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Critical Temperature of Heisenberg Models on Regular Trees, via Random Loops

  • Jakob E. Björnberg
  • Daniel Ueltschi
Open Access
Article

Abstract

We estimate the critical temperature of a family of quantum spin systems on regular trees of large degree. The systems include the spin-\(\frac{1}{2}\) XXZ model and the spin-1 nematic model. Our formula is conjectured to be valid for large-dimensional cubic lattices. Our method of proof uses a probabilistic representation in terms of random loops.

Keywords

Random loop model Quantum Heisenberg Critical temperature 

Mathematics Subject Classification

60K35 82B20 82B26 82B31 

1 Introduction and Main Result

The main goal of this study is to predict an expression for the critical temperature of a family of quantum spin systems on the cubic lattice \({{\mathbb {Z}}}^\nu \) that holds asymptotically for large dimension \(\nu \). More precisely, we propose the first two terms in the expansion in powers of \(\nu ^{-1}\). The family of quantum spin systems includes the spin \(\frac{1}{2}\) ferromagnetic and antiferromagnetic Heisenberg models and the XXZ model. We also consider spin 1 quantum nematic systems. Our results are expected to be exact but they are not rigorous on \({{\mathbb {Z}}}^\nu \). In fact we do not perform calculations with the cubic lattice but we consider the model on regular trees with d descendants; we obtain the first two terms of the critical inverse temperature in powers of \(d^{-1}\). For trees our computations are completely rigorous. We conjecture that our expression applies to \({{\mathbb {Z}}}^\nu \) when taking \(d = 2\nu -1\).

1.1 Random-Loop Model

Our method is based on using a random loop representation, which we now describe. The relevant model of random loops may be defined for arbitrary finite graphs, here we consider mainly trees. Let T denote an infinite rooted tree where each vertex has \(d\ge 2\) offspring, and write \(\rho \) for its root. We sometimes refer to the number of offspring of vertex as its outdegree. For \(m\ge 0\) let \(T_m\) denote the subtree of T consisting of the first m generations (\(\rho \) being generation zero). Write \(V_m\) and \(E_m\) for the vertex- and edge sets of \(T_m\).

Let \(\mathbb {P}_m(\cdot )\) denote a probability measure governing a collection \(\omega =(\omega _{xy}:xy\in E_m)\) of independent Poisson processes on the interval [0, 1], indexed by the edge-set \(E_m\), each having rate \(\beta \) (the inverse-temperature). We refer to realizations of \(\omega \) as a collections of links, and to \(\omega _{xy}\) as the links supported by the edge xy. Thus, disjoint sub-intervals \(I,J\subseteq [0,1]\) independently receive uniformly placed links, their number being Poisson-distributed with mean \(\beta |I|\) and \(\beta |J|\), respectively. We write \(\mathbb {E}_m[\cdot ]\) for expectation under \(\mathbb {P}_m(\cdot )\).

A given link is assigned to be a cross with probability u, otherwise a double-bar, independently between different links. The collection of links then decomposes \(T_m\times [0,1]\) into a collection of disjoint loops in a natural way. Rather than giving a formal definition here, we refer to Fig. 1. A formal definition may be found e.g. in [17, Sect. 2.1].
Fig. 1

Random loops coming from a configuration \(\omega \) of crosses and bars, in the case when the underlying graph is a line with seven vertices. To each vertex corresponds a vertical line segment which is a copy of the interval [0, 1]. On following a loop one reverses direction when traversing a double-bar, maintains direction when traversing a cross, and proceeds periodically in the vertical direction. In this example there are \(\ell (\omega )=4\) loops

The total number of loops is denoted \(\ell =\ell (\omega )\). We actually work with a weighted version of \(\mathbb {P}_m(\cdot )\), denoted \(\mathbb {P}^{(\theta )}_m(\cdot )\) with a positive parameter \(\theta \). This is the probability measure whose expectation operator \(\mathbb {E}_m^{(\theta )}[\cdot ]\) is given by
$$\begin{aligned} \mathbb {E}^{(\theta )}_m[X]=\frac{\mathbb {E}_m[X\theta ^{\ell (\omega )}]}{\mathbb {E}_m[\theta ^{\ell (\omega )}]}. \end{aligned}$$
Note that \(\mathbb {P}^{(1)}_m=\mathbb {P}_m\).
All loops are small when \(\beta \) is small, this may be shown e.g. as in [9, Thm. 6.1]. But it is expected that there exists \(\beta _\mathrm {c}\), that depends on the parameter \(\theta \) and the outdegree d, such that a given points lies in an infinite loop with positive probability for \(\beta > \beta _\mathrm {c}\). Our main result is a formula for \(\beta _\mathrm {c}\); it is asymptotic in the outdegree \(d\rightarrow \infty \), namely
$$\begin{aligned} \frac{\beta _\mathrm {c}}{\theta }=\frac{1}{d}+ \frac{1-\theta u(1-u)-\tfrac{1}{6}\theta ^2(1-u)^2}{d^2} +o(d^{-2}), \end{aligned}$$
(1.1)
and we can prove that there are infinite loops for \(\beta >\beta _\mathrm {c}\) in the vicinity of \(\beta _\mathrm {c}\). For a more precise statement, see Theorem 1.1 below.

The first study of this model on trees is due to Angel [2], who established the presence of long loops for a range of parameters \(\beta \) when \(d\ge 4\); he only considered the case \(u=1\) and \(\theta =1\). Angel’s results were extended by Hammond [10, 11]; he gave a precise characterisation of the critical parameter \(\beta _\mathrm {c}\) for large enough d. The Formula (1.1) was established in [6] in the case \(\theta =1\), our study following a suggestion of Hammond. Very recently Hammond and Hegde [12] proved that the Formula (1.1) for \(\theta =1\) truly identifies the critical point, not only in the local sense considered here and in [6]; their results hold for large d. Another extension to \(\theta \ne 1\) has independently been proposed by Betz et al. [4].

1.2 Quantum Spin Systems

Let \((\Lambda ,{{\mathcal {E}}})\) denote a graph, with \(\Lambda \) the set of vertices and \({{\mathcal {E}}}\) the set of edges. The main examples to bear in mind here are regular trees, and finite subsets of \({{\mathbb {Z}}}^d\) with nearest-neighbour edges. The spin-\(\frac{1}{2}\) systems have Hilbert space \({{\mathcal {H}}}_\Lambda = \otimes _{x\in \Lambda } {{\mathbb {C}}}^2\) and the hamiltonian is
$$\begin{aligned} H_\Lambda = -2 \sum _{\{x,y\} \in {{\mathcal {E}}}} \Bigl ( S_x^{(1)} S_y^{(1)} + S_x^{(2)} S_y^{(2)} + \Delta S_x^{(3)} S_y^{(3)} \Bigr ), \end{aligned}$$
where \(S_x^{(i)}\), \(i=1,2,3\) denotes the ith spin operator at site \(x \in \Lambda \). Here, \(\Delta \in [-1,1]\) is a parameter.
As was progressively understood in [1, 15, 17], this quantum system is represented by the model of random loops with \(\theta =2\) and \(u = \frac{1}{2} (1+\Delta )\). Indeed, the quantum two-point correlation function is given by loop correlations,
$$\begin{aligned} \langle S^{(1)}_x S^{(1)}_y\rangle := \frac{{{\text {tr}}\,}\big (S^{(1)}_xS^{(1)}_y\,\mathrm{e}^{-\beta H_\Lambda }\,\big )}{{{\text {tr}}\,}\big (\,\mathrm{e}^{-\beta H_\Lambda }\,\big )} =\tfrac{1}{4} \mathbb {P}_\Lambda ^{(\theta =2)}(x\leftrightarrow y), \end{aligned}$$
(1.2)
where \(\{x\leftrightarrow y\}\) is the event that (x, 0) and (y, 0) belong to the same loop. It follows that magnetic long-range order is related to the occurrence of large loops.
On \({{\mathbb {Z}}}^3\), the critical inverse temperature has been computed numerically; it was found that
$$\begin{aligned} \beta _{\mathrm{c}}^{(\nu =3)}(\Delta ) = {\left\{ \begin{array}{ll} 0.596 \dots &{} \text {if } \Delta = 1;\text { Troyer et al. [16]} \\ 0.4960 \dots &{} \text {if }\Delta =0;\text { Wessel, private communication in [3]} \\ 0.530 \dots &{} \text {if }\Delta =-1;\text { Sandvik [13], Troyer et al. [16]} \end{array}\right. } \end{aligned}$$
(1.3)
For large \(\nu \), the lattice \({{\mathbb {Z}}}^\nu \) behaves like a tree of outdegree \(d=2\nu -1\). Our Formula (1.1) gives
$$\begin{aligned} \beta _{\mathrm{c}}^{(\nu )}(\Delta ) = \tfrac{1}{\nu }+ \tfrac{1}{\nu ^2} \bigl [ 1 - \tfrac{1}{6} (1-\Delta )(2+\Delta ) \bigr ] + o\left( \tfrac{1}{\nu ^2}\right) . \end{aligned}$$
(1.4)
With \(\nu =3\), the values for \(\Delta = 1,0,-1\) are \(\frac{4}{9}, \frac{11}{27}, \frac{11}{27}\) respectively. They corroborate the numerical values (1.3) to some extent. Of course, the Formula (1.4) gets more accurate in high dimensions.
In the case of spin-1 systems, the Hilbert space is \({{\mathcal {H}}}_\Lambda = \otimes _{x\in \Lambda } {{\mathbb {C}}}^3\) and the hamiltonian is
$$\begin{aligned} H_\Lambda = -\sum _{\{x,y\} \in {{\mathcal {E}}}} \Bigl ( u \vec S_x \cdot S_y + (\vec S_x \cdot S_y)^2 \Bigr ), \end{aligned}$$
(1.5)
See [17]. The phase diagram of this model was determined in [7]. For \(0<u<1\) the system displays nematic long-range order at low temperatures (if \(d\ge 3\); also in the ground state when \(d=2\)). This was rigorously proved in [14, 17]. The corresponding loop model has parameter \(\theta =3\), and the same u as in (1.5). Loop correlations are related to nematic long-range order, namely
$$\begin{aligned} \langle A_x A_y \rangle = \tfrac{2}{9} \mathbb {P}_\Lambda ^{(\theta =3)}(x\leftrightarrow y), \end{aligned}$$
(1.6)
with \(A_x = (S_x^{(3)})^2 - \frac{2}{3}\). We are not aware of numerical calculations of the critical inverse temperature \(\beta _{\mathrm{c}}\) for this model on \({{\mathbb {Z}}}^3\). With \(\theta =3\) and \(d=2\nu -1\), the Formula (1.1) gives
$$\begin{aligned} \beta _{\mathrm{c}}^{(\nu )}(u) = \tfrac{3}{2\nu } + \tfrac{3}{2\nu ^2} \bigl [ 1 - \tfrac{3}{4} (1-u^2) \bigr ] + o\left( \tfrac{1}{\nu ^2}\right) . \end{aligned}$$

