Abstract
We study scaling limits of a family of planar random growth processes in which clusters grow by the successive aggregation of small particles. In these models, clusters are encoded as a composition of conformal maps and the location of each successive particle is distributed according to the density of harmonic measure on the cluster boundary, raised to some power. We show that, when this power lies within a particular range, the macroscopic shape of the cluster converges to a disk, but that as the power approaches the edge of this range the fluctuations approach a critical point, which is a limit of stability. The methodology developed in this paper provides a blueprint for analysing more general random growth models, such as the Hastings-Levitov family.
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1 Introduction
We study a family of planar random growth processes in which clusters grow by the successive aggregation of particles. Clusters are encoded as a composition of conformal maps, following an approach first introduced by Carleson and Makarov [5] and Hastings and Levitov [8]. The specific models that we study fall into the class of Laplacian growth models in which the growth rate of the cluster boundary is determined by the density of harmonic measure of the boundary as seen from infinity. In our case, the location of each successive particle is distributed according to the density of harmonic measure raised to some power. Our set-up is closely related to that of the Hastings-Levitov family of models, HL(\({\alpha }\)), \({\alpha }\in [0,\infty )\) [8], which includes off-lattice versions of the physically occurring dielectric-breakdown models [13], in particular the Eden model for biological growth [6] and diffusion-limited aggregation (DLA) [20]. Our family of models shares with the HL(0) model the unphysical feature that new particles are distorted by the conformal map which encodes the current cluster. However, in subsequent work [14], we show that these models share common behaviour with the HL(\({\alpha }\)) models when \({\alpha }\ne 0\), so the present paper serves to develop methods applicable to these more physical models.
We establish scaling limits of the growth processes in the small-particle scaling regime where the size of each particle converges to zero as the number of particles becomes large. We show that, when the power of harmonic measure is chosen within a particular range, the macroscopic shape of the cluster converges to a disk, but that as the power approaches the edge of this range the fluctuations approach a critical point, which is a limit of stability. This phase transition in fluctuations can be interpreted as the beginnings of a macroscopic phase transition, from disks to non-disks.
1.1 Description of the model
Our clusters will grow from the unit disk by the aggregation of many small particles. Let
We fix a non-empty subset P of \(D_0\) and set
We assume that P is chosen so that K is compact and simply connected. Then we call P a basic particle.
We will call a conformal map F, defined on \(D_0\) and having values in \(D_0\), a basic map if it is univalent and satisfies, as \(z\rightarrow \infty \),
From now on, we will express this last condition by writing \(F(\infty )=\infty \) and \(F'(\infty )=e^c\). By the Riemann mapping theorem, there is a one-to-one correspondence between basic particles and basic maps given by
For convenience, we will assume throughout that F has a continuous extension to the unit circle. It is well understood geometrically when this holds. The map F has the form
for some \(c>0\) and sequence \((a_k:k\ge 0)\) in \({{\mathbb {C}}}\). The value \(e^c\) is called the logarithmic capacity of the cluster K. We define the capacity of the particle P (or, interchangeably, of the map F) by
For the purpose of this introduction, we will assume that we have chosen a family of basic particles \((P^{(c)}:c\in (0,\infty ))\), such that \({\text {cap}}(P^{(c)})=c\). Figure 1 shows four representative particles from some families we have in mind. Write \((F^{(c)}:c\in (0,\infty ))\) for the family of associated basic maps. Given a sequence of attachment angles \((\Theta _n:n\ge 1)\) and capacities \((c_n:n\ge 1)\), set
Define a process \((\Phi _n:n\ge 0)\) of conformal maps on \(D_0\) as follows: set \(\Phi _0(z)=z\) and for \(n\ge 1\) define recursively
Then \(\Phi _n\) encodes a compact set \(K_n\subseteq {{\mathbb {C}}}\), given by
and \(\Phi _n\) is the unique conformal map \(D_0\rightarrow D_n\) such that
where \(D_n={{\mathbb {C}}}\setminus K_n\). It is straightforward to see that
and that \(K_n\) may be written as the following disjoint union
We think of the compact set \(K_n\) as a cluster, formed from the unit disk \(K_0\) by the addition of n particles.
