\mathcal {F}_t\}_{0\le t \!<\! T_0}\) semi-martingale \(\{J(t)\}_{0 \le t < T_0}\) of the form
where \(M\) is a \(\{\mathcal {F}_t\}\) local martingale and \(\gamma \) is a progressively measurable process, we write \(\alpha = \mathbf{d}(J)\), \(M = \varvec{M}(J)\) and \(\gamma = \varvec{v}(J)\).
Organization: The rest of this section is organized as follows. In Sect. 6.1, we state a recent result on BSR models and certain deterministic maps associated with the evolution of \(\mathbf{{BSR}}_t^*\) from [8] that will be used in this work. In Sect. 6.2, we will study the asymptotics of \(\bar{s}_{2,\varpi }\) and \(\bar{s}_{3,\varpi }\). In Sect. 6.3, we will complete the proof of Theorem 3.2(i). In Sect. 6.4, we will obtain some useful semi-martingale decompositions for certain functionals of \(\bar{s}_2\) and \(\bar{s}_3\). In Sect. 6.5, we will prove parts (ii) and (iii) of Theorem 3.2.
6.1 Evolution of \(\mathbf{{BSR}}^*_t\)
We begin with the following lemma from [8] (see also [27]).
Lemma 6.1
-
(a)
For each \(i \in \Omega _K\), there exists a continuously differentiable function \(x_i:[0,T] \rightarrow [0,1]\) such that for any \(\delta \in (0,1/2)\), there exist \(C_1, C_2 \in (0, \infty )\) such that for all \(n\),
$$\begin{aligned} \mathbb {P}\left( \sup _{i \in \Omega _K} \sup _{s \in [0,T]} |\bar{x}_i(t)-x_i(t) | > n^{-\delta }\right) <C_1 \exp \left( -C_2 n^{1-2\delta }\right) . \end{aligned}$$ -
(b)
There exist polynomials \(\{F^x_i({\mathbf{x}})\}_{i\in \Omega _K}\), \({\mathbf{x}}=(x_i)_{i \in \Omega _K} \in \mathbb {R}^{K+1}\), such that \({\mathbf{x}}(t)= (x_i(t))_{i \in \Omega _K}\) is the unique solution to the differential equations:
$$\begin{aligned} x_i^\prime (t)\!=\! F^x_i({\mathbf{x}}(t)), \;\; i \!\in \! \Omega _K, \;\; t \!\in \! [0,T] \hbox { with initial values } {\mathbf{x}}(0)=(1,0,\ldots , 0).\nonumber \\ \end{aligned}$$(6.4)
Furthermore, \(\bar{x}_i\) is a \(\{\mathcal {F}_t\}_{0\le t < T}\) semi-martingale of the form (6.3) and
Also, for all \(i \in \Omega _K\) and \(t \in (0,T]\), we have \(x_i(t)>0\) and \( \sum _{i\in \Omega _K} x_i(t) = 1\).
Recall that \(\mathbf{{BSR}}^*(t)\) is the subgraph of \(\mathbf{{BSR}}(t)\) consisting of all components of size greater than \(K\). The evolution of this graph is governed by three type of events:
Type 1 (Immigrating vertices): This corresponds to the merger of two components of size bounded by \(K\) into a component of size larger than \(K\). Such an event leads to the appearance of a new component in \(\mathbf{{BSR}}^*(t)\) which we view as the immigration of a ‘vertex’ into \(\mathbf{{BSR}}^*(t)\). Denote by \(n a^*_i(t)\) the rate at which a component of size \(K+i\) immigrates into \(\mathbf{{BSR}}_t^*\) at time \(t\). In [8] it is shown that there are polynomials \(F_i^a({\mathbf{x}})\) for \(1\le i\le K\) such that, with \({\bar{{\mathbf{x}}}}(t) = (\bar{x}_i(t))_{i \in \Omega _K}\)
We define, with \({\mathbf{x}}(t)\) as in Lemma 6.1,
Type 2 (Attachments): This event corresponds to a component of size at most \(K\) getting linked with some component of size larger than \(K\). For \(1\le i \le K\), denote by \(|\mathcal {C}| c^*_i(t)\) the rate at which a component of size \(i\) attaches to a component \(\mathcal {C}\) in \(\mathbf{{BSR}}^*_{t-}\). Then (see [8]) there exist polynomials \(F^c_i({\mathbf{x}})\) for \(1\le i\le K\), such that \(c_i^*(t) = F^c_i({\bar{{\mathbf{x}}}}(t))\). Define
Type 3 (Edge formation): This event corresponds to the addition of an edge between components in \(\mathbf{{BSR}}^*_t\). The occurrence of this event adds one edge between two vertices in \(\mathbf{{BSR}}^*_{t-}\), the vertex set stays unchanged, whereas the edge set has one additional element. From [8], there is a polynomial \(F^b({\mathbf{x}})\) such that, defining \(b^*(t) = F^b({\bar{{\mathbf{x}}}}(t))\), the rate at which each pair of components \(\mathcal {C}_1 \ne \mathcal {C}_2 \in \mathbf{{BSR}}_t^*\) merge at time \(t\), equals \(|\mathcal {C}_1| |\mathcal {C}_2| b^*(t)/n\). Furthermore, define
From [8], \(F_i^a({\mathbf{x}})\), \(F_i^c({\mathbf{x}})\) and \(F^b({\mathbf{x}})\) are polynomials with positive coefficients, thus from the last statement in Lemma 6.1 we have that \(b(t_c) \in (0, \infty )\).
6.2 Analysis of \(\bar{s}_{2,\varpi }(t)\) and \(\bar{s}_{3,\varpi }(t)\)
Define functions \(F^s_{2,\varpi } : [0,1]^{K+1}\times \mathbb {R}\rightarrow \mathbb {R}\) and \(F^s_{3,\varpi } : [0,1]^{K+1}\times \mathbb {R}^2 \rightarrow \mathbb {R}\) as
for \(({\mathbf{x}}, s_2) \in [0,1]^{K+1}\times \mathbb {R}\) and, for \(({\mathbf{x}}, s_2, s_3) \in [0,1]^{K+1}\times \mathbb {R}^2 \)
The following lemma relates the evolution of \(\bar{s}_{j,\varpi }(\cdot )\) to that of \(F^s_{j,\varpi }({\bar{{\mathbf{x}}}}(\cdot ), \bar{s}_{2,\varpi }(\cdot ))\). By definition, \(\bar{s}_{j,\varpi }(t)\) is a non-decreasing process with RCLL paths, and therefore a semi-martingale. For \(T_0 \in [0, T]\), a stochastic process \(\{\xi (t)\}_{0 \le t < T_0}\), and a nonnegative sequence \(\alpha (n)\), the quantity \(O_{T_0}(\xi (t)\alpha (n))\) will represent a stochastic process \(\{\eta (t)\}_{0\le t < T_0}\) such that for some \(d_1 \in (0, \infty )\), \(\eta (t) \le d_1 \xi (t)\alpha (n)\), for all \(0 \le t < T_0\) and \(n \ge 1\).