1.3 Main Result

In the rest of this paper we deal only with the probabilistic model of random loops defined above, and we allow \(\theta \) to be any (fixed) positive real number. Our main result is that, as the distance between x and y goes to \(\infty \), the two-point function vanishes or stays positive, according to whether \(\beta \) is smaller or larger than \(\beta _\mathrm {c}\) given above. Let us say that a loop visits a vertex x of \(T_m\) if the loop contains a point (xt) for some \(t\in [0,1]\). Motivated by (1.2) and (1.6) we consider
$$\begin{aligned} \sigma _m = \mathbb {P}^{(\theta )}_m(\rho \leftrightarrow m), \end{aligned}$$
that is, \(\sigma _m\) is the \(\mathbb {P}^{(\theta )}_m\)-probability that \((\rho ,0)\) belongs to a loop which visits some vertex in generation m in \(T_m\).
Throughout this paper we work with \(\beta \) of the form
$$\begin{aligned} \frac{\beta }{\theta }=\frac{1}{d}+\frac{\alpha }{d^2}, \qquad \text{ where } |\alpha |\le \alpha _0 \end{aligned}$$
(1.7)
for some fixed but arbitrary \(\alpha _0>0\). All error terms \(O(\cdot )\), \(o(\cdot )\) and constants may depend on \(\alpha _0\) but are otherwise uniform in \(\alpha \).

Theorem 1.1

Consider \(\beta \) of the form (1.7), and write
$$\begin{aligned} \alpha _*=\alpha _*(\theta ,u)=1-\theta u(1-u)-\tfrac{1}{6}\theta ^2(1-u)^2. \end{aligned}$$
For any \(\delta >0\) there exists \(d_0=d_0(\theta ,u,\alpha _0,\delta )\) such that for \(d\ge d_0\) we have:
  • if \(\alpha \le \alpha _*-\delta \) then \(\lim _{m\rightarrow \infty } \sigma _m=0\);

  • if \(\alpha \ge \alpha _*+\delta \) then \(\liminf _{m\rightarrow \infty } \sigma _m>0\).

Let us remark that for \(\theta =1\) the result was shown in our previous work [6]. The arguments presented here are strengthened versions of those arguments. The basic strategy is to establish recursion inequalities for the sequence \(\sigma _m\), see Proposition 2.1. These are obtained by analyzing the local configuration around the root \(\rho \), in particular we identify two events \(A_1\) and \(A_2\) which together contribute most of the probability in the regime we consider (\(d\rightarrow \infty \) and \(\beta \) as in (1.7)).

2 Proof of the Main Result

The indicator function of an event A will be written \(\mathbb {1}_A\) or \(\mathbb {1}\{A\}\). The partition function for the loop model on \(T_m\) is written \(Z_m=\mathbb {E}_m[\theta ^\ell ]\). For convenience we also define
$$\begin{aligned} z_m=e^{-d\beta (1-1/\theta )}\frac{\theta Z_{m-1}^d}{Z_m}. \end{aligned}$$
(2.1)
For given \(m\ge 1\) and \({\varepsilon }>0\) we define
$$\begin{aligned} {\tilde{\sigma }}_m = \sigma _m\wedge \sigma _{m-1}\wedge \left( \tfrac{{\varepsilon }}{d}\right) . \end{aligned}$$
(A priori we need not have \(\sigma _m\le \sigma _{m-1}\) since they are computed using different measures.) In this section we will prove the following recursion-inequalities.

Proposition 2.1

For all \(m\ge 1\) we have
$$\begin{aligned} \sigma _m\ge {\tilde{\sigma }}_{m-1}+ \tfrac{{\tilde{\sigma }}_{m-1}}{d}\big (\alpha -\alpha _*\big )-\tfrac{1}{2}{\tilde{\sigma }}_{m-1}^2 + O(d^{-3}), \end{aligned}$$
(2.2)
and
$$\begin{aligned} \sigma _m\le (\sigma _{m-1}\vee \sigma _{m-2})\big [1+\tfrac{1}{d}\big (\alpha -\alpha _*\big )+O(d^{-2}) \big ]. \end{aligned}$$
(2.3)
Here the \(O(d^{-3})\) and \(O(d^{-2})\) are uniform in m.

Our main result follows easily:

Proof of Theorem 1.1

First suppose \(\alpha <\alpha _*\). For d large enough the factor in square brackets in (2.3) is strictly smaller than 1. This easily gives that \(\sigma _m\) decays to 0 exponentially fast.

Now suppose \(\alpha >\alpha _*\). Clearly \(\sigma _0=1\), and it is not hard to see that there exists a constant \(c_1>0\) such that \(\sigma _1\ge c_1\) for all d. This implies that \({\tilde{\sigma }}_1={\varepsilon }/d\) if \({\varepsilon }<c_1\). If also \({\varepsilon }<2(\alpha -\alpha _*)\) and d is large enough then (2.2) and induction on m give that \(\sigma _m\ge {\tilde{\sigma }}_{m}= {\varepsilon }/d\) for all \(m\ge 1\).

\(\square \)

Before turning to the proof of Proposition 2.1, let us describe some of the main ideas and also what new input is required compared to our previous work [6] on the case \(\theta =1\). For the lower bound (2.2) we will estimate the probability of certain local configurations near \(\rho \) which guarantee that \(\rho \) is connected to generation m if certain of its children (or grandchildren) are. For the upper bound we similarly estimate \(\mathbb {P}_m^{(\theta )}(\rho \not \leftrightarrow m)\) in terms of the probability that certain of \(\rho \)’s children (or grandchildren) are blocked from generation m. When \(\theta \ne 1\), the configurations in the subtrees rooted at the children of \(\rho \) are not independent of the local configuration adjacent to \(\rho \). Thus we must deal carefully with the factor \(\theta ^{\ell (\omega )}\) and how it behaves in the local configurations which we consider. This involves obtaining estimates for the partition function \(Z_m\) in terms of the partition function \(Z_{m-1}\) in the smaller tree, which is where the number \(z_m\) in (2.1) becomes relevant.

As was the case in [6], the hardest part is the upper bound (2.3). This is because we must rule out connections due to ‘lower order events’ (\((A_1\cup A_2)^c\) in the notation below) where the loop structure is too complicated to handle directly. The main technical advance compared to [6] started with a simplification of the argument used there to deal with this difficulty. Having this simpler version allowed us to deal also with the correlations caused by the factor \(\theta ^{\ell (\omega )}\), see Proposition 2.5.

2.1 Preliminary Calculations

Let us first introduce some notations and prove some facts that will be used for establishing both bounds in Proposition 2.1.