By choosing the sequences \((\Theta _n:n\ge 1)\) and \((c_n:n\ge 1)\) in different ways, we can obtain a wide variety of growth processes. The aggregate Loewner evolution (ALE) model, which was introduced in [19] for slit particles, fits into this scheme and is dependent on the parameters \({\alpha }\in {{\mathbb {R}}}\), \(\eta \in {{\mathbb {R}}}\), \(c\in (0,\infty )\) and \(\sigma \in [0,\infty )\). In this model, which is abbreviated as ALE(\({\alpha },\eta \)), we set
where
Conditional on \({{\mathcal {F}}}_{n-1}={\sigma }(\Phi _1,\dots ,\Phi _{n-1})\), \({\Theta }_n\) is taken to be a random variable whose distribution given by
and we set
We complete the recursive definition of \(\Phi \) using equation (2). Observe that, with these choices, \({{\mathcal {F}}}_n={\sigma }({\Theta }_1,\dots ,{\Theta }_n)\).
In this paper, we will consider only the case where \({\alpha }=0\), which takes as data a single basic map \(F=F^{(c)}\) and a choice of \(\eta \in {{\mathbb {R}}}\) and \({\sigma }\in [0,\infty )\). For simplicity, we refer to this model here as the ALE\((\eta )\) model with basic map F and regularization parameter \({\sigma }\).
If, on the other hand, we were to take \(\eta ={\sigma }=0\) and fix \({\alpha }\in [0,\infty )\), then we would obtain the HL(\({\alpha }\)) model considered by Hastings and Levitov [8]. The parameters \({\alpha }\) and \(\eta \) play a similar role in adjusting the ‘local growth rate of capacity’ as a function of the current cluster shape. Indeed, in the subsequent paper [14] we show that, modulo a deterministic time-change and under the same restrictions on the parameter \(\sigma \) as will be used in this paper, the scaling limit of ALE(\({\alpha },\eta \)) depends primarily on the sum \({\alpha }+\eta \) provided that \({\alpha }+\eta \le 1\). This means that ALE(\(\eta \)) and regularized HL(\(\alpha \)) have qualitatively similar behaviour when \({\alpha }=\eta \). Moreover, the range of the attachment densities considered in ALE(\(\eta \)) corresponds exactly to those used to define the dielectric-breakdown models, so the full family ALE(\({\alpha },\eta \)) is of wider interest than HL(\({\alpha }\)) alone. See [19] for a comprehensive discussion of other models related to ALE, and [7] for a survey of Laplacian growth.
One of the challenges of studying HL(\({\alpha }\)) when \({\alpha }\ne 0\) is that the capacity of the cluster \(K_n\) is random and could be quite badly behaved. It is therefore a priori unclear how to tune parameters in order to obtain non-trivial scaling limits. One way in which ALE(\(\eta \)) is simpler is that the capacity of the cluster \(K_n\) is always cn, where \(c=\log F'(\infty )\). Nevertheless, the models have much in common, and it has turned out that the framework developed here for ALE(\(\eta \)) provides useful ideas for the analysis of HL(\({\alpha }\)). In this paper we will focus on the case where \(\eta \in (-\infty ,1]\). We will establish scaling limits and fluctuations for ALE(\(\eta \)) in the small-particle regime, where simultaneously \(c\rightarrow 0\), \({\sigma }\rightarrow 0\) and \(n\rightarrow \infty \) with n tuned so that \(nc \rightarrow t\), for some fixed \(t \in {\mathbb {R}}\), thereby giving clusters of macroscopic capacity.
1.2 Review of related work
Much effort has been devoted to the analysis of lattice-based random growth models. These are models in which, at each step, a lattice site adjacent to the current cluster is added, chosen according to a distribution determined by the current cluster. Examples include the Eden model [6], diffusion limited aggregation (DLA) [20] and the family of dielectric-breakdown models [13]. Around 20 years ago, Carleson and Makarov [5] and Hastings and Levitov [8] introduced an alternative approach in the planar case, which allows the formulation of a discrete particle model directly in the continuum by encoding clusters in terms of conformal maps, as described in the preceding subsection. In [5], the authors obtained a growth estimate for a deterministic analogue of DLA which is formulated in terms of the Loewner equation. In [8], the HL(\({\alpha }\)) model was studied numerically and experimental evidence was shown for a phase transition in behaviour at \(\alpha =1\): when \(\alpha <1\), clusters appeared to converge to disks; on the other hand, when \(\alpha >1\), a turbulent growth regime emerged, in which clusters behaved randomly at large scale. Hastings and Levitov argued that HL(1) is a candidate for an off-lattice version of the Eden model, and HL(2) corresponds to DLA. Establishing the existence of this phase transition rigorously is one of the main open problems in this area.