Lemma 6.2
The processes \(\bar{s}_{j,\varpi }\), \(j=2,3\), are \(\{\mathcal {F}_t\}_{0\le t < t_c}\) semi-martingales of the form (6.3) and
Proof
Note that \(\mathcal {S}_{2, \varpi }\) and \(\mathcal {S}_{3, \varpi }\) have jumps at time instant \(t\) with rates and values \( \Delta \mathcal {S}_{2, \varpi }(t)\), \(\Delta \mathcal {S}_{3, \varpi }(t)\), respectively, given as follows.
-
for each \(1 \le i \le K\), with rate \(n a^*_i(t)\),
$$\begin{aligned} \Delta \mathcal {S}_{2, \varpi }(t) = (K+i)^2, \;\; \Delta \mathcal {S}_{3, \varpi }(t) = (K+i)^3. \end{aligned}$$ -
for each \(1 \le i \le K\) and \(\mathcal {C}\subset \mathbf{{BSR}}_{t-}^*\), at rate \(|\mathcal {C}| c^*_i(t)\),
$$\begin{aligned} \Delta \mathcal {S}_{2, \varpi }(t) = 2|\mathcal {C}| i+ i^2, \;\; \Delta \mathcal {S}_{3, \varpi }(t) = 3|\mathcal {C}|^2 i + 3 |\mathcal {C}| i^2 +i^3. \end{aligned}$$ -
for all unordered pair \(\mathcal {C},{\tilde{\mathcal {C}}} \subset \mathbf{{BSR}}_{t-}^*\), such that \(\mathcal {C}\ne {\tilde{\mathcal {C}}}\), at rate \( |\mathcal {C}| |{\tilde{\mathcal {C}}}| b^*(t)/n\),
$$\begin{aligned} \Delta \mathcal {S}_{2, \varpi }(t) = 2|\mathcal {C}| |{\tilde{\mathcal {C}}}| , \;\; \Delta \mathcal {S}_{3, \varpi }(t) = 3|\mathcal {C}|^2|{\tilde{\mathcal {C}}}| + 3 |\mathcal {C}| |{\tilde{\mathcal {C}}}|^2. \end{aligned}$$
Thus
and
The result follows. \(\square \)
6.3 Proof of Theorem 3.2(i)
In this section we prove Theorem 3.2(i). We begin with the following differential equations, whose solutions play an important role in defining \(s_2(\cdot )\) and \(s_3(\cdot )\) that appear in Theorem 3.2.
Consider the equations
The following lemma describes some properties of solutions to the above differential equations.
Lemma 6.3
Equations (6.12) and (6.13) have unique solutions \(s_{2,\varpi }(t)\) and \(s_{3,\varpi }(t)\) for \( t \in [0,t_c)\). Furthermore \(s_{2,\varpi }(t)\) and \(s_{3,\varpi }(t)\) are non-negative, increasing and \(\lim _{t \uparrow t_c} s_{2,\varpi }(t)= \lim _{t \uparrow t_c} s_{3,\varpi }(t)=\infty \).
Proof
The existence and uniqueness of solution to (6.12) follows from Lemma 6.1(b) and Theorem 2.2 in [27]. Furthermore, Theorem 2.2 in [27] also implies \(\lim _{t \uparrow t_c} s_{2,\varpi }(t) = \infty \).
Note that \(F^s_{3,\varpi }\) is a polynomial and the right hand side of (6.13) is linear in \(s_{3,\varpi }\), thus (6.13) has a unique solution on \([0, t_c)\) as well. Since \( s_{2,\varpi }(t)\) explodes at \(t_c\), we have \(\lim _{t \uparrow t_c} s_{3,\varpi }(t) = \infty \). The monotonicity of \(s_{2,\varpi }(t)\) and \(s_{3,\varpi }(t)\) follow from the positivity of the functions \(\{F^a_j: 1\le j\le K\}, F^b, \{F^c_j: 1\le j\le K\}\) that appear in the definition of the functions \(F^s_{2,\varpi }\) and \( F^s_{3,\varpi }\) in (6.9) and (6.10) respectively.
Define \(s_k : [0,t_c) \rightarrow [0,\infty )\), \(k=2,3\), as follows.
Then using (6.12)–(6.13) we get the following differential equations for \(s_2\) and \(s_3\).
Lemma 6.4
The functions \(s_2,s_3\) are continuously differentiable on \([0, t_c)\), \(\lim _{t \uparrow t_c}s_2(t) =\lim _{t \uparrow t_c}s_3(t)=\infty \) , and can be characterized as the unique solutions of the following differential equations
where the function \(F^s_2(\cdot )\) and \(F^s_3(\cdot )\) are defined as
Proof
The differentiability of \(s_2(\cdot )\), \(s_3(\cdot )\) on \([0, t_c)\) and the form of \(F^s_2\) and \(F^s_3\) follow from (6.14) and the differential equations (6.12), (6.13) and (6.4). The uniqueness of the solution follows from the fact that \(F^s_2\) and \(F^s_3\) are polynomials. The limit behavior as \(t \uparrow t_c\) follows from the definition of \(s_k\).\(\square \)
The following lemma defines the two parameters \(\alpha \) and \(\beta \) that appear in Theorems 3.2 and 3.3. Recall from Sect. 6.1 that \(b(t_c) \in (0, \infty )\).
Lemma 6.5
The following two limits exist,
Furthermore, \(\alpha , \beta \in (0, \infty )\) and \(\alpha = 1/{b(t_c)} \).
Proof
By (6.14), for \(k = 2,3\), \(|s_k(t)-s_{k,\varpi }(t)| \le K^{k}\). Since \(\lim _{t \rightarrow t_c-}s_{k,\varpi }(t) = \infty \), we thus have that \(\lim _{t\rightarrow t_c-} s_k(t)/s_{k,\varpi }(t) = 1\). Write \(y_\varpi (t)=1/s_{2,\varpi }(t)\) and \(z_\varpi (t)=y^3_\varpi (t) s_{3,\varpi }(t)\), it suffices to show that:
Define \(A_l(t)=\sum _{i=1}^K(K+i)^l a_i(t)\) and \(C_l(t)=\sum _{i=1}^{K}i^l c_i(t)\) for \(l=1,2,3\). Then by (6.12), (6.13), (6.9) and (6.10), the derivative of \(y_\varpi (t)\) and \(z_\varpi (t)\) can be written as follows (we omit \(t\) from the notation):
where \(B_1(t) = (3y_\varpi (t) A_2(t)+3y_\varpi (t) C_2(t)x_\varpi (t) +3C_1(t))\) and \(B_2(t) = (y_\varpi ^3(t)A_3(t)+ 3y_\varpi ^2(t)C_2(t) +y_\varpi ^3(t)C_3(t) x_\varpi (t)) \). Since \(\lim _{t \rightarrow t_c-} y_\varpi (t)=0\), we have \(\lim _{t \rightarrow t_c-}y'_\varpi (t) = - b(t_c)\) which proves the first statement in (6.15).
Choose \(t_1 \in (0,t_c)\) such that \(y_{\varpi }(t), z_{\varpi }(t) \in (0,\infty )\) for all \(t \in (t_1, t_c)\). Then from (6.17), for all such \(t\)
Since \(B_1, B_2\) are nonnegative and \(\sup _{t \in [t_1, t_c]} \{ B_1(t) + B_2(t)\} < \infty \), we have \(\lim _{t \rightarrow t_c-} z_\varpi (t) \in (0, \infty )\). This completes the proof of (6.15). The result follows.\(\square \)
We now complete the proof of Theorem 3.2(i).