Write \(A_1\) for the event that, for each child x of \(\rho \), there is at most one link between \(\rho \) and x. Write \(A_2\) for the event that: (i) there is a unique child x of \(\rho \) with exactly 2 links between \(\rho \) and x, (ii) for all siblings \(x'\) of x there is at most one link between \(\rho \) and \(x'\), and (iii) for all children y of x there is at most one link between x and y. See Fig. 2.
Fig. 2

Illustrations of the two events \(A_1\) and \(A_2\). Numbers on edges indicate the number of links

Let \(\zeta _m=1-\sigma _m\) and let \(B^\rho _m\) be the event that \((\rho ,0)\) does not belong to a loop which reaches generation m in \(T_m\), thus \(\mathbb {P}^{(\theta )}_m(B^\rho _m)=\zeta _m\). Clearly we have that
$$\begin{aligned} \zeta _m=\mathbb {P}^{(\theta )}_m(B^\rho _m)= \mathbb {P}^{(\theta )}_m(B_m^\rho \cap A_1) +\mathbb {P}^{(\theta )}_m(B_m^\rho \cap A_2) +\mathbb {P}^{(\theta )}_m(B_m^\rho \setminus (A_1\cup A_2)). \end{aligned}$$
(2.4)
Let us enumerate the children of \(\rho \) by \(i=1,\cdots ,d\) and let \(\ell _i\) denote the number of loops in the restriction of \(\omega \) to the subtree to distance m rooted at child i. On the event \(A_1\), and if there are k links from \(\rho \), the number \(\ell \) of loops satisfies
$$\begin{aligned} \ell =\textstyle \sum _{i=1}^d \ell _i -k +1. \end{aligned}$$
(2.5)
To see this, one may imagine that the k links to \(\rho \) are put in last, one at a time. Each such link then merges some loop in the corresponding subtree with a loop visiting \(\rho \). (This uses the tree-structure of the underlying graph, which implies that there can be no connections between \(\rho \) and the subtree until the link is put in.) It follows that
$$\begin{aligned} \begin{aligned} \mathbb {E}_m[\theta ^\ell \mathbb {1}_{A_1}]&=\sum _{k=0}^d \theta ^{-k+1}\mathbb {E}_m\Big [\theta ^{\sum _i\ell _i} \mathbb {1}_{A_1}\mathbb {1}\{k \text{ links } \text{ at } \rho \}\Big ] \\&=\theta \sum _{k=0}^d \left( {\begin{array}{c}d\\ k\end{array}}\right) \left( e^{-\beta }\right) ^{d-k} \left( e^{-\beta }\tfrac{\beta }{\theta }\right) ^k Z_{m-1}^{d}\\&=\theta Z_{m-1}^d\left( e^{-\beta }\left( 1+\tfrac{\beta }{\theta }\right) \right) ^d \end{aligned} \end{aligned}$$
and hence (recalling \(z_m\) from (2.1))
$$\begin{aligned} \mathbb {P}^{(\theta )}_m(A_1)= z_m \left( e^{-\beta /\theta }\left( 1+\tfrac{\beta }{\theta }\right) \right) ^d. \end{aligned}$$
(2.6)
Similarly, since the k children with links would need to be blocked from reaching distance \(m-1\), we also have
$$\begin{aligned} \begin{aligned} \mathbb {P}^{(\theta )}_m(A_1\cap B_m^\rho )&= \frac{\theta }{Z_m}\sum _{k=0}^d \left( {\begin{array}{c}d\\ k\end{array}}\right) \left( e^{-\beta }\right) ^{d-k} \left( e^{-\beta }\tfrac{\beta }{\theta }\right) ^k Z_{m-1}^{d-k}\mathbb {E}_{m-1}\left[ \theta ^\ell \mathbb {1}_{B^\rho _{m-1}}\right] ^k\\&=z_m \left( e^{-\beta /\theta }\left( 1+\zeta _{m-1}\tfrac{\beta }{\theta }\right) \right) ^d. \end{aligned} \end{aligned}$$
(2.7)
Fig. 3

Illustration of the possibilities for \(\omega _{\rho x}\) on the event \(A_2\). On \(A_2^{\mathrm {same}}\) there are two loops, one of which contains \((\rho ,0)\); on \(A_2^{\mathrm {mix}}\) only one. The latter is thus more advantageous for long connections. The random variable X has mean \(\tfrac{2}{3}\)

For the event \(A_2\), we decompose it as \(A_2=A_2^{\mathrm {mix}}\cup A_2^{\mathrm {same}}\), according as the 2 links from \(\rho \) to x are different sorts (crosses/double-bars) or the same (Fig 3). If we look at the restriction of \(\omega \) to the link \(\rho x\) only (i.e., at \(\omega _{\rho x}\)) then it has two loops on \(A_2^{\mathrm {same}}\) and a single loop on \(A_2^{\mathrm {mix}}\). Let us number the children of x together with the children of \(\rho \) excepting x by \(i=1,\cdots ,2d-1\). Then we have that
$$\begin{aligned} \ell =\textstyle \sum _{i=1}^{2d-1} \ell _i -k + \left\{ \begin{array}{ll} 1 &{}\quad \text{ on } A_2^{\mathrm {mix}},\\ 2 &{}\quad \text{ on } A_2^{\mathrm {same}}, \end{array} \right. \end{aligned}$$
(2.8)
where k denotes the total number of 1-links at \(\rho \) and at x. To see this one may again imagine that the 1-links are placed last, one at a time. If \(k=0\) then (2.8) holds due to our observation about \(A_2^{\mathrm {mix}}\) and \(A_2^{\mathrm {same}}\) above, if \(k>0\) then each link we place merges two previously disjoint loops.
Let \(\Lambda \) denote the loop in \(\omega _{\rho x}\) containing \((\rho ,0)\), and let \(\Lambda _\rho =\Lambda \cap (\{\rho \}\times [0,1])\) and \(\Lambda _x=\Lambda \cap (\{x\}\times [0,1])\) denote the parts of \(\Lambda \) at \(\rho \) and at x, respectively. For \(B_m^\rho \) to happen, children of \(\rho \) which link to \(\Lambda \) need to be blocked from distance \(m-1\) and children of x which link to \(\Lambda \) need to be blocked from distance \(m-2\); the remaining children of \(\rho \) and x do not need to be blocked. In particular, on \(A_2^{\mathrm {mix}}\) all children which link to either \(\rho \) or x need to be blocked. Write \(A_2^{\mathrm {mix}}(x,k_0,k_1)\) for the event that (i) \(\rho x\) supports one link of each sort, (ii) among the remaining children of \(\rho \) exactly \(k_0\) support 1 link and the rest 0, and (iii) among the children of x exactly \(k_1\) support 1 link and the rest 0. Using (2.8) with \(k=k_0+k_1\) and a calculation similar to (2.7) we get
$$\begin{aligned} \begin{aligned} \mathbb {E}_m[\theta ^\ell \mathbb {1}_{B^\rho _m} \mathbb {1}_{A_2^\mathrm {mix}}]&= \sum _{x\sim \rho }\sum _{k_0=0}^{d-1}\sum _{k_1=0}^d \theta ^{-k_0-k_1+1} \mathbb {E}_m\Big [\theta ^{\sum _i\ell _i} \mathbb {1}_{A_2^{\mathrm {mix}}(x,k_0,k_1)}\mathbb {1}_{B^\rho _{m}}\Big ]\\&=\theta Z_{m-1}^{d-1}{Z_{m-2}^d} \tfrac{d\beta ^2e^{-\beta }}{2} 2u(1-u) \left( e^{-\beta }\left( 1+\tfrac{\beta }{\theta }\zeta _{m-1}\right) \right) ^{d-1}\\&\quad \times \left( e^{-\beta }\left( 1+\tfrac{\beta }{\theta }\zeta _{m-2}\right) \right) ^{d}. \end{aligned} \end{aligned}$$
(2.9)
For the case of \(A_2^{\mathrm {same}}\) we may start with a similar decomposition,
$$\begin{aligned} \mathbb {E}_m\left[ \theta ^\ell \mathbb {1}_{B^\rho _m} \mathbb {1}_{A_2^\mathrm {same}}\right] =\sum _{x\sim \rho }\sum _{k_0=0}^{d-1}\sum _{k_1=0}^d \theta ^{-k_0-k_1+2} \mathbb {E}_m\Big [\theta ^{\sum _i\ell _i} \mathbb {1}_{A_2^{\mathrm {same}}(x,k_0,k_1)}\mathbb {1}_{B^\rho _m}\Big ], \end{aligned}$$
where \(A_2^{\mathrm {same}}(x,k_0,k_1)\) is defined as \(A_2^{\mathrm {mix}}(x,k_0,k_1)\) except for requiring the two links supported by \(\rho x\) to be of the same sort instead. Here we may then further consider the number \(j_0\in \{0,\cdots ,k_0\}\) of links with an endpoint in \(\Lambda _\rho \) as well as the number \(j_1\in \{0,\cdots ,k_1\}\) of links with an endpoint in \(\Lambda _x\). As mentioned above, these links need to be blocked, but the remaining do not. Recalling that the locations of links are uniform on [0, 1] this means that we obtain a factor \(|\Lambda _\rho |\) (respectively \(|\Lambda _x|\)) for each of these \(j_0\) (respectively, \(j_1\)) links, and hence
$$\begin{aligned} \begin{aligned} \mathbb {E}_m[\theta ^\ell \mathbb {1}_{B^\rho _m} \mathbb {1}_{A_2^\mathrm {same}}]&=\theta ^2 Z_{m-1}^{d-1}{Z_{m-2}^d} \tfrac{d\beta ^2e^{-\beta }}{2} (u^2+(1-u)^2) \\&\quad \mathbb {E}\big [ \left( e^{-\beta }\left( 1+\tfrac{\beta }{\theta }\zeta _{m-1}|\Lambda _\rho | +\tfrac{\beta }{\theta }(1-|\Lambda _\rho |)\right) \right) ^{d-1}\\&\quad \times \left( e^{-\beta }\left( 1+\tfrac{\beta }{\theta }\zeta _{m-2}|\Lambda _x| +\tfrac{\beta }{\theta }(1-|\Lambda _x|)\right) \right) ^{d} \big ]. \end{aligned} \end{aligned}$$
(2.10)
Here we have simply written \(\mathbb {E}[\cdot ]\) for \(\mathbb {E}_m[\cdot \mid A_2]\), this expectation is over the choice of crosses or double-bars and over the lengths \(|\Lambda _\rho |\) and \(|\Lambda _x|\) only.