In [19], Sola, Turner and Viklund showed the existence of a phase transition in the ALE(\(\eta \)) model. They showed that, for \(\eta >1\), if particles are taken to be slits, and the regularisation parameter \(\sigma \) is sufficiently small then, in the small-particle limit, the clusters themselves grow from the unit disk by the emergence of a radial slit, at a random angle. In this case, harmonic measure is concentrated at the tip of the slit, where derivative of the slit map vanishes. The derivative of the slit map also has two poles on either side of the base of the slit. In the case \(\eta <-2\), Higgs [9] shows that, when the regularisation parameter \(\sigma \) is very small, ALE clusters converge to an SLE curve.
Both of these limits are qualitatively different to the known behaviour of ALE(0), that is to say HL(0), in the same scaling regime. In [15], Norris and Turner showed that the HL(0) clusters converge to disks with internal branching structure given by the Brownian web. More recently, Silvestri [18] analysed the fluctuations in HL(0) and showed that these converge to a log-correlated fractional Gaussian field. Several other papers consider modifications of the HL(0) model [1, 2, 10,11,12, 17].
In this paper, we approach the question of the phase transition in ALE\((\eta )\) at \(\eta =1\) from the opposite direction to that in [19] by showing convergence to a disk for ALE(\(\eta \)) for all \(\eta \le 1\), provided that \({\sigma }\) does not converge to zero too fast. Further, we prove convergence of the associated fluctuations to an explicit limit, which depends on \(\eta \), and which would exhibit unstable behaviour if one took \(\eta >1\). Our results apply in a different regime to that considered in [19]. We require that the regularization parameter \({\sigma }\gg c^{1/2}\) (and sometimes more), which enables us to show that, for each \(\eta \le 1\), the disk limit and the fluctuations hold universally for a wide class of particle shapes. By contrast, in [19] the parameter \({\sigma }\ll c\) and the results rely heavily on the slit particle being non-differentiable at its tip.
1.3 Statement of results
Our main results will be proved under the technical assumption (4) below, which we will show in Appendix A to be satisfied for small particles of any given shape. This assumption expresses that the basic particle P is concentrated near the point 1 on the unit circle in a certain controlled way. Let F be a basic map of capacity \(c\in (0,1]\), in the sense of Sect. 1.1, that is to say, a univalent conformal map from \(\{|z|>1\}\) into \(\{|z|>1\}\) such that \(F(z)/z\rightarrow e^c\) as \(z\rightarrow \infty \). We say that F has regularity \({\Lambda }\in [0,\infty )\) if, for all \(|z|>1\),
Here and below we choose the branch of the logarithm so that \(\log (F(z)/z)\) is continuous on \(\{|z|>1\}\) with limit c at \(\infty \). Our results will concern the limit \(c\rightarrow 0\) with \({\Lambda }\) fixed, but are otherwise universal in the choice of particle. We will show that, for \(\eta \in (-\infty ,1]\), in this limit, provided the regularisation parameter \({\sigma }\) does not converge to 0 too fast, the cluster \(K_n\) converges to a disk of radius \(e^{cn}\), and the fluctuations, suitably rescaled, converge to the solution of a certain stochastic partial differential equation.