Proof of Theorem 3.2(i)
Let \(\alpha , \beta \) be as introduced in Lemma 6.5. From Lemma 6.4 it follows that \(y(t) = 1/ s_2(t)\) and \(z(t) = y^3(t) s_3(t)\), for \(0 \le t < t_c\), solve the differential equations
where \(F^y: [0,1]^{K+2} \rightarrow \mathbb {R}\) and \(F^z: [0,1]^{K+2} \times \mathbb {R}\rightarrow \mathbb {R}\) are defined as
\(({\mathbf{x}}, y, z) \in [0,1]^{K+2} \times \mathbb {R}\rightarrow \mathbb {R}\). We claim that \(F^y({\mathbf{x}},y)\) and \(F^z({\mathbf{x}},y,z)\) are polynomials.
To see this note that, by (6.9), for any \(u \in \mathbb {R}\)
From the definition of \(F_2^s\) in Lemma 6.4 it is now clear that \(F^y({\mathbf{x}},y)\) is a polynomial. To check that \(F^z({\mathbf{x}},y,z)\) is a polynomial, one only needs to examine the expression
Note that the non-polynomial part in the first term is \(-3yz \cdot \frac{1}{y^2}F^b({\mathbf{x}})\), which gets cancelled with the non-polynomial part of the second term, namely \(y^3 \cdot \frac{3}{y} \cdot \frac{z}{y^3}F^b({\mathbf{x}})\). This proves the claim. Also, from (6.10) it is clear that the order of \(z\) in \(F^z({\mathbf{x}},y,z)\) is at most 1.
Thus (6.18) has a unique solution. Also, defining \(y(t_c) = \lim _{t\rightarrow t_c-}y(t) = \lim _{t\rightarrow t_c-}y_{\varpi }(t)\) and \(z(t_c) = \lim _{t\rightarrow t_c-}z(t) = \lim _{t\rightarrow t_c-}z_{\varpi }(t)\), we see that \(y, z\) are twice continuously differentiable (from the left) at \(t_c\). Furthermore, \(y'(t_c-) = -\alpha ^{-1}\) and \(z(t_c-) = \beta \). Thus we have
The result follows. \(\square \)
6.4 Asymptotic analysis of \(\bar{s}_2(t)\) and \(\bar{s}_3(t)\)
In preparation for the proof of Theorem 3.2(ii), in this section we will obtain some useful semi-martingale decompositions for \( Y(t):=\frac{1}{\bar{s}_2(t)}\) and \(Z(t):=\frac{\bar{s}_3(t)}{(\bar{s}_2(t))^3}\). Throughout this section and next we will denote \( |\mathcal {C}_1^{\scriptscriptstyle (n)}(t)|\) as \(I(t)\). Recall the functions \(F^s_2\), \(F^s_3\) introduced in Lemma 6.4.
Lemma 6.6
The processes \(\bar{s}_2\) and \(\bar{s}_3\) are \(\{\mathcal {F}_t\}_{0\le t < t_c}\) semi-martingales of the form (6.3) and the following equations hold.
-
(a)
\(\mathbf{d}(\bar{s}_2)(t) = F^s_2( {\bar{{\mathbf{x}}}}(t), \bar{s}_2(t)) + O_{t_c}\left( I^2(t)\bar{s}_2(t)/{n}\right) .\)
-
(b)
\(\mathbf{d}(\bar{s}_3)(t) = F^s_3({\bar{{\mathbf{x}}}}(t), \bar{s}_2(t) ,\bar{s}_3(t)) + O_{t_c}\left( {I^3(t) \bar{s}_2(t)}/{n}\right) .\)
-
(c)
\(\varvec{v}(\bar{s}_2)(t) = O_{t_c}(I^2(t)\bar{s}_2^2(t)/n).\)
Proof
Parts (a) and (b) are immediate from (6.2), Lemma 6.1(b) and Lemma 6.2. For part (c), recall the three types of events described in Sect. 6.1. For type 1, \(\Delta \bar{s}_2(t)\) is bounded by \(2K^2/n\) and the total rate of such events is bounded by \(n/2\). For type 2, the attachment of a size \(j\) component, \(1\le j \le K\), to a component \(\mathcal {C}\) in \(\mathbf{{BSR}}^*_{t-}\) occurs at rate \(|\mathcal {C}| c^*_j(t)\) and produces a jump \(\Delta \bar{s}_2(t) = 2j|\mathcal {C}|/n\). For type 3, components \(\mathcal {C}\) and \({\tilde{\mathcal {C}}}\) merge at rate \(|\mathcal {C}| |{\tilde{\mathcal {C}}}| b^*(t)/n\) and produce a jump \( \Delta \bar{s}_2(t) = 2 |\mathcal {C}| |{\tilde{\mathcal {C}}}|/n\). Thus for \(t \in [0, t_c)\), \(\varvec{v}(\bar{s}_2)(t)\) can be estimated as
This proves (c).\(\square \)
In the next lemma, we obtain a semi-martingale decomposition for \(Y\).
Lemma 6.7
The process \(Y(t)=1/\bar{s}_2(t)\) is a \(\{\mathcal {F}_t\}_{0\le t < t_c}\) semi-martingale of the form (6.3) and
-
(i)
With \(F^y(\cdot )\) as defined in (6.19),
$$\begin{aligned} \mathbf{d}(Y)(t) = F^y({\bar{{\mathbf{x}}}}(t), Y(t)) + O_{t_c}\left( \frac{I^2(t)Y(t)}{n}\right) . \end{aligned}$$(6.20) -
(ii)
$$\begin{aligned} \varvec{v}(Y)(t) = O_{t_c}\left( \frac{ I^2(t)Y^2(t)}{n} \right) . \end{aligned}$$
Proof
Note that
Thus by Lemma 6.6(a), we have,
This proves (i). For (ii), note that (6.21) also implies \( (\Delta Y(t))^2 \le \frac{(\Delta \bar{s}_2)^2}{\bar{s}_2^4}\). We then have
The result follows. \(\square \)
We now give a semi-martingale decomposition for \(Z(t)=\bar{s}_3(t)/(\bar{s}_2(t))^3\).
Lemma 6.8
The process \(Z(t)=\bar{s}_3(t)/(\bar{s}_2(t))^3\) is a \(\{\mathcal {F}_t\}_{0\le t < t_c}\) semi-martingale of the form (6.3) and
-
(i)
With \(F^z(\cdot )\) as defined in (6.19),
$$\begin{aligned} \mathbf{d}(Z)(t) = F^z ({\bar{{\mathbf{x}}}}(t), Y(t), Z(t)) + O_{t_c}\left( \frac{I^3(t)Y^2(t)}{n}\right) . \end{aligned}$$ -
(ii)
$$\begin{aligned} \varvec{v}(Z)(t) = O_{t_c}\left( \frac{ I^4(t)Y^4(t)}{n} + \frac{ I^6(t)Y^6(t)}{n}\right) . \end{aligned}$$
Proof
Note that
where \(R(\Delta Y, \Delta \bar{s}_3)\) is the error term which, using the observations that \( \bar{s}_3 \le I \bar{s}_2\), \(\Delta \bar{s}_3 \le 3 I \Delta \bar{s}_2\) and \(|\Delta Y| \le Y^2 \Delta \bar{s}_2\), can be bounded as follows.