We note here that the joint expectations of \(|\Lambda _\rho |\) and \(|\Lambda _x|\) may be computed explicitly. Indeed, as illustrated in Fig. 3, there is a random variable X such that \(\Lambda _\rho \) and \(\Lambda _x\) have respective lengths X and \(1-X\) in the case of two crosses; X and X in the case of two double-bars; and \(|\Lambda _\rho |=|\Lambda _x|=X=1\) in the case of a mixture. We have the following:

Lemma 2.2

\(\mathbb {E}_m[X\mid A_2^{\mathrm {same}}]=\tfrac{2}{3}\).

Proof

The conditional distribution of X given \(A_2^{\mathrm {same}}\) is that of that of \(1-(U_1\vee U_2-U_1\wedge U_2)\) where \(U_1,U_2\) are independent random variables, uniformly distributed on [0, 1]. Using that \(U_1\wedge U_2 = U_1+U_2-U_1\vee U_2\) and \(\mathbb {P}(U_1\vee U_2\le t)=t^2\) for \(t\in [0,1]\) gives the result. \(\square \)

At this point, let us mention the following asymptotics, which will be useful several times: if \(\sigma =O(d^{-1})\) and \(x\in \mathbb {R}\) then we have
$$\begin{aligned}&\left( e^{-\beta /\theta }\left( 1+\tfrac{\beta }{\theta }-\sigma x \tfrac{\beta }{\theta }\right) \right) ^d=1- \tfrac{1}{d}(1/2+x\sigma d)\nonumber \\&\quad +\tfrac{1}{d^2}\left( 1/3-\alpha +x\sigma d-\alpha x \sigma d +\tfrac{1}{2}(1/2+x\sigma d)^2\right) +O(d^{-3}). \end{aligned}$$
(2.11)
To compute \(\mathbb {P}(A_2)\) we may remove the enforcement of \(B^\rho _m\) in (2.9) and (2.10) by setting \(\zeta _{m-1}\) and \(\zeta _{m-2}\) to 1 and summing the results together, giving
$$\begin{aligned} \begin{aligned} \mathbb {P}^{(\theta )}_m({A_2})&= z_m z_{m-1} \tfrac{d\beta ^2e^{-\beta /\theta }}{2\theta } \left( e^{-\beta /\theta }\left( 1+\tfrac{\beta }{\theta }\right) \right) ^{2d-1} \left( 2u(1-u)+\theta (u^2+(1-u)^2)\right) \\&= z_mz_{m-1} \tfrac{\theta }{2d} \big [1-\tfrac{1}{d}\big ] \big (2u(1-u)+\theta (u^2+(1-u)^2) + O(d^{-2})\big ). \end{aligned} \end{aligned}$$
(2.12)
For the last step we used (2.11) to first order, and that
$$\begin{aligned} \tfrac{d\beta ^2e^{-\beta /\theta }}{2\theta }=\tfrac{\theta }{2d}+O(d^{-2}). \end{aligned}$$
(2.13)

2.2 Stochastic Domination

In some estimates we will want to approximate the complicated measure \(\mathbb {P}_m^{(\theta )}(\cdot )\), which involves counting loops, by some simpler measure. For this we use stochastic domination. Let us define \(\beta ^+=(\beta \theta )\vee (\beta /\theta )\). Also let us define \(\mathbb {E}^+_m\) in the same way as \(\mathbb {E}_m\) but with \(\beta \) replaced by \(\beta ^+\); thus the links form independent Poisson processes with rate \(\beta ^+\). We say that an event A is increasing if it cannot be destroyed by adding more links; examples of increasing events include \(A_1^c\) and \((A_1\cup A_2)^c\) where \(A_1\) and \(A_2\) are as defined above. Stochastic domination tells us that
$$\begin{aligned} A \text{ increasing }\quad \Rightarrow \quad \mathbb {P}_m^{(\theta )}(A)\le \mathbb {P}_m^+(A). \end{aligned}$$
(2.14)

Proof of (2.14)

We apply [8, Thm. 1.1]. Note that \(\mathbb {P}_m^{(\theta )}\ll \mathbb {P}^+_m\) and the density \(f(\omega )=\tfrac{d \mathbb {P}_m^{(\theta )}}{d\mathbb {P}^+_m} \propto \theta ^{\ell (\omega )}(\tfrac{\beta }{\beta ^+})^{|\omega |}\) where \(|\omega |\) denotes the number of links. Let \(\omega '\) be obtained from \(\omega \) by adding a single link. This link either splits a loop, merges two loops, or does not change the number of loops, hence
$$\begin{aligned} \tfrac{f(\omega ')}{f(\omega )}=\theta ^{\ell (\omega ')-\ell (\omega )}\tfrac{\beta }{\beta ^+} \in \left\{ \tfrac{\beta \theta }{\beta ^+},\tfrac{\beta /\theta }{\beta ^+},\tfrac{\beta }{\beta ^+}\right\} . \end{aligned}$$
The result follows since all three possible values are \(\le 1\). \(\square \)
An immediate consequence of (2.14) is that there is some constant \(c>0\) such that
$$\begin{aligned} \mathbb {P}_m^{(\theta )}(A_1^c\cap A_2^c)\le c/d^2 \quad \text{ for } \text{ all } m, d\ge 1. \end{aligned}$$
(2.15)
We now deduce some information about the asymptotic behaviour of the numbers \(z_m=e^{-d\beta (1-1/\theta )}\theta Z_{m-1}^d/Z_m\). We write
$$\begin{aligned} q= & {} q(\theta ,u)=\tfrac{\theta }{2}\left( 2u(1-u)+\theta \left( u^2+(1-u)^2\right) \right) \end{aligned}$$
(2.16)
$$\begin{aligned} r= & {} r(\theta ,u)=2\theta u(1-u)+\tfrac{1}{2}\theta ^2\left( u^2+\tfrac{4}{3}(1-u)^2\right) \end{aligned}$$
(2.17)
so that \(\alpha _*=1+q-r\).