Theorem 1.1
Let \(\eta \in (-\infty ,1]\), \({\Lambda }\in [0,\infty )\) and \({\varepsilon }\in (0,1/2)\) be given. Let \((\Phi _n:n\ge 0)\) be an ALE\((\eta )\) process with basic map F and regularization parameter \({\sigma }\). Assume that F has capacity c and regularity \({\Lambda }\), and that \(e^{\sigma }\ge 1+c^{1/2-{\varepsilon }}\). For all \(\eta \in (-\infty ,1)\), \(m\in {{\mathbb {N}}}\) and \(T\in [0,\infty )\), there is a constant \(C=C(\eta ,{\varepsilon },{\Lambda },m,T)<\infty \) with the following property. There is an event \({\Omega }_1\) of probability exceeding \(1-c^m\) on which, for all \(n\le T/c\) and all \(|z|\ge 1+c^{1/2-{\varepsilon }}\),
Moreover, in the case where \(\eta =1\), provided \({\varepsilon }\in (0,1/5)\) and \(e^{\sigma }\ge 1+c^{1/5-{\varepsilon }}\), there is also a constant \(C=C({\varepsilon },{\Lambda },m,T)<\infty \) with the following property. There is an event \({\Omega }_1\) of probability exceeding \(1-c^m\) on which, for all \(n\le T/c\) and all \(|z|\ge 1+c^{1/5-{\varepsilon }}\),
We remark that Theorem 1.1 can be recast in terms of a regularized particle \(P^{({\sigma })}\) given by
Note that \(P^{({\sigma })}\) also has capacity c and is associated to the conformal map
Let \((\Phi _n^{({\sigma })}:n\ge 0)\) be an ALE process with basic map \(F^{({\sigma })}\) and regularization parameter 0. Then
for an ALE process \((\Phi _n:n\ge 0)\) with basic map F and regularization parameter \({\sigma }\). Hence, if we replace \(\Phi _n\) by \(\Phi _n^{({\sigma })}\) in Theorem 1.1, then under the same restrictions on \({\sigma }\), the same estimates are valid but now for all \(|z|\ge 1\) and without regularization in the density of attachment angles.
The simulations on the left side of Fig. 2 illustrate the conjectured phase transition in macroscopic shape from disks to non-disks at \(\eta =1\). The simulations on the right show the sensitivity of the fluctuations of the level lines \({\theta }\mapsto \Phi _n(re^{i{\theta }})\) in ALE(0) to taking \(r-1\approx c^{1/2}\) versus \(r-1\gg c^{1/2}\). This provides evidence that the speed at which \({\sigma }\rightarrow 0\) as \(c\rightarrow 0\) in ALE\((\eta )\) significantly affects cluster behaviour.
We also establish the following characterization of the limiting fluctuations, which shows in particular that they are universal within the class of particles considered.
Theorem 1.2
Let \(\eta \in (-\infty ,1]\), \({\Lambda }\in [0,\infty )\) and \({\varepsilon }\in (0,1/6)\) be given. Let \((\Phi _n:n\ge 0)\) be an ALE\((\eta )\) process with basic map F and regularization parameter \({\sigma }\). Assume that F has capacity c and regularity \({\Lambda }\). Assume further that
Set \(n(t)=\lfloor t/c\rfloor \). Then, in the limit \(c\rightarrow 0\) with \({\sigma }\rightarrow 0\), uniformly in F,
in distribution on \(D([0,\infty ),{{\mathcal {H}}})\), where \({{\mathcal {H}}}\) is the set of analytic functions on \(\{|z|>1\}\) vanishing at \(\infty \), equipped with the metric of uniform convergence on compacts, and where \({{\mathcal {F}}}\) is given by the following stochastic PDE driven by the analytic extension \(\xi \) in \(D_0\) of space-time white noise on the unit circle,
The space \({{\mathcal {H}}}\) and the meaning of this PDE are discussed in more detail in Sect. 7. For \(\eta =0\) we recover the fluctuation result in [18]. The solution to the above stochastic PDE is an Ornstein–Uhlenbeck process in \({{\mathcal {H}}}\). This process converges to equilibrium as \(t\rightarrow \infty \). When \(\eta <1\), the equilibrium distribution is given by the analytic extension in \(D_0\) of a log-correlated Gaussian field defined on the unit circle. In the case \(\eta =0\), this is known as the augmented Gaussian Free Field. When \(\eta =1\), the equilibrium distribution is the analytic extension of complex white noise on the unit circle. The equation (6) can be interpreted as a family of independent equations for the Laurent coefficients of \({{\mathcal {F}}}(t,.)\), given in (51). These equations may be considered also for \(\eta >1\) but now the equation for the kth Laurent coefficient shows exponential growth of solutions at rate \((\eta -1)k\), so there is no solution to (6) in \({{\mathcal {H}}}\), indicating a destabilization of dynamics as \(\eta \) passes through 1. The mathematical formulation of universal limits for cluster shapes when \(\eta >1\) remains an open problem.