From Lemma 6.6(b), Lemma 6.7(i) and Lemma 6.6(c), we have
This proves (i). For (ii), note that
Thus,
Applying Lemma 6.6(c) we now have,
The result follows. \(\square \)
6.5 Proof of Theorem 3.2(ii)
We begin with an upper bound on the size of the largest component at time \(t \le t_n = t_c - n^{-\gamma }\) for \(\gamma \in (0,1/4)\), which has been proved in [8], and will play an important role in the proof of Theorem 3.2(ii).
Theorem 6.9
([8, Theorem 1.2] Barely subcritical regime) Fix \(\gamma \in (0,1/4)\). Then there exists \(C_3 \in (0, \infty )\) such that, as \(n \rightarrow \infty \),
The next lemma is an elementary consequence of Gronwall’s inequality.
Lemma 6.10
Let \(\{t_n\}\) be a sequence of positive reals such that \(t_n \in [0, t_c)\) for all \(n\). Suppose that \(U^{\scriptscriptstyle (n)}\) is a semi-martingale of the form (6.3) with values in \({\mathbb {D}}\subset \mathbb {R}\). Let \(g:[0,t_c)\times {\mathbb {D}}\rightarrow {\mathbb {R}}\) be such that, for some \(C_4(g) \in (0, \infty )\),
Let \(\{u(t)\}_{t \in [0,t_c)}\) be the unique solution of the differential equation
Further suppose that there exist positive sequences:
-
(i)
\(\{\theta _1(n)\}\) such that, whp, \(|U^{\scriptscriptstyle (n)}(0)-u_0| \le \theta _1(n)\).
-
(ii)
\(\{\theta _2(n)\}\) such that, whp,
$$\begin{aligned} \int _0^{t_n}\left| \mathbf{d}(U^{\scriptscriptstyle (n)})(t) - g(t,U^{\scriptscriptstyle (n)}(t))\right| dt \le \theta _2(n). \end{aligned}$$ -
(iii)
\(\{\theta _3(n)\}\) such that, whp, \(\langle \varvec{M}(U^{\scriptscriptstyle (n)}), \varvec{M}(U^{\scriptscriptstyle (n)})\rangle _{t_n} \le \theta _3(n)\).
Then, whp,
where \(\theta _4=\theta _4(n)\) is any sequence satisfying \( \sqrt{\theta _3(n)} = o(\theta _4(n))\).
Proof
We suppress \(n\) from the notation unless needed. Using the Lipschitz property of \(g\), we have, for all \(t \in [0, t_n]\),
Then by Gronwall’s lemma
Let \(\tau ^{\scriptscriptstyle (n)} = \inf \{ t\ge 0: \langle \varvec{M}(U), \varvec{M}(U) \rangle _t > \theta _3(n) \}\). By Doob’s inequality
Then for any \(\theta _4(n)\) such that \( \theta _3=o((\theta _4)^2)\), we have
The result now follows on using the above observation in (6.23). \(\square \)
Proof of Theorem 3.2(ii)
Let \(y\) and \(z\) be as in the proof of Theorem 3.2(i). It suffices to show
We begin by proving the following weaker result than (6.24):
Recalling from the proof of Theorem 3.2(i) that \({\mathbf{x}}\mapsto F^y({\mathbf{x}}, y)\) is Lipschitz for \({\mathbf{x}}\in [0,1]^{K+1}\), uniformly for all \(y \in [0,1]\), we get for some \(d_1 \in (0, \infty )\)
From Lemma 6.7(i) and Lemma 6.1(a) we now get for some \(d_2 \in (0, \infty )\), whp,
Thus, from Theorem 6.9 and recalling that \(\gamma < 1/5\), we have whp,
Next, by Lemma 6.7(ii) and using the fact \(Y(t) \le 1\) for all \(t \in [0, t_c)\),
The statement in (6.26) now follows on observing that \(((\log n)^8n^{3\gamma -1})^{1/2} = o(n^{-1/5})\) and applying Lemma 6.10 with \({\mathbb {D}}:=[0,1]\), \(g(t,y):=F^y({\mathbf{x}}(t),y)\), \(\theta _1=0\), \(\theta _2 = n^{-2/5}\) and \(\theta _3 = (\log n)^8n^{3\gamma -1}\).
We now strengthen the estimate in (6.26) to prove (6.24). From Theorem 3.2(i) it follows that \(y(t_n) = 1/s_2(t_n) = \Theta (n^{-\gamma })\). Since \(\gamma < 1/5\), from (6.26) we have, whp, \(Y(t) \le 2 y(t)\) for all \(t \le t_n\). Thus from the first equality in (6.27) and Theorem 3.2(i) we get, whp,
Since \(((\log n)^8 n^{\gamma -1})^{1/2}=o(n^{-2/5})\), applying Lemma 6.10 again gives
This proves (6.24).
We now prove (6.25). We will apply Lemma 6.10 to \({\mathbb {D}}:= \mathbb {R}\) and \( g(t,z):= F^z({\mathbf{x}}(t),y(t),z). \) As noted in the proof of Theorem 3.2(i), \(g\) defined as above satisfies (6.22).
We now verify the three assumptions in Lemma 6.10. Note that (i) is satisfied with \(\theta _1 = 0\), since \(Z(0)=z(0)=1\). Next, by Lemma 6.8(ii) and the fact \(Y(t) \le 2y(t)\) for \(t \le t_n\), whp, we have
Since \(\gamma < 1/5\), we can find \(\theta _4(n) \rightarrow 0\) such that \(\sqrt{(\log n)^{24}n^{5\gamma -1} }= o(\theta _4(n))\) Thus (iii) in Lemma 6.10 is satisfied. Next recall from the proof of Theorem 3.2(i) that \(g(t,z)\) is linear in \(z\). Also, \(Z(t) \le I(t)\). Thus from Lemma 6.1 and (6.28), for some \(d_3 \in (0, \infty )\) whp, for all \(t \le t_n\)
By Lemma 6.8(i) and the above bound,
This verifies (ii) in Lemma 6.10 with \(\theta _2(n) = O((\log n)^{12} n^{3\gamma -1})\). From Lemma 6.10 we now have
The result follows. \(\square \)
Part (iii) of Theorem 3.2 is a simple consequence of part (ii).
Proof of Theorem 3.2(iii)
The convergence when \(t=0\) is trivial. Consider now \(t > 0\). Since \(t_n \rightarrow t_c\) as \(n \rightarrow \infty \), we have from part (ii) of the theorem that, for fixed \(t \in (0,t_c)\),
Also, since \(t >0\), we have that \(x_i(t) > 0\) for all \(i \in \Omega _K\), thus \(s_2(t)>0\). Theorem 3.2(iii) is now immediate.\(\square \)
7 Coupling with the multiplicative coalescent
In this section we prove Theorem 3.3. Throughout this section we fix \(\gamma \in (1/6,1/5)\). The basic idea of the proof is as follows. Recall \(\alpha , \beta \in (0, \infty )\) from Theorem 3.2 (see also Lemma 6.5). We begin by approximating the BSR random graph process by a process which until time \(t_n := t_c - n^{-\gamma }\) is identical to the BSR process and in the time interval \([t_n, t_c + \alpha \beta ^{2/3}\frac{\lambda }{n^{1/3}}]\) evolves as an Erdős–Rényi process, namely over this interval edges between any pair of vertices appear at rate \(1/\alpha n\), and self loops at any given vertex appear at rate \(1/2\alpha n\). Asymptotic behavior of this random graph is analyzed using Theorem 5.1. Theorems 3.2 and 6.9 help in verifying the conditions (5.1) and (5.2) in the statement of Theorem 5.1. We then complete the proof of Theorem 3.3 by arguing that the ‘difference’ between the BSR process and the modified random graph process is asymptotically negligible.