Proposition 2.3

There is a constant C and there are functions \({\varepsilon }^{(j)}_m(d)\), \(j\in \{1,2,3\}\), satisfying
$$\begin{aligned} |{\varepsilon }^{(j)}_m(d)|\le C/d^2 \hbox { for all}\ m,d\ge 1, \end{aligned}$$
(2.18)
such that
$$\begin{aligned} z_m=1-\tfrac{1}{d}(q-1/2)+{\varepsilon }^{(1)}_m(d) \end{aligned}$$
(2.19)
and
$$\begin{aligned} z_m \big (1+\tfrac{1}{d}(q-1/2)+{\varepsilon }^{(2)}_m(d)\big ) =1-{\varepsilon }^{(3)}_m(d). \end{aligned}$$
(2.20)

Proof

Note that (2.19) and (2.20) are equivalent, hence one may proceed by induction on m, proving (2.20) with the induction hypothesis provided by (2.19). For the base case \(m=1\) one may establish (2.19) directly, splitting into the cases \(A_1\), \(A_2\) and \((A_1\cup A_2)^c\) to get
$$\begin{aligned} Z_1=\theta ^{d+1} e^{-d\beta (1-1/\theta )}\Big [ \big (e^{-\beta /\theta }\big (1+\tfrac{\beta }{\theta }\big )\big )^d+ q(\theta ,u) \tfrac{d\beta ^2e^{-\beta /\theta }}{\theta ^2} \big (e^{-\beta /\theta }\big (1+\tfrac{\beta }{\theta }\big )\big )^{d-1} +{\varepsilon }_1(d) \Big ], \end{aligned}$$
where \(0\le {\varepsilon }_1(d)\le e^{d\beta (1-1/\theta )}\mathbb {P}_1(A_1^c\cap A_2^c)\) satisfies (2.18).
For \(m>1\), write \({\varepsilon }^{(3)}_m(d)=\mathbb {P}^{(\theta )}_m(A_1^c\cap A_2^c)\), this satisfies (2.18) by (2.15). From the expressions (2.6) and (2.12) we have
$$\begin{aligned} \begin{aligned} 1&= \mathbb {P}^{(\theta )}_m(A_1)+\mathbb {P}^{(\theta )}_m(A_2)+ \mathbb {P}^{(\theta )}_m(A_1^c\cap A_2^c)\\&= z_m\big (e^{-\beta /\theta }\big (1+\tfrac{\beta }{\theta }\big )\big )^d+ z_m z_{m-1} \tfrac{d\beta ^2e^{-\beta /\theta }}{\theta ^2} \big (e^{-\beta /\theta }\big (1+\tfrac{\beta }{\theta }\big )\big )^{2d-1} q(\theta ,u) +{\varepsilon }_m^{(3)}(d) \end{aligned} \end{aligned}$$
Hence, using the asymptotics (2.11) and (2.13),
$$\begin{aligned} 1-{\varepsilon }^{(3)}_m(d)= z_m \Big [ 1-\tfrac{1}{2d}+ z_{m-1} \tfrac{1}{d}\big (1-\tfrac{1}{d}\big ) q +{\varepsilon }^{(4)}(d) \Big ] \end{aligned}$$
for a function \({\varepsilon }^{(4)}(d)\) not depending on m but otherwise satisfying the bounds (2.18). Using the induction hypothesis we get
$$\begin{aligned} 1-{\varepsilon }^{(3)}_m(d)= z_m \Big [ 1+\tfrac{1}{d}\big (q-\tfrac{1}{2}\big ) +{\varepsilon }_m^{(2)}(d) \Big ], \end{aligned}$$
where
$$\begin{aligned} {\varepsilon }^{(2)}_m(d) = {\varepsilon }^{(4)}(d) - \tfrac{q}{d^2} \big ( q + \tfrac{1}{2} \bigr ) + \tfrac{q}{d^3} \big ( q - \tfrac{1}{2} \bigr ) + \tfrac{q}{d} \bigl ( 1 - \tfrac{1}{d} \bigr ) {\varepsilon }^{(1)}_{m-1}(d) \end{aligned}$$
is easily seen to satisfy (2.18). \(\square \)

Remark 2.4

From the proposition it follows that
$$\begin{aligned} z_mz_{m-1} = 1+O(d^{-1}) \end{aligned}$$
(2.21)
where the \(O(\cdot )\) is uniform in m.

We now turn to the details of the proof of Proposition 2.1.

2.3 Proof of the Lower Bound (2.2)

We have from the definition \(\sigma _m=1-\mathbb {P}^{(\theta )}_m(B_m^\rho )\) that
$$\begin{aligned} \sigma _m\ge \mathbb {P}^{(\theta )}_m(A_1) -\mathbb {P}^{(\theta )}_m(B_m^\rho \cap A_1) +\mathbb {P}^{(\theta )}_m(A_2) -\mathbb {P}^{(\theta )}_m(B_m^\rho \cap A_2), \end{aligned}$$
(2.22)
where we have simply bounded the remaining difference involving the event \((A_1\cup A_2)^c\) from below by 0. Consider first the terms involving \(A_1\). From (2.6) and (2.7), bounding \(\sigma _{m-1}\ge {\tilde{\sigma }}_{m-1}\), and using the asymptotics (2.11) as well as the estimates Proposition 2.3 on \(z_m\) we get
$$\begin{aligned} \begin{aligned}&\mathbb {P}^{(\theta )}_m(A_1)-\mathbb {P}^{(\theta )}_m(B_m^\rho \cap A_1)\ge z_m \big ({\tilde{\sigma }}_{m-1}-\tfrac{{\tilde{\sigma }}_{m-1}}{d}(3/2-\alpha )- \tfrac{1}{2}{\tilde{\sigma }}_{m-1}^2+ O(d^{-3})\big )\\&\qquad =\big (1-\tfrac{1}{d} (q-1/2)\big ) \big ({\tilde{\sigma }}_{m-1}-\tfrac{{\tilde{\sigma }}_{m-1}}{d}(3/2-\alpha )- \tfrac{1}{2}{\tilde{\sigma }}_{m-1}^2\big ) + O(d^{-3})\\&\qquad = {\tilde{\sigma }}_{m-1}+\tfrac{{\tilde{\sigma }}_{m-1}}{d}(\alpha -q-1)- \tfrac{1}{2}{\tilde{\sigma }}_{m-1}^2+ O(d^{-3}). \end{aligned} \end{aligned}$$
(2.23)
Now consider the terms involving \(A_2\). Using that \(\zeta _{m-1},\zeta _{m-2}\le 1-{\tilde{\sigma }}_{m-1}\), as well as the asymptotics (2.11) to order \(d^{-1}\), we deduce from (2.9) that
$$\begin{aligned} \mathbb {P}^{(\theta )}_m({B_m^\rho } \cap {A_2^\mathrm {mix}})\le & {} z_m z_{m-1} \tfrac{d\beta ^2 e^{-\beta /\theta }}{2\theta } 2u(1-u) \big (e^{-\beta /\theta }\big (1+\tfrac{\beta }{\theta }-{\tilde{\sigma }}_{m-1}\tfrac{\beta }{\theta }\big ) \big )^{2d-1}\nonumber \\= & {} z_m z_{m-1} \frac{\theta }{2d} \Big [\big (1-\tfrac{1}{d}\big )2u(1-u)- {\tilde{\sigma }}_{m-1}4u(1-u)+ O(d^{-2}) \Big ]\nonumber \\ \end{aligned}$$
(2.24)
and from (2.10) that
$$\begin{aligned} \begin{aligned} \mathbb {P}^{(\theta )}_m({B_m^\rho } \cap {A_2^\mathrm {same}})&\le z_m z_{m-1} \tfrac{d\beta ^2 e^{-\beta /\theta }}{2} (u^2+(1-u)^2) \\&\qquad \mathbb {E}\Big [ \big (e^{-\beta /\theta }\big (1{+}\tfrac{\beta }{\theta }{-} {\tilde{\sigma }}_{m-1}|\Lambda _\rho | \tfrac{\beta }{\theta }\big )\big )^{d-1} \big (e^{-\beta /\theta }\big (1{+}\tfrac{\beta }{\theta }{-}{\tilde{\sigma }}_{m-1}|\Lambda _x|\tfrac{\beta }{\theta }\big )\big )^{d} \Big ]\\&= z_m z_{m-1}\frac{\theta ^2}{2d} (u^2{+}(1{-}u)^2) \Big [\big (1{-}\tfrac{1}{d}\big ) {-} {\tilde{\sigma }}_{m{-}1} \mathbb {E}\big (|\Lambda _\rho |{+}|\Lambda _x|\big )+O(d^{-2}) \Big ]\\&= z_m z_{m{-}1} \frac{\theta ^2}{2d} \Big [\big (1{-}\tfrac{1}{d}\big )(u^2{+}(1-u)^2){-} {\tilde{\sigma }}_{m{-}1}\big (u^2{+}\tfrac{4}{3}(1-u)^2\big ){+} O(d^{-2}) \Big ]. \end{aligned} \end{aligned}$$
(2.25)
Here we used the properties of \(|\Lambda _\rho |\) and \(|\Lambda _x|\) stated below (2.10) (see also Fig. 3). Using also (2.12) and (2.21) we get
$$\begin{aligned}\begin{aligned}&\mathbb {P}^{(\theta )}_m(A_2)-\mathbb {P}^{(\theta )}_m(B_m^\rho \cap A_2)\\&\quad \ge z_m z_{m-1} \frac{\theta }{2d} \left\{ 2u(1-u)\left( \left[ 1-\tfrac{1}{d}\right] -\left[ 1-\tfrac{1}{d}-2{\tilde{\sigma }}_{m-1}\right] \right) \right. \\&\left. \qquad +\theta (u^2+(1-u)^2) \left( \left[ 1-\tfrac{1}{d}\right] - \left[ 1-\tfrac{1}{d}-{\tilde{\sigma }}_{m-1}\frac{u^2+\tfrac{4}{3}(1-u)^2}{u^2+(1-u)^2}\right] \right) +O(d^{-2}) \right\} \\&\quad =r(\theta ,u) \tfrac{{\tilde{\sigma }}_{m-1}}{d} + O(d^{-3}), \end{aligned}\end{aligned}$$
where r is defined in (2.17). Putting this together in (2.22) gives
$$\begin{aligned} \sigma _m\ge {\tilde{\sigma }}_{m-1}+\tfrac{{\tilde{\sigma }}_{m-1}}{d} (\alpha -[1+q-r])- \tfrac{1}{2}{\tilde{\sigma }}_{m-1}^2+ O(d^{-3}). \end{aligned}$$
Since \(\alpha _*=1+q-r\) this gives (2.2). \(\square \)