Although we have stated our theorems above for \(\eta \in (-\infty ,1]\), in many of our arguments we restrict to the case \(\eta \in [0,1]\). The proofs are largely similar when \(\eta <0\) except in the way that we decompose the operator in Sect. 4. We remark on the correct decomposition in the case \(\eta <0\) at the relevant point.
1.4 Remarks on context and scope of results
The process of conformal maps \((\Phi _n:n\ge 0)\) is Markov and takes values in an infinite-dimensional vector space. In the limit considered, where \(c\rightarrow 0\), the jumps of this process become small, while we speed up the discrete time-scale to obtain a non-trivial limiting drift. So we are in the domain of fluid limits for Markov processes. The analysis of such limits, and of the renormalized fluctuations around them, is well understood in finite dimensions. However, while the formal lines of this analysis transfer readily to infinite dimensions, its detailed implementation is not so clear, not least because it is necessary to choose a norm, which should be well adapted to the dynamics, and the limiting drift will in general be a non-linear and unbounded operator.
In the case at hand, there are a number of special features which are important to the analysis. First, while the limiting dynamics is not in equilibrium, it is an explicit steady state, which allows us to handle convergence of the Markov process in terms of linearizations around this steady state: we find that the difference \(\Phi _n(z)-e^{cn}z\) may usefully be expressed by an interpolation in time, in which each term describes the error introduced by a single added particle. Second, the map \(\Phi _n\) is determined by its restriction to the unit circle \((\Phi _n(e^{i{\theta }}):{\theta }\in [0,2\pi ))\) and the action of each jump, besides being small, also becomes localized in \({\theta }\) in the limit \(c\rightarrow 0\). This is one of the features contributing to the explicit form found for the limiting fluctuations. Third, we have at our disposal, not only the usual tools of stochastic analysis, but also a range of tools from complex analysis, including distortion estimates, and \(L^p\)-estimates for multiplier operators, which turn out to mesh well with \(L^p\)-martingale inequalities.
We have tried to optimise, as far as our present techniques allow, the constraints in our results on the regularization parameter \({\sigma }\). In the case \(\eta <1\), we establish the disk limit for \({\sigma }\gg c^{1/2}\). Indeed, for \(\eta <1\), in the limit considered, we show that the derivative of the fluctuations at radius \(e^{\sigma }\), which controls the scale of \(h_n({\theta })-1\), is at most of order \(c^{1/2}/(e^{\sigma }-1)\). Therefore, to leading order, the distribution of each attachment angle is approximately uniform and the bulk dynamics of our process resemble that of HL(0). As seen in Proposition A.2, the scale of individual particles is \(c^{1/2}\), so for \({\sigma }\sim c^{1/2}\) the fluctuations of \(e^{-c}f'(e^{{\sigma }+ i{\theta }})\) around 1 are scale-invariant. With that choice of \(\sigma \) we would expect to see macroscopic variations of \(h_n({\theta })\), so the attachment distributions would no longer be well approximated by the uniform distribution. We therefore believe our constraint on \({\sigma }\) is close to optimal within this regime and it remains a challenging open problem to allow \({\sigma }\sim c^{1/2}\). When \(\eta = 1\), on the other hand, we show that the derivative of the fluctuations at radius \(e^{\sigma }\) is at most of order \(c^{1/2}/(e^{\sigma }-1)^{3/2}\). The break-down in the uniform approximation may therefore well happen for larger \({\sigma }\) than \({\sigma }\sim c^{1/2}\) and the form of the fluctuations is suggestive of \({\sigma }\sim c^{1/3}\). Although we need a stronger regularization for the fluctuation result (cf. Theorem 1.2), we find that the fluctuations develop variations on all spatial scales, so the modification of dynamics from HL(0) to ALE\((\eta )\), even with the averaging enforced by our choice of regularization, results in a feedback which affects the limiting evolution, and which identifies the case \(\eta =1\) as critical.