Let
Throughout this section, for \(\lambda \in \mathbb {R}\), we denote \(t^{\lambda } = t_c + \alpha \beta ^{2/3}{\lambda }/{n^{1/3}}\). Recall the random graph process \(\mathbf{{BSR}}^*(t)\) introduced in Sect. 6. Denote by \((|\mathcal {C}_{i}^{*}(t)|, \xi _{i}^{*}(t))_{i\ge 1}\) the vector of ordered component size and corresponding surplus in \(\mathbf{{BSR}}^*(t)\) (the components are denoted by \(\mathcal {C}_{i}^{*}(t)\) ). Let, for \(\lambda \in \mathbb {R}\),
For \(i \ge 1\), denote \({\bar{\varvec{C}}}_i^{{\scriptscriptstyle (n)},*}(\lambda )\) and \({\bar{\varvec{Y}}}_i^{{\scriptscriptstyle (n)},*}(\lambda )\) for the \(i\)th coordinate of \({\bar{\varvec{C}}}^{{\scriptscriptstyle (n)},*}(\lambda )\) and \({\bar{\varvec{Y}}}^{{\scriptscriptstyle (n)},*}(\lambda )\) respectively. Write \({\bar{\varvec{Y}}}_i^{{\scriptscriptstyle (n)},*} = \tilde{\xi }_{i}^{\scriptscriptstyle (n)} + \hat{\xi }_{i}^{\scriptscriptstyle (n)}\) where \(\tilde{\xi }_{i}^{\scriptscriptstyle (n)}(\lambda )\) represents the surplus in \(\mathbf{{BSR}}^*(t^{\lambda })\) that is created before time \(t_n\), namely
In Sect. 7.2 we will show that the contribution from \(\tilde{\xi }^{\scriptscriptstyle (n)}(\lambda ):= (\tilde{\xi }_i^{\scriptscriptstyle (n)}(\lambda ): i \ge 1)\) is asymptotically negligible. First, in Sect. 7.1 below we analyze the contribution from the ‘new surplus’, i.e. \(\hat{\xi }^{\scriptscriptstyle (n)}:= (\hat{\xi }_i^{\scriptscriptstyle (n)} : i \ge 1)\).
7.1 Surplus created after time \(t_n\)
The main result of this section is as follows. Recall the process \(\varvec{Z}(\lambda ) = (\varvec{X}(\lambda ), \varvec{Y}(\lambda ))\) introduced in Theorem 3.1.
Theorem 7.1
For every \(\lambda \in \mathbb {R}\), as \(n\rightarrow \infty \), \(({\bar{\varvec{C}}}^{{\scriptscriptstyle (n)},*}(\lambda ), \hat{\xi }^{\scriptscriptstyle (n)}(\lambda ))\) converges in distribution, in \(\mathbb {U}_{\downarrow }\), to \(({\varvec{X}}(\lambda ), \varvec{Y}(\lambda ))\).
The basic idea in the proof of the above theorem is to argue that \(\mathbf{{BSR}}^*(t^{\lambda })\) ‘lies between’ two Erdős–Rényi random graph processes \({\mathbf{G}}^{\scriptscriptstyle (n),-}(t^\lambda )\) and \({\mathbf{G}}^{\scriptscriptstyle (n),+}(t^\lambda )\), whp, and then apply Theorem 5.1 to each of these processes. For a graph \({\mathbf{G}}\), denote by \(|\mathcal {C}_i({\mathbf{G}})|\) and \(\xi _i({\mathbf{G}})\) the size and surplus, respectively, of the \(i\)th largest component, \(\mathcal {C}_i({\mathbf{G}})\) of graph \({\mathbf{G}}\). We begin with the following lemma. Recall \(\lambda _n\) from (7.1).
Lemma 7.2
There exists a construction of \(\{\mathbf{{BSR}}^*(t)\}_{t\ge 0}\) along with two other random graph processes \(\{{\mathbf{G}}^{\scriptscriptstyle (n),-}(t)\}_{t\ge 0}\) and \(\{{\mathbf{G}}^{\scriptscriptstyle (n),+}(t)\}_{t\ge 0}\) such that:
-
(i)
With high probability,
$$\begin{aligned} {\mathbf{G}}^{\scriptscriptstyle (n),-}(t^\lambda ) \subset \mathbf{{BSR}}^*(t^\lambda )\subset {\mathbf{G}}^{\scriptscriptstyle (n),+}(t^\lambda ) \quad \hbox { for all } \lambda \in [-\lambda _n, \lambda _n]. \end{aligned}$$(7.2) -
(ii)
Let for \(i \ge 1\), \({\bar{\varvec{C}}}^{\scriptscriptstyle (n),\mp }_i(\lambda ) = \frac{\beta ^{1/3}}{n^{2/3}}|\mathcal {C}_i\left( {\mathbf{G}}^{\scriptscriptstyle (n),\mp }(t^\lambda )\right) |\) and
$$\begin{aligned} {\bar{\varvec{Y}}}^{\scriptscriptstyle (n),\mp }_{i}(\lambda ) = \xi _i\left( {\mathbf{G}}^{\scriptscriptstyle (n),\mp }(t^\lambda )\right) - \sum _{j: \mathcal {C}_j\left( {\mathbf{G}}^{\scriptscriptstyle (n),\mp }(t_n)\right) \subset \mathcal {C}_i\left( {\mathbf{G}}^{\scriptscriptstyle (n),\mp }(t^{\lambda })\right) } \xi _j\left( {\mathbf{G}}^{\scriptscriptstyle (n),\mp }(t_n)\right) . \end{aligned}$$Then, for all \(\lambda \in \mathbb {R}\)
$$\begin{aligned} ({\bar{\varvec{C}}}^{\scriptscriptstyle (n),\bullet }(\lambda ), {\bar{\varvec{Y}}}^{\scriptscriptstyle (n),\bullet }(\lambda )) \mathop {\longrightarrow }\limits ^{d}(\varvec{X}(\lambda ), \varvec{Y}(\lambda )), \; \bullet = -, +, \end{aligned}$$where \(\mathop {\longrightarrow }\limits ^{d}\) denotes weak convergence in \(\mathbb {U}_{\downarrow }\).
We remark that \({\bar{\varvec{Y}}}^{\scriptscriptstyle (n),\mp }(\lambda )\) represents the surplus in \({\mathbf{G}}^{\scriptscriptstyle (n),\mp }(t^\lambda )\) created after time instant \(t_n\). Proof of the lemma relies on the following proposition which is an immediate consequence of Theorem 3.2 and Theorem 6.9.
Proposition 7.3
There exists a \(\kappa \in (0, \frac{1}{3} - \gamma )\) such that
We now prove Lemma 7.2.