2.4 Proof of the Upper Bound (2.3)

Write \(\Sigma _m^\rho \) for the complement of \(B_m^\rho \), so that \(\sigma _m=\mathbb {P}^{(\theta )}_m(\Sigma _m^\rho )\). Clearly
$$\begin{aligned} \sigma _m=\mathbb {P}^{(\theta )}_m(A_1\cap \Sigma _m^\rho )+ \mathbb {P}^{(\theta )}_m(A_2\cap \Sigma _m^\rho )+ \mathbb {P}^{(\theta )}_m(A_1^c\cap A_2^c \cap \Sigma _m^\rho ). \end{aligned}$$
The following will be proved at the end of this section:

Proposition 2.5

For all d large enough there is a constant C such that
$$\begin{aligned} \mathbb {P}^{(\theta )}_m(A_1^c\cap A_2^c \cap \Sigma _m^\rho ) \le \frac{C}{d^2} (\sigma _{m-1}\vee \sigma _{m-2}). \end{aligned}$$
Before proving this we show how to deduce (2.3). We have by taking the difference of the expressions (2.6) and (2.7) that
$$\begin{aligned} \begin{aligned} \mathbb {P}^{(\theta )}_m(A_1\cap \Sigma _m^\rho )&= z_m \Big \{\big (e^{-\beta /\theta }\big (1+\tfrac{\beta }{\theta }\big )\big )^d- \big (e^{-\beta /\theta }\big (1+\tfrac{\beta }{\theta }-\tfrac{\beta }{\theta }\sigma _{m-1}\big )\big )^d \Big \}\\&=z_m \big (e^{-\beta /\theta }\big (1+\tfrac{\beta }{\theta }\big )\big )^d \Big \{1- \Big (1-\frac{\tfrac{\beta }{\theta }\sigma _{m-1}}{1+\beta /\theta }\Big )^d \Big \}\\&\le z_m \big (e^{-\beta /\theta }\big (1+\tfrac{\beta }{\theta }\big )\big )^d \tfrac{d\beta }{\theta }\big (1+\tfrac{\beta }{\theta }\big )^{-1}\sigma _{m-1}. \end{aligned} \end{aligned}$$
(2.26)
In the last step we used the concavity of the function \(f(x)=1-(1-x)^d\) to bound \(f(x)\le x f'(0)\).
Similarly using (2.9) and concavity of \(f(x,y)=1-(1-x)^{d-1}(1-y)^d\) (for \(d\ge 3\)),
$$\begin{aligned} \begin{aligned} \mathbb {P}^{(\theta )}_m({\Sigma _m^\rho } \cap {A_2^\mathrm {mix}})&= z_m z_{m-1} \tfrac{d\beta ^2e^{-\beta /\theta }}{2} \tfrac{2}{\theta }u(1-u) \big (e^{-\beta /\theta }\big (1+\tfrac{\beta }{\theta }\big )\big )^{2d-1} \\&\quad \times \left\{ 1-\left( 1-\frac{\tfrac{\beta }{\theta }\sigma _{m-1}}{1+\tfrac{\beta }{\theta }}\right) ^{d-1} \left( 1-\frac{\tfrac{\beta }{\theta }\sigma _{m-2}}{1+\tfrac{\beta }{\theta }}\right) ^{d} \right\} \\&\le z_m z_{m-1} \tfrac{d\beta ^2e^{-\beta /\theta }}{2} \tfrac{2}{\theta }u(1-u) \big (e^{-\beta /\theta }\big (1+\tfrac{\beta }{\theta }\big )\big )^{2d-1}\\&\quad \times \tfrac{\beta }{\theta }\big (1+\tfrac{\beta }{\theta }\big )^{-1} \{(d-1)\sigma _{m-1}+d\sigma _{m-2}\}\\&\le z_m z_{m-1} \tfrac{d\beta ^2e^{-\beta /\theta }}{2} \big (e^{-\beta /\theta }\big (1+\tfrac{\beta }{\theta }\big )\big )^{2d-1}\\&\quad \times \tfrac{d\beta }{\theta }\big (1+\tfrac{\beta }{\theta }\big )^{-1} \tfrac{2}{\theta }u(1-u) (\sigma _{m-1}+\sigma _{m-2}). \end{aligned} \end{aligned}$$
(2.27)
The same argument applied to (2.10) gives (with the notation \(\mathbb {E}\) used there and using Lemma 2.2)
$$\begin{aligned} \begin{aligned} \mathbb {P}^{(\theta )}_m\big ({\Sigma _m^\rho } \cap {A_2^\mathrm {same}}\big )&= z_m z_{m-1} \tfrac{d\beta ^2e^{-\beta /\theta }}{2} (u^2+(1-u)^2) \big (e^{-\beta /\theta }\big (1+\tfrac{\beta }{\theta }\big )\big )^{2d-1}\\&\qquad \cdot \mathbb {E}\left[ 1-\left( 1-\frac{\tfrac{\beta }{\theta }\sigma _{m-1}|\Lambda _\rho |}{1+\tfrac{\beta }{\theta }}\right) ^{d-1} \left( 1-\frac{\tfrac{\beta }{\theta }\sigma _{m-2}|\Lambda _x|}{1+\tfrac{\beta }{\theta }}\right) ^{d} \right] \\&\le z_m z_{m-1} \tfrac{d\beta ^2e^{-\beta /\theta }}{2} (u^2+(1-u)^2) \big (e^{-\beta /\theta }\big (1+\tfrac{\beta }{\theta }\big )\big )^{2d-1} \tfrac{\beta }{\theta }\big (1+\tfrac{\beta }{\theta }\big )^{-1}\\&\qquad \cdot \{(d-1)\sigma _{m-1}\mathbb {E}|\Lambda _\rho | +d\sigma _{m-2}\mathbb {E}|\Lambda _x|\}\\&\le z_m z_{m-1} \tfrac{d\beta ^2e^{-\beta /\theta }}{2} \big (e^{-\beta /\theta }\big (1+\tfrac{\beta }{\theta }\big )\big )^{2d-1} \tfrac{d\beta }{\theta }\big (1+\tfrac{\beta }{\theta }\big )^{-1}\\&\qquad \cdot \big \{\sigma _{m-1}\tfrac{2}{3} (u^2+(1-u)^2) +\sigma _{m-2}\left( \tfrac{1}{3} u^2+\tfrac{2}{3} (1-u)^2\right) \big \} \end{aligned} \end{aligned}$$
(2.28)
Using Proposition 2.3 to estimate \(z_m\), the asymptotics (2.11), as well as \(\tfrac{d\beta }{\theta }\big (1+\tfrac{\beta }{\theta }\big )^{-1}=1+\tfrac{\alpha -1}{d}+O(d^{-2})\) we see that the right-hand side of (2.26) satisfies
$$\begin{aligned} z_m \big (e^{-\beta /\theta }\big (1+\tfrac{\beta }{\theta }\big )\big )^d \tfrac{d\beta }{\theta }\big (1+\tfrac{\beta }{\theta }\big )^{-1}\sigma _{m-1} =\Big (1+\frac{\alpha -(1+q)}{d}+O(d^{-2})\Big )\sigma _{m-1}, \end{aligned}$$
where \(q=q(\theta ,u)\) was defined in (2.16). Similarly, using (2.13) and (2.21), in the right-hand-sides of (2.27) and (2.28) we have the factors
$$\begin{aligned} z_m z_{m-1} \tfrac{d\beta ^2e^{-\beta /\theta }}{2} \big (e^{-\beta /\theta }\big (1+\tfrac{\beta }{\theta }\big )\big )^{2d-1} \tfrac{d\beta }{\theta }\big (1+\tfrac{\beta }{\theta }\big )^{-1} =\tfrac{\theta ^2}{2d}(1+O(d^{-1})). \end{aligned}$$
Hence, bounding also \(\sigma _{m-1}\) and \(\sigma _{m-2}\) by their maximum, we have that
$$\begin{aligned} \begin{aligned} \sigma _m&\le (\sigma _{m-1}\vee \sigma _{m-2})\Big [1+\frac{\alpha -(1+q)}{d} + \frac{\theta ^2}{2d}\Big ( \tfrac{4}{\theta }u(1-u)+\tfrac{2}{3} (u^2+(1-u)^2) +\tfrac{1}{3} u^2\\&\quad +\tfrac{2}{3} (1-u)^2\Big )+O(d^{-2})\Big ] + \mathbb {P}^{(\theta )}_m(A_1^c\cap A_2^c \cap \Sigma _m^\rho )\\&= (\sigma _{m-1}\vee \sigma _{m-2})\Big [1+\frac{\alpha -(1+q-r)}{d}+O(d^{-2}) \Big ]+\mathbb {P}^{(\theta )}_m(A_1^c\cap A_2^c \cap \Sigma _m^\rho ), \end{aligned} \end{aligned}$$
where \(r=r(\theta ,u)\) was defined in (2.17). In the above, all \(O(d^{-2})\) terms are uniform in m. Since \(1+q-r=\alpha _*\) we see that (2.3) follows once we prove Proposition 2.5.