1.5 Organisation of the paper
The structure of the paper is as follows. In Sect. 2, we give a simplified proof of convergence to a disk in the case \(\eta =0\), corresponding to HL(0). This is followed by an overview of the proof when \(\eta \ne 0\). In Sect. 3, we decompose the increment \(\Phi _n(z)-\Phi _{n-1}(e^cz)\) as a sum of martingale difference and drift terms, which we expand to leading order in c with error estimates. In Sect. 4 we obtain the evolution equation and decomposition for the fluctuations. The remainder of the paper analyses this equation. Specifically, in Sect. 5 we use the estimates from Sect. 3 to obtain bounds on the terms arising in the decomposition of the differentiated fluctuations. These bounds are then used in Sect. 6 to obtain our disk limit Theorem 1.1. Finally the fluctuation limit Theorem 1.2 is derived in Sect. 7.
Some necessary but technical estimates are deferred to appendices. In Appendix A we show that our main assumption (4) is satisfied for small particles of any given shape. Appendix B contains the estimates for multiplier operators used in the paper. In Appendices C and D we derive the specific estimates on ALE(\(\eta \)) used in our main results. We believe that some of the results and estimates in the appendices may be of independent use in related work. For example, the “spread out” particle in Appendix A.2, has a very convenient form which makes it a useful test case when trying to prove general theorems about particle aggregation.
2 HL(0) and overview of the proof of Theorem 1.1
In this section we give a quick argument for the scaling limit of HL(0) (which is the same as ALE(0)), where the attachment angles \((\Theta _n:n\ge 1)\) are independent and uniformly distributed. Then we discuss the structure of the proof of Theorem 1.1, some aspects of which follow the argument used for HL(0).
For a measurable function f on \(\{|z|>1\}\), for \(p\in [1,\infty )\) and \(r>1\), we will write
In the case where f is analytic and is bounded at \(\infty \), we have, for \(\rho \in (1,r)\),
The notation \(\Vert \cdot \Vert _p\) will be reserved for the \(L^p ({\mathbb {P}})\)-norm on the probability space.
2.1 Disk limit for \(\eta =0\)
We now show that HL(0) converges to a disk in the small-particle limit. A weaker form of this result was shown in [15] by fluid limit estimates on the Markov processes \((\Phi _n^{-1}(z):n\ge 0)\). Here, we will use a new method, based on estimating directly the conformal maps \(\Phi _n\). This both gives a simpler argument and leads to a stronger result.
Theorem 2.1
Let \((\Phi _n:n\ge 0)\) be an HL(0) process with basic map F. Assume that F has capacity \(c\in (0,1]\) and regularity \({\Lambda }\in [0,\infty )\). Then, for all \(p\in [2,\infty )\), there is a constant \(C=C({\Lambda },p)<\infty \) such that, for all \(r>1\) and \(n\ge 0\), we have
We remark that by taking p large enough it is possible to deduce that, for all \({\varepsilon }\in (0,1/2)\) and \(T\ge 0\), we have
in probability as \(c\rightarrow 0\). As this is spelled out more generally in Sect. 6.2, we omit the details at this stage. Indeed, on applying Theorem 1.1 to HL(0), say with \({\sigma }=1\), we obtain the stronger estimate
with high probability as \(c\rightarrow 0\). This improvement can be traced to the iterative argument used in the proof of Proposition 6.1.
Proof of Theorem 2.1 It will suffice to consider the case where \(r\ge 1+\sqrt{c}\). Set
Note that \(\Phi _{n-1}(e^cz)\) is the map we would obtain after n steps if we substituted \(F_n(z)\) by \(e^cz\) in (2). As we aim to show that \(\Phi _n(z)\) is close to \(e^{cn}z\), \(\Delta _n(z)\) can be understood as the error due to the nth particle. We can write \(\Phi _n\) as a telescoping sum
The functions F and \(\Phi _{j-1}\) are analytic in \(\{|z|>1\}\) and \(F(z)/z\rightarrow e^c\) as \(z\rightarrow \infty \), so the function
is analytic in \(\{0<|w|<|z|\}\) and extends analytically to \(\{|w|<|z|\}\). Hence, almost surely, by Cauchy’s theorem,
There is a constant \(C=C({\Lambda })<\infty \) such that, for all \(|z|>1+\sqrt{c}/2\),
Since \(\Phi _{j-1}\) is univalent on \(\{|z|>1\}\) and \(\Phi _{j-1}(z)/z\rightarrow e^{c(j-1)}\) as \(z\rightarrow \infty \), by a standard distortion estimate, for all \(|z|=r>1\),