Proof of Lemma 7.2
We suppress \(n\) in the notation for the random graph processes. Write \(t_n^+:=t_c+n^{-\gamma }\). Let \(\mathbf{{BSR}}(t)\) for \(t \in [0, t_n^+]\) be constructed as in Sect. 2.2 and define \(\mathbf{{BSR}}^*(t)\) for \(t \in [0, t_n)\) as in Sect. 6. Set
We now give the construction of these processes for \(t \in [t_n, t_n^+]\).
The construction is done in two rounds. In the first round, we construct processes \({\mathbf{G}}^{I,-}(t)\), \(\mathbf{{BSR}}^{I,*}(t)\) and \({\mathbf{G}}^{I,+}(t)\) for \( t \in [t_n ,t_n^+]\) by using only the information about immigrations and attachments in \(\mathbf{{BSR}}(t)\), while the edge formation between large components is ignored. We first construct the process \(\{\mathbf{{BSR}}^{ I}(t)\}_{t \in [t_n, t_n^+] }\) as follows. Let \(\mathbf{{BSR}}^{ I}(t_n) := \mathbf{{BSR}}(t_n)\). For \(t > t_n\), \(\mathbf{{BSR}}^{ I}(t)\) is constructed along with and same as \(\mathbf{{BSR}}(t)\), except for when
in which case no edge is added to \(\mathbf{{BSR}}^{ I}(t)\).
Let \(\bar{x}_i(t), a_i^*(t),b^*(t), c^*_i(t)\), \(1 \le i \le K\), \(t \in [t_n, t_n^+]\), be the processes determined from \(\{\mathbf{{BSR}}(t)\}_{t\in [t_n, t_n^+] }\) as in Sect. 6. These processes will be used in the second round of the construction.
Now define \(\mathbf{{BSR}}^{I,*}(t)\) to be the subgraph that consists of all large components (components of size greater than \(K\)) in \(\mathbf{{BSR}}^{I}(t)\), and then define \({\mathbf{G}}^{I,-}(t)\) and \({\mathbf{G}}^{I,+}(t)\) for \(t \in [t_n, t_n^+]\) as follows:
Then
We now proceed to the second round of the construction. Let
Note that Lemma 6.1 and (6.8) implies that with probability at least \(1- C_1 e^{-C_2 n^{1/5}}\),
Thus \(\mathbb {P}\{E_n^c\} \rightarrow 0\) as \(n \rightarrow \infty \). Since we only need the coupling to be good with high probability, it suffices to construct the coupling of the three processes until the first time \(t \in [t_n, t_n^+]\) when \(b^*(t) \not \in [b(t_c) -n^{-1/6}, b(t_c) + n^{-1/6}]\). Equivalently, we can assume without loss of generality that \(b^*(t) \in [b(t_c) -n^{-1/6}, b(t_c) + n^{-1/6}]\), for all \(t \in [t_n, t_n^+]\), a.s.
We will construct \({\mathbf{G}}^{ +}(t)\), \(\mathbf{{BSR}}^*(t)\) and \({\mathbf{G}}^{ -}(t)\) by adding new edges between components in the three random graph processes \({\mathbf{G}}^{I,-}(t)\), \(\mathbf{{BSR}}^{I,*}(t)\) and \({\mathbf{G}}^{I,+}(t)\) such that, at time \(t \in [t_n ,t_n^+]\) edges are added between each pair of vertices in \({\mathbf{G}}^{I,-}(t)\), \(\mathbf{{BSR}}^{I,*}(t)\) and \({\mathbf{G}}^{I,+}(t)\), at rates \(\frac{1}{n}(b(t_c)-n^{-1/6})\), \(\frac{1}{n}b^*(t)\) and \(\frac{1}{n}(b(t_c)+n^{-1/6})\), respectively. The precise mechanism is as follows.
We first construct \({\mathbf{G}}^{+}(t)\) for \(t \in (t_n,t_n^+]\) by adding edges between every pair of vertices in \({\mathbf{G}}^{I,+}(t)\) at the rate \(\frac{1}{n}(b(t_c)+n^{-1/6})\) and creating self-loops at the rate \(\frac{1}{2n}(b(t_c)+n^{-1/6})\) for each vertex in \({\mathbf{G}}^{I,+}(t)\).
Next, we construct \(\mathbf{{BSR}}^*(t)\) and \({\mathbf{G}}^{-}(t)\) through successive thinning of \({\mathbf{G}}^{+}(t)\), thus obtaining the desired coupling. Let \((e_1,e_2, \ldots )\) be the sequence of edges that are added to \({\mathbf{G}}^{I,+}(t)\) to obtain \({\mathbf{G}}^{+}(t)\). Let \((u_1, u_2, \ldots )\) be i.i.d Uniform\([0,1]\) random variables that are also independent of the random variables used to construct \({\mathbf{G}}^{I,-}, \mathbf{{BSR}}^{I,*}, {\mathbf{G}}^{I,+}, {\mathbf{G}}^{+}\). Suppose at time \(t_k\), we have \({\mathbf{G}}^{+}(t_k)={\mathbf{G}}^{+}(t_k-) \cup \{ e_k\}\), where \(e_k = \{v_1,v_2\}\). We set \(\mathbf{{BSR}}^*(t_k) = \mathbf{{BSR}}^*(t_k-) \cup \{ e_k\}\) if and only if
otherwise let \(\mathbf{{BSR}}^*(t_k) = \mathbf{{BSR}}^*(t_k-)\). This defines the process \(\mathbf{{BSR}}^*(t)\) (with the correct probability law) such that the second inclusion in (7.2) is satisfied. Finally, construct \({\mathbf{G}}^{-}(t)\) by a thinning of \(\mathbf{{BSR}}^*(t)\) exactly as above by replacing \(\frac{b^*(t_k)}{b(t_c) + n^{-1/6}}\) with \(\frac{b(t_c) - n^{-1/6}}{b^*(t_k)}\). Then \({\mathbf{G}}^{-}(t)\), for \(t \in [t_n ,t_n^+]\) is an Erdős–Rényi type processes and the first inclusion in (7.2) is satisfied. This completes the proof of the first part of the lemma.
We now prove (ii). Consider first the case \(\bullet = -\). We will apply Theorem 5.1. With notation as in that theorem, it follows from the Erdős–Rényi dynamics of \({\mathbf{G}}^{\scriptscriptstyle (n),-}(t)\) that, the distribution of \(({\bar{\varvec{C}}}^{\scriptscriptstyle (n),-}(\lambda ), {\bar{\varvec{Y}}}^{\scriptscriptstyle (n),-}(\lambda ))\), conditioned on \(\{\mathcal {P}_{\varvec{v}}(t),\, t \le t_n \; \varvec{v} \in [n]^4\}\), for each \(\lambda \in [-\lambda _n, \lambda _n]\), is same as the distribution of \(\varvec{Z}(z^{\scriptscriptstyle (n)}, q^{\scriptscriptstyle (n)})\), where \(z^{\scriptscriptstyle (n)} = ({\bar{\varvec{C}}}^{\scriptscriptstyle (n),-}(-\lambda _n), \mathbf {0})\), \(\mathbf {0}\) denotes the vector \((0, 0, \ldots )\) and \(q^{\scriptscriptstyle (n)}\) is determined by the equality
for \(i \ne j\). Recalling that \(\alpha b(t_c)=1\) it then follows that \(q^{\scriptscriptstyle (n)} = \lambda + \frac{n^{1/3-\gamma }}{\alpha \beta ^{2/3}} + O(n^{1/6-\gamma }).\) We now verify the conditions of Theorem 5.1. Taking \(x^{\scriptscriptstyle (n)} = {\bar{\varvec{C}}}^{\scriptscriptstyle (n),-}(-\lambda _n)\) we see with, \(x^*, s_k\), \(k = 1,2,3\) as in Theorem 5.1,
Recall the definition of \(\bar{s}_k\) and \(\bar{s}_{k,\varpi }\) from (3.1) and Sect. 6. Then
From the first two convergences in Proposition 7.3 and recalling that, for \(k=1,2\), \(|\bar{s}_{k, \varpi }-\bar{s}_k| \le K^k\), we immediately get that the first two convergences in (5.1) hold. Also,
where the second equality is consequence of the second convergence in Proposition 7.3, and the convergence of the last term follows from the third convergence in Proposition 7.3. This proves the third convergence in (5.1).