In the following argument we will examine the subtree \({\check{T}}\) of \(T_m\) which contains the root and is spanned by edges supporting at least two links. In \({\check{T}}\), the loop-structure is very complicated and we will not attempt to keep track of it. Instead we use that \({\check{T}}\) is likely to be small, and that a loop exiting it must do so across an edge supporting exactly one link, which is a simpler situation to analyze. Roughly speaking, the enforcement of the event \(A_1^c\cap A_2^c\) will give rise to the factor \(d^{-2}\), and the requirement that the loop exits \({\check{T}}\) will give a factor \(\sigma _{m-k}\) for some \(k\ge 1\), which can then be bounded in terms of \(\sigma _{m-1}\vee \sigma _{m-2}\). The details are quite technical.

Proof of Proposition 2.5

We begin by defining \({\check{T}}\) carefully: we let \({\check{T}}\) be the (random) subtree of \(T_m\) containing
  1. (1)

    The root \(\rho \)

     
  2. (2)

    Any vertex in generation 1 with \(\ge 2\) links to \(\rho \),

     
  3. (3)

    In general, any vertex in generation k with \(\ge 2\) links to some vertex of \({\check{T}}\) in generation \(k-1\).

     
Note that \(A_1^c\cap A_2^c\) is precisely the event that \({\check{T}}\) has at least two edges. Let \(V_k({\check{T}})\) denote the set of vertices in \({\check{T}}\) in generation k. For x a vertex of \({\check{T}}\), \(x\not \in V_m({\check{T}})\), let \(d_x\) denote its number of descendants not in \({\check{T}}\). Thus x has \(d_x\) outgoing edges carrying only 0 or 1 links of \(\omega \). For \(0\le k\le m-1\) we let \(\mathcal {E}_k\) denote the set of outgoing edges from generation k (to generation \(k+1\)) which carry precisely 1 link.
Note that if the loop of \((\rho ,0)\) reaches generation m then either it reaches generation m within \({\check{T}}\), or it passes some link of \(\cup _{k=0}^{m-1} \mathcal {E}_k\). Let us by convention set \(\sigma _{-1}=1\) and \(|\mathcal {E}_{m}|=|V_m({\check{T}})|\). We claim that
$$\begin{aligned} \mathbb {P}^{(\theta )}_m(A_1^c\cap A_2^c \cap \Sigma _m^\rho ) \le \sum _{k=0}^{m} \sigma _{m-k-1} \mathbb {E}_m^{(\theta )}[|\mathcal {E}_k| \mathbb {1}_{A_1^c\cap A_2^c}]. \end{aligned}$$
(2.29)
Intuitively, this is because if the loop exits \({\check{T}}\) through some edge in \(\mathcal {E}_k\), then it has distance \(m-k-1\) left to go to reach the \(m{\text {th}}\) generation of \(T_m\). A detailed justification of (2.29) requires dealing with the dependencies caused by the factor \(\theta ^\ell \).
To do this, let us introduce the following notation. First, let \({\check{\omega }}\) denote the restriction of \(\omega \) to \({\check{T}}\). Next, let \(\partial ^+{\check{T}}\) denote the set of vertices \(y\in T_m\setminus {\check{T}}\) whose parent belongs to \({\check{T}}\), and write \(\omega _y\) for the restriction of \(\omega \) to the subtree rooted at y. For simplicity, in the rest of this proof we simply write \(\mathbb {E}\) for \(\mathbb {E}_m\). We will make use of the fact that, given \({\check{\omega }}\), the random collections \((\mathcal {E}_j)_{j=0}^{m-1}\) and \((\omega _y)_{y\in \partial ^+{\check{T}}}\) are conditionally independent under \(\mathbb {E}\). This implies that for three functions
$$\begin{aligned} F_1({\check{\omega }}),\quad F_2({\check{\omega }},(\mathcal {E}_j)_{j=0}^{m-1}), \quad F_3({\check{\omega }},(\omega _y)_{y\in \partial ^+{\check{T}}}) \end{aligned}$$
(2.30)
we have
$$\begin{aligned}&\mathbb {E}\big [F_1({\check{\omega }}) F_2({\check{\omega }},(\mathcal {E}_j)_{j=0}^{m-1}) F_3({\check{\omega }},(\omega _y)_{y\in \partial ^+{\check{T}}})\big ]\nonumber \\&\quad =\mathbb {E}\big [F_1({\check{\omega }}) \mathbb {E}[F_2({\check{\omega }},(\mathcal {E}_j)_{j=0}^{m-1})\mid {\check{\omega }}] \mathbb {E}[F_3({\check{\omega }},(\omega _y)_{y\in \partial ^+{\check{T}}})\mid {\check{\omega }}]\big ]. \end{aligned}$$
(2.31)
Note that we have the decomposition (similar to (2.5))
$$\begin{aligned} \ell ={\check{\ell }}+\sum _{j=0}^{m-1}\left[ \sum _{x\in V_j({\check{T}})} \sum _{i=1}^{d_x} \ell _i^{(x)} -|\mathcal {E}_j| \right] , \end{aligned}$$
where \({\check{\ell }}\) denotes the number of loops in the configuration \({\check{\omega }}\), and \(\ell _i^{(x)}\) denotes the number of loops in the subtree rooted at the \(i{\text {th}}\) descendant of x not belonging to \({\check{T}}\) (in some numbering of these descendants). Hence
$$\begin{aligned} \theta ^\ell =\theta ^{{\check{\ell }}} \left( \prod _{j=0}^{m-1} \theta ^{-|\mathcal {E}_j|}\right) \left( \prod _{j=0}^{m-1} \prod _{x\in V_j({\check{T}})} \prod _{i=1}^{d_x} \theta ^{\ell _i^{(x)}}\right) \end{aligned}$$
(2.32)
is a factorization into three functions as in (2.30). Turning to (2.29), by considering the possibilities that either \({\check{T}}\) reaches generation m (meaning \(V_m({\check{T}})\ne \varnothing \)) or that loop of \((\rho ,0)\) passes some edge \(e\in \cup _{k=0}^{m-1}\mathcal {E}_k\), we have
$$\begin{aligned}&\mathbb {P}^{(\theta )}_m(A_1^c\cap A_2^c \cap \Sigma _m^\rho )\le \mathbb {E}^{(\theta )}_m[|V_m({\check{T}})|\mathbb {1}_{A_1^c\cap A_2^c}]\\&\nonumber \quad + \sum _{e} \sum _{k=0}^{m-1} \mathbb {P}^{(\theta )}_m(A_1^c\cap A_2^c\cap \{e\in \mathcal {E}_k\} \cap \{(e^+,t^+)\leftrightarrow m\}], \end{aligned}$$
(2.33)
where the first sum is over all edges e of \(T_m\), and \(\{(e^+,t^+)\leftrightarrow m\}\) denotes the event that the further (from \(\rho \)) endpoint \((e^+,t^+)\) of the unique link at e lies in a loop of \(\omega _{e^+}\) reaching the \(m{\text {th}}\) generation of \(T_m\). Applying (2.31) and (2.32) we have
$$\begin{aligned}&\mathbb {P}^{(\theta )}_m(A_1^c\cap A_2^c\cap \{e\in \mathcal {E}_k\} \cap \{(e^+,t^+)\leftrightarrow m\}]\\&\quad = \frac{1}{Z_m} \mathbb {E}\left[ \mathbb {1}_{A_1^c\cap A_2^c} \theta ^{{\check{\ell }}} \mathbb {E}\left[ \mathbb {1}\{e\in \mathcal {E}_k\}\prod _{j=0}^{m-1} \theta ^{-|\mathcal {E}_j|}\mid {\check{T}}\right] \sigma _{m-k-1} \prod _{j=0}^{m-1} \prod _{x\in V_j({\check{T}})} Z_{m-j-1}^{d_x} \right] . \end{aligned}$$