Hence, for \(|z|=r>1+\sqrt{c}/2\), we have
and so
Burkholder’s inequality (see Sect. B.1) applies to the sum of martingale differences (10), to give that for all \(p\in [2,\infty )\) there is a constant \(C=C(\Lambda , p)<\infty \), such that
Hence, for \(|z|\ge 1+\sqrt{c}/2\),
where we used an integral comparison for the last inequality. Set
and write \(\rho =(r+1)/2\). Then, for \(|z|\ge 1+\sqrt{c}\), we have \(\rho \ge 1+\sqrt{c}/2\), so
and the claimed estimate follows. \(\square \)
2.2 Overview of the proof of Theorem 1.1
We now discuss how the above strategy can be adapted to the case where \(\eta \in (-\infty ,1]\). Write
with \(\Delta _j(z)=\Phi _j(z)-\Phi _{j-1}(e^cz)\) as in (9). We split \(\Delta _j(z)\) as the sum of a martingale difference
and a drift term (which vanished in the case \(\eta =0\))
Set \({{\tilde{\Phi }}}_n(z)=e^{-cn}\Phi _n(z)-z\) as above. We start by identifying the leading term in the drift, showing that
where \(R_j(z)\) is small provided \(\Vert {{\tilde{\Phi }}}'_{j-1}\Vert _{\infty ,e^{\sigma }}\) is sufficiently small. This gives the following decomposition
where P is the operator which acts on analytic functions on \(\{|z|>1\}\) by
The reader is alerted to the fact that, while we used P to denote our basic particle in Sects. 1 and Appendix A, in the rest of the paper, P will refer to this operator instead. Solving the recursion we end up with
Note that for \(\eta =0\) the operator P has the simple form \(Pf(z)=e^{-c}f(e^cz)\) and we recover (10). We treat the general case \(\eta \in (-\infty ,1]\) by observing that P acts diagonally on the Laurent coefficients, thus is a Fourier multiplier operator, which we can bound in \(\Vert \cdot \Vert _{p,r}\)-norm by means of the Marcinkiewicz multiplier theorem (see Appendix B.2).
The proof strategy for the disk theorem then goes as follows. For \({\delta }={\delta }(c)\) small, to be specified, introduce the stopping time
Then for all \(n\le N({\delta })\) the angle density \(h_n\) defined in (3) is approximately uniform. This, together with the multiplier theorem, can be used to bound both the martingale term (the first term in (15)) and the remainder term (the second term in (15)), thus leading to a bound for the map \({{\tilde{\Phi }}}_n\). At this point it remains to show that we can pick \({\delta }_0\) such that \(N({\delta }_0)\ge \lfloor T/c\rfloor \) with high probability to conclude the proof. To this end, it turns out to be convenient to work instead with the differentiated dynamics
for which a decomposition similar to (15) holds (see (32) below). We use it to show that \(\Vert \Psi _n1_{\{n\le N_0\}}\Vert _{p,r}\) is small in \(L^p({\mathbb {P}})\) (see Proposition 6.1), where we have set \(N_0 = N({\delta }_0)\) to ease the notation slightly. The analyticity of \(\Psi _n\) then allows us to make this bound into a high probability statement on the supremum norm of \(\Psi _n1_{\{n\le N_0\}}\), at the price of taking p large enough (see Proposition 6.2). By showing that this bound is smaller than \({\delta }\) for all \(n\le N_0\), we deduce that in fact we must have \(N_0\ge \lfloor T/c\rfloor \), thus concluding the proof.
2.3 Choice of state variables
The sequence of conformal maps \((\Phi _n)_{n\ge 0}\) is a Markov process. This allows an approach to the desired scaling limits using martingale estimates. Above, we introduced the analytic function \(\Psi _n\) on \(\{|z|>1\}\) given by
where we set \(Df(z)= zf'(z)\) and \({{\tilde{\Phi }}}_n\) is the process of fluctuations given by
Then the process \((\Psi _n)_{n\ge 0}\) is also Markov and it proves more convenient to use this as our primary state variable. In doing this, we forget the limiting values \((\Phi _n(\infty ))_{n\ge 0}\), so we see the clusters only up to an unknown displacement. Otherwise, the use of \((\Psi _n)_{n\ge 0}\) may be considered as a particular choice of coordinates for the sequence of clusters. The function \(\Phi _n\) has a Laurent expansion in \(\{|z|>1\}\) of the form
so \(\Psi _n\) has expansion
In the final section of the paper, we will characterise the limit distribution of the fluctuations, suitably rescaled, by analysing the Laurent coefficients.