Finally we note that the convergence in (5.2) holds with \(\varsigma = \frac{1}{1 - 3(\gamma + \kappa )}\), where \(\kappa \) is as in Proposition 7.3, since
where the last equality follows from our choice of \(\varsigma \) and the convergence is a consequence of Proposition 7.3. Thus we have verified all the conditions in Theorem 5.1 and therefore we have from this result that \(({\bar{\varvec{C}}}^{\scriptscriptstyle (n),-}(\lambda ), {\bar{\varvec{Y}}}^{\scriptscriptstyle (n),-}(\lambda ))\) converges in distribution, in \(\mathbb {U}_{\downarrow }\), to \((\varvec{X}^*(\lambda ), \varvec{Y}^*(\lambda ))\) proving part (ii) of the lemma for \(\bullet = -\).
To prove part (ii) of the lemma for \(\bullet = +\), one needs slightly more work. Once more we will apply Theorem 5.1. As before, conditioned on \(\{{\bar{\varvec{C}}}^{\scriptscriptstyle (n),+}(\lambda _0): \lambda _0 \le -\lambda _n\}\), for each \(\lambda \in [-\lambda _n, \lambda _n]\), the distribution of \(({\bar{\varvec{C}}}^{\scriptscriptstyle (n),+}(\lambda ), {\bar{\varvec{Y}}}^{\scriptscriptstyle (n),+}(\lambda ))\) is same as the distribution of \(\varvec{Z}(\bar{z}^{\scriptscriptstyle (n)}, \bar{q}^{\scriptscriptstyle (n)})\), where \(\bar{z}^{\scriptscriptstyle (n)} = ({\bar{\varvec{C}}}^{\scriptscriptstyle (n),+}(-\lambda _n), \mathbf {0})\) and \(\bar{q}^{\scriptscriptstyle (n)} = \lambda + \frac{n^{1/3-\gamma }}{\alpha \beta ^{2/3}} + O(n^{1/6-\gamma }).\) Taking \(x^{\scriptscriptstyle (n)} = {\bar{\varvec{C}}}^{\scriptscriptstyle (n),+}(-\lambda _n)\) we see with, \(x^*, s_k\), \(k = 1,2,3\) as in Theorem 5.1,
Next note that for any component \(\mathcal {C}\subset {\mathbf{G}}^-(t_n)= \mathbf{{BSR}}^{I,*}(t_n) \) there is a unique component \( \mathcal {C}^+ \subset {\mathbf{G}}^+(t_n) =\mathbf{{BSR}}^{I,*}(t_n^+)\), such that \(\mathcal {C}\subset \mathcal {C}^+\). Denote by \(\mathcal {C}_i\) the \(i\)th largest component in \(\mathbf{{BSR}}^{I,*}(t_n)\), and let \(\mathcal {C}_i^+\) be the corresponding component in \(\mathbf{{BSR}}^{I,*}(t_n^+)\) such that \(\mathcal {C}_i \subset \mathcal {C}_i^+\). Denote by \(N\) the number of immigrations that occur during \([t_n, t_n^+]\) in \(\mathbf{{BSR}}^{I,*}\), and denote by \(\{{\tilde{\mathcal {C}}}_i^+\}_{i=1}^N\) the components in \(\mathbf{{BSR}}^{I,*}(t_n^+)\) resulting from these immigrations. Then
where
To complete the proof it suffices to show that the statement in Proposition 7.3 holds with \((\bar{s}_2(t_n), \bar{s}_3(t_n), I^{\scriptscriptstyle (n)}(t_n))\) replaced with \((\bar{s}_2^+, \bar{s}_3^+, I^+)\). This follows from Proposition 7.4 given below and hence completes the proof of the lemma.\(\square \)
Proposition 7.4
With notation as in the proof of Lemma 7.2, as \(n \rightarrow \infty \), we have
Proof
The proof is similar to that of Proposition 8.1 in [7] thus we only give a sketch.
Observe that the total rate of attachments is \(\sum _{i=1}^K c^*_i(t) \le 1\) and each attachment has size no bigger than \(K\). Recall that \(\mathcal {C}_i\) denotes the \(i\)th largest component in \(\mathbf{{BSR}}^{I,*}(t_n)\). Denote by \(V_i(t)\), \(t \in [t_n, t_n^+]\), the stochastic process defining the size of the component containing \(\mathcal {C}_i\) in \(\mathbf{{BSR}}^{I,*}(t)\). Note that \(V_i(t_n) = |\mathcal {C}_i|\) and \(V_i(t_n^+)=|\mathcal {C}_i^+|\). Then \({V_i(t)}/{K}\) can be stochastically dominated by a Yule process starting with \(\lceil |\mathcal {C}_i|/K\rceil \) particles and birth rate \(K\). Using this and an argument similar to [7], it follows that,
Next, note that the immigrations are of size no bigger than \(2K\), and thus for the same reason, we have the bound,
Since the total number of vertices is \(n\), the number of immigrations \(N\) can be bounded by \(n/K\).
With the above three bounds the proof of the proposition follows exactly as the proof of Proposition 8.1 in [7] with obvious changes needed due to the constant \(K\) that appears in the above bounds. Details are omitted.
The next proposition says that the inclusion in (7.2) can be strengthened to component-wise inclusion.
Proposition 7.5
Fix \(\lambda \in {\mathbb {R}}\) and \(i_0\ge 1\). Then, as \(n\rightarrow \infty \),
Proof
From Lemma 7.2 and Lemma 15 in [6] (see also Section 8.2 of [7] for a similar argument), we have, as \(n\rightarrow \infty \),
in \(l^2_{\downarrow }\times l^2_{\downarrow }\times l^2_{\downarrow }\), where \(\varvec{X}\) is as in Theorem 3.1. Define events \(E_n, F_n\) as
Then on the set \(E_n \cap F_n\)
From Lemma 7.2 (i) \(\mathbb {P}\{ F_n^c\} \rightarrow 1\). Also
This shows that \(\mathbb {P}(E_n \cap F_n) \rightarrow 1\) as \(n \rightarrow \infty \). The result follows.\(\square \)
We will also need the following elementary lemma. Proof is omitted.