Taking out the factor \(\sigma _{m-k-1}\), applying (2.31) again in reverse, and putting back into (2.33), we obtain (2.29).
We proceed by bounding the expectations
$$\begin{aligned} \mathbb {E}_m^{(\theta )}[|\mathcal {E}_k| \mathbb {1}_{A_1^c\cap A_2^c}]= \frac{1}{Z_m} \mathbb {E}[\theta ^\ell |\mathcal {E}_k| \mathbb {1}_{A_1^c\cap A_2^c}]. \end{aligned}$$
Arguing as above we get:
$$\begin{aligned} \mathbb {E}[\theta ^\ell |\mathcal {E}_k| \mathbb {1}_{A_1^c\cap A_2^c}]= \mathbb {E}\left[ \mathbb {1}_{A_1^c\cap A_2^c} \theta ^{{\check{\ell }}} \prod _{j=0}^{m-1} \prod _{x\in V_j({\check{T}})} Z_{m-j-1}^{d_x} \mathbb {E}\left[ |\mathcal {E}_k|\prod _{j=0}^{m-1} \theta ^{-|\mathcal {E}_j|}\mid {\check{T}}\right] \right] . \end{aligned}$$
The \(|\mathcal {E}_j|\) are conditionally independent given \({\check{T}}\), hence
$$\begin{aligned} \mathbb {E}\left[ |\mathcal {E}_k|\prod _{j=0}^{m-1} \theta ^{-|\mathcal {E}_j|}\mid {\check{T}}\right] =\mathbb {E}\left[ |\mathcal {E}_k| \theta ^{-|\mathcal {E}_k|}\mid {\check{T}}\right] \prod _{j\ne k} \mathbb {E}\left[ \theta ^{-|\mathcal {E}_j|}\mid {\check{T}}\right] . \end{aligned}$$
Let \(p_i=e^{-\beta }\beta ^i/i!\) denote the probabilities of a Poisson(\(\beta \)) random variable. Direct computation gives
$$\begin{aligned} \mathbb {E}\big [\theta ^{-|\mathcal {E}_k|}\mid {\check{T}}\big ] = \prod _{x\in V_k({\check{T}})} \left( \frac{p_0+p_1/\theta }{p_0+p_1}\right) ^{d_x} \end{aligned}$$
and (e.g. by differentiating the previous expression)
$$\begin{aligned} \mathbb {E}\big [|\mathcal {E}_k| \theta ^{-|\mathcal {E}_k|}\mid {\check{T}}\big ] = \frac{p_1/\theta }{p_0+p_1/\theta } \left( \sum _{x\in V_k({\check{T}})}d_x\right) \prod _{x\in V_k({\check{T}})} \left( \frac{p_0+p_1/\theta }{p_0+p_1}\right) ^{d-d_x}. \end{aligned}$$
Hence
$$\begin{aligned} \frac{\mathbb {E}\big [|\mathcal {E}_k| \theta ^{-|\mathcal {E}_k|}\mid {\check{T}}\big ]}{\mathbb {E}\big [\theta ^{-|\mathcal {E}_k|}\mid {\check{T}}\big ]}\le \frac{d p_1/\theta }{p_0+p_1/\theta } |V_k({\check{T}})|, \end{aligned}$$
and
$$\begin{aligned}&\mathbb {E}[\theta ^\ell |\mathcal {E}_k| \mathbb {1}_{A_1^c\cap A_2^c}]\\&\quad {\le } \frac{d p_1/\theta }{p_0{+}p_1/\theta } \mathbb {E}\left[ |V_k({\check{T}})|\mathbb {1}_{A_1^c\cap A_2^c} \theta ^{{\check{\ell }}} \prod _{j=0}^{m{-}1} \prod _{x\in V_j({\check{T}})} Z_{m-j-1}^{d_x} \mathbb {E}\left[ \prod _{j{=}0}^{m{-}1} \theta ^{-|\mathcal {E}_j|}\mid {\check{T}}\right] \right] . \end{aligned}$$
Applying (2.31) in reverse it follows that
$$\begin{aligned} \mathbb {E}_m^{(\theta )}[|\mathcal {E}_k| \mathbb {1}_{A_1^c\cap A_2^c}]\le \frac{d p_1/\theta }{p_0+p_1/\theta } \mathbb {E}^{(\theta )}_m\big [|V_k({\check{T}})|\mathbb {1}_{A_1^c\cap A_2^c} \big ]. \end{aligned}$$
We bound the last expectation using stochastic domination. Indeed, both \(|V_k({\check{T}})|\) and \(\mathbb {1}_{A_1^c\cap A_2^c}\) are increasing functions of \(\omega \). Hence from (2.14)
$$\begin{aligned} \mathbb {E}^{(\theta )}_m\big [|V_k({\check{T}})|\mathbb {1}_{A_1^c\cap A_2^c} \big ]\le \mathbb {E}^+\big [|V_k({\check{T}})|\mathbb {1}_{A_1^c\cap A_2^c} \big ]. \end{aligned}$$
Write \(p_i^+=e^{-\beta ^+}(\beta ^+)^i/i!\) for the Poisson probabilities with parameter \(\beta ^+\), and \(p^+_{\ge i}=p^+_i+p^+_{i+1}+\cdots \). By a recursive computation using independence we see that
$$\begin{aligned} \mathbb {E}^+|V_k({\check{T}})|=(d p_{\ge 2}^+)\mathbb {E}^+|V_{k-1}({\check{T}})|= \cdots =(d p_{\ge 2}^+)^k. \end{aligned}$$
We also have
$$\begin{aligned} |V_k({\check{T}})|\mathbb {1}_{A_1}=\delta _{k,0}\mathbb {1}_{A_1},\qquad |V_k({\check{T}})|\mathbb {1}_{A_2}=(\delta _{k,0}+\delta _{k,1})\mathbb {1}_{A_2}. \end{aligned}$$
Using (2.29) we find that
$$\begin{aligned} \begin{aligned} \mathbb {P}^{(\theta )}_m(A_1^c\cap A_2^c \cap \Sigma _m^\rho )&\le \frac{d p_1/\theta }{p_0+p_1/\theta } \sum _{k=0}^{m} \sigma _{m-k-1} \mathbb {E}^+[|V_k({\check{T}})| \mathbb {1}_{A_1^c\cap A_2^c}]\\&= \frac{d p_1/\theta }{p_0+p_1/\theta } \Big ( \sigma _{m-1}(1-\mathbb {P}^+(A_1\cup A_2))+ \sigma _{m-2} (dp_{\ge 2}^+ -\mathbb {P}^+(A_2))\\&\qquad \qquad \qquad +\sum _{k=2}^{m} \sigma _{m-k-1} (dp_{\ge 2}^+)^k\Big )\\&\le c_0\left( \sigma _{m-1}\frac{c_1}{d^2}+ \sigma _{m-2}\frac{c_2}{d^2}+ \sum _{k=2}^{m} \sigma _{m-k-1}\left( \frac{c_3}{d}\right) ^{k} \right) , \end{aligned} \end{aligned}$$
for constants \(c_0,\cdots ,c_3\) uniform in d. Now we use that there is some \(c_4>0\), uniform in d, such that \(\sigma _{m-1}\le c_4 \sigma _m\) for all \(m\ge 0\). (This can be seen e.g. by considering the event that \(A_1\) occurs and that \((x,t_x)\) lies in a loop reaching generation m in its subtree, where x is some fixed child of \(\rho \) and \(t_x\) is the ‘time’ of the incoming link from \(\rho \). This gives \(\sigma _m\ge \sigma _{m-1} \mathbb {P}_m^{(\theta )}(A_1)\).) It follows that
$$\begin{aligned} \mathbb {P}^{(\theta )}_m(A_1^c\cap A_2^c \cap \Sigma _m^\rho )\le \frac{C'}{d^2}\left( \sigma _{m-1}+\sigma _{m-2}+ \sigma _{m-2}\sum _{k=2}^{\infty } \left( \frac{c_3c_4}{d}\right) ^{k-2}. \right) . \end{aligned}$$
The last sum converges if d is large enough, and this establishes Proposition 2.5. \(\square \)

Notes

Acknowledgements

We thank the anonymous referees for several helpful suggestions to improve the presentation. The research of JEB is supported by Vetenskapsrådet Grant 2015-05195.

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Authors and Affiliations

  1. 1.Department of Mathematical SciencesChalmers University of Technology and the University of GothenburgGothenburgSweden
  2. 2.Department of MathematicsUniversity of WarwickCoventryUnited Kingdom

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