3 Expansions to first order and error estimates
In this section we identify the leading order behaviour of several quantities of interest and gather together bounds on the error terms which hold while the differentiated fluctuation process \(({{\tilde{\Phi }}}'_n)_{n\ge 0}\) is well-behaved. Our main objective is to justify (13).
Fix \({\delta }_0\in (0,1/8]\) and consider the stopping time \(N_0 = N(\delta _0)\) where \(N(\delta )\) is defined in (16). Several of our estimates will be made under the assumption that . In fact, in this section, we only use that \(|{{\tilde{\Phi }}}'_{j}(e^{{\sigma }+i{\theta }})|\le {\delta }_0\le 1/8\) when \(j=n-1\). However, we will need this to hold for all in the remainder of the paper and it simplifies notation to make the assumption here. This assumption guarantees that \(h_n\), defined in (3), can be bounded above and below by absolute constants. Bounding very crudely,
A more refined analysis shows that, for all ,
where \(C = C(\eta )\) is a constant depending only on the value of \(\eta \). As the precise computation consists of elementary manipulations, it is deferred to Appendix C (see (67) and (68)).
Recall the definitions of \(\Delta _n({\theta },z)\) and \(\Delta _n(z)\) from (9) and the definitions of \(A_n(z)\) and \(B_n(z)\) from (12) and (11). Then
Furthermore, \(A_n\) and \(B_n\) are analytic in \(\{|z|>1\}\) and, almost surely,
As we showed in the proof of Theorem 2.1, by Cauchy’s theorem,
so
We now identify the leading order terms in \(\Delta _n(z)\) and \(A_n(z)\), in the limit \(c\rightarrow 0\). Where the computations add little to the intuition, these are also deferred to Appendix C.
Given \({\theta }\in [0,2\pi )\) and \(|z|>1\), define, for \(s\in [0,1]\),
Note that \(F_{0,{\theta }}(z)=e^cz\) and \(F_{1,{\theta }}(z)=e^{i{\theta }}F(e^{-i{\theta }}z)\). Note also that \(|F_{s,{\theta }}(z)|\ge |z|\) for all \(s\in [0,1]\) and
Then
where \(2 c {\beta }= a_0\), the coefficient of \(z^0\) in the Laurent expansion (1). We factorize \(a_0\) in this way to highlight that \(a_0 \sim 2c\) (see Proposition A.1 in the appendix). The first term in the decomposition (21) captures the leading order of the increment and will determine the evolution of the process; the second term is a recursive error arising from the fluctuations of \(\Phi _{n-1}\); and the third term is an error term dependent just on the class of particle chosen (see Appendix A.1 and in particular (59)). It will be convenient to write
for the leading term and to set
Note that \(w_n({\theta },\infty )=0\) and for all \(|z|\ge 1+\sqrt{c}\)
for some constant \(C=C(\eta ,{\Lambda })<\infty \) (see (69) and (71) in the appendix).
Using (19), (18), (21) and that (cf. Proposition A.1), the leading term of \(A_n(z)\) is
where the equality follows by Cauchy’s integral formula. To be precise, set
Then, by the argument in Appendix C, for and \(|z|=r\) with \(r\ge 1+\sqrt{c}\),
and
for some constant \(C=C(\eta ,\Lambda )<\infty \) (possibly different to the constant C obtained earlier). By the maximum principle, it follows that provided one takes \(\delta _0 \ge \sqrt{c}/(e^{\sigma }- 1)\) and \(r \ge e^{\sigma }\ge 1 + \sqrt{c}\),
From this bound, it can be easily seen that \(R_n(z)\) is small if \(\Vert {{\tilde{\Phi }}}'_{n-1}\Vert _{\infty ,e^{\sigma }}\) is sufficiently small, which is what we wanted to show. However, the assumption that \(r \ge e^{\sigma }\) is too restrictive for our needs, so in subsequent analysis we revert to the more general estimate (27).
4 Linear evolution equation for the fluctuations
In this section, our objective is to justify the expansion (15). In fact, we obtain an analogous