Lemma 7.6
Let \( \eta ^{\scriptscriptstyle (n),-}, \eta ^{\scriptscriptstyle (n), +}, \eta ^*\) be real random variables such that \(\eta ^{\scriptscriptstyle (n),-} \le \eta ^{\scriptscriptstyle (n),+}\) with high probability. Further assume \(\eta ^{\scriptscriptstyle (n),-} \mathop {\longrightarrow }\limits ^{d}\eta ^*\) and \(\eta ^{\scriptscriptstyle (n),+} \mathop {\longrightarrow }\limits ^{d}\eta ^*\). Then \( \eta ^{\scriptscriptstyle (n),+} - \eta ^{\scriptscriptstyle (n),-} \mathop {\longrightarrow }\limits ^{\scriptscriptstyle {\mathbb P}}0. \) Furthermore, if \(\eta ^{\scriptscriptstyle (n)}\) are random variables such that \(\eta ^{\scriptscriptstyle (n),-} \le \eta ^{\scriptscriptstyle (n)} \le \eta ^{\scriptscriptstyle (n),+}\) with high probability, then \( \eta ^{\scriptscriptstyle (n)} \mathop {\longrightarrow }\limits ^{d}\eta ^*\) and \(\eta ^{\scriptscriptstyle (n)} - \eta ^{\scriptscriptstyle (n),-} \mathop {\longrightarrow }\limits ^{\scriptscriptstyle {\mathbb P}}0.\)
We now complete the proof of Theorem 7.1.
Proof of Theorem 7.1
From Lemma 7.2 (ii) we have that
in \(l^2_{\downarrow }\times \mathbb {N}^{\infty } \times \mathbb {R}\), where on \(\mathbb {N}^{\infty }\) we consider the product topology.
In order to prove the theorem it suffices, in view of Lemma 4.11, to show that
in \(l^2_{\downarrow }\times \mathbb {N}^{\infty } \times \mathbb {R}\).
From Proposition 7.5, we have for any \(i_0 \in \mathbb {N}\), with high probability
Also, from Lemma 7.2 (i), whp,
From Lemma 7.6 and Lemma 7.2 (ii), we then have
in \(\mathbb {N}^{\infty } \times \mathbb {R}\), where for \(y = (y_1, y_2, \ldots ) \in \mathbb {Z}^{\infty }\), \(|y| = (|y_1|, |y_2|, \ldots )\). The convergence in (7.5) now follows on combining (7.4) and (7.3). The result follows.\(\square \)
7.2 Proof of Theorem 3.3
As a first step towards the proof we show the following convergence result for one dimensional distributions.
Theorem 7.7
For every \(\lambda \in \mathbb {R}\), as \(n\rightarrow \infty \), \((\bar{\varvec{C}}^{\scriptscriptstyle (n)}(\lambda ), {\bar{\varvec{Y}}}^{\scriptscriptstyle (n)}(\lambda ))\) converges in distribution, in \(\mathbb {U}_{\downarrow }\), to \(({\varvec{X}}(\lambda ), \varvec{Y}(\lambda ))\).
Proof
Fix \(\lambda \in \mathbb {R}\). We first argue that
For this, it suffices to show that
Define
By Theorem 6.9, \(\mathbb {P}\{E_n^c\} \rightarrow 0\) and \(E_n \in {\tilde{\mathcal {F}}}(\lambda ) := \sigma \{ |\mathcal {C}_i(s)|: i\ge 1, s \le t^{\lambda }\}\) for all \(\lambda \ge -\lambda _n\). We begin by showing that there exists \(d_1 \in (0, \infty )\) such that, for all \(i \in \mathbb {N}\),
Note that at any time \(s < t^{\lambda }\), for a component \(\mathcal {C}\subset \mathbf{{BSR}}^{\scriptscriptstyle (n)}(s)\), there are at most \(2 |\mathcal {C}|^2 n^2\) quadruples of vertices which may provide a surplus edge within \(\mathcal {C}\). Since edges are formed at rate \(2/n^3\), we have that
Thus, for some \(d_0, d_1 \in (0, \infty )\)
This proves (7.8). As an immediate consequence of this inequality we have that
Observing that \(\gamma - 1/3 < 0\) and, from Theorem 7.1, that \(\sum _{i}\left( \bar{\varvec{C}}_i^{{\scriptscriptstyle (n)},*}(\lambda ) \right) ^2\) converges in distribution, we have that
Since \(\mathbb {P}\{E_n\} \rightarrow 1\), letting \(\eta ^{\scriptscriptstyle (n)} = \sum _{i}\tilde{\xi }_i^{\scriptscriptstyle (n)}(\lambda )\bar{\varvec{C}}_i^{{\scriptscriptstyle (n)},*}(\lambda )\), we have that \(\mathbb {E}(\eta ^{\scriptscriptstyle (n)} \mid {\tilde{\mathcal {F}}}(\lambda )) \rightarrow 0\) in probability. Convergence in (7.7) now follows from Markov’s inequality on noting that, as \(n\rightarrow \infty \),
This proves (7.6). Next note that
Also,
Thus, as \(n \rightarrow \infty \),
The result now follows on combining the above convergence with (7.9) and (7.6).\(\square \)
Remark 7.8
The proofs of Theorems 7.1 and 7.7 in fact establish the following stronger statement: For all \(\lambda \in \mathbb {R}\),
in probability, in \(\mathbb {N}^{\infty }\times \mathbb {R}\times \mathbb {R}\).
Proof of Theorem 3.3
For simplicity we present the proof for the case \(m=2\). The general case can be treated similarly. Fix \(-\infty < \lambda _1 < \lambda _2 < \infty \). Denote, for \(\lambda \in \mathbb {R}\), \({\bar{\varvec{Z}}}^{\scriptscriptstyle (n),-}(\lambda ) = (\bar{\varvec{C}}^{\scriptscriptstyle (n),-}(\lambda ), {\bar{\varvec{Y}}}^{\scriptscriptstyle (n),-}(\lambda ))\). In view of Remark 7.8 it suffices to show that, as \(n \rightarrow \infty \),
for which it is enough to show that for all \(f_1, f_2 \in C_b(\mathbb {U}_{\downarrow }^0)\)
Note that the left side of (7.10) equals
which using Theorem 3.1 (2), Lemma 7.2 (ii) and the fact that \(\varvec{Z}(\lambda ) \in \mathbb {U}_{\downarrow }^1\) a.s., converges to
where the last equality follows from Theorem 3.1 (3). This proves (7.10) and the result follows.\(\square \)
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Acknowledgments
We thank the referee for a very careful reading of the manuscript and for the many suggestions that significantly improved the presentation. AB and XW has been supported in part by the National Science Foundation (DMS-1004418, DMS-1016441, DMS-13051120), the Army Research Office (W911NF-0-1-0080, W911NF-10-1-0158) and the US-Israel Binational Science Foundation (2008466). SB and XW have been supported in part by the National Science Foundation (DMS-1105581, DMS-1310002).
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Bhamidi, S., Budhiraja, A. & Wang, X. The augmented multiplicative coalescent, bounded size rules and critical dynamics of random graphs. Probab. Theory Relat. Fields 160, 733–796 (2014). https://doi.org/10.1007/s00440-013-0540-x
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DOI: https://doi.org/10.1007/s00440-013-0540-x
Keywords
- Bounded-size rules
- Surplus
- Critical random graphs
- Scaling window
- Multiplicative coalescent
- Entrance boundary
- Giant component
- Branching processes
- Inhomogeneous random graphs
- Differential equation method
- Dynamic random graph models
Mathematics Subject Classification (2000)
- Primary 60C05
- 05C80
- 90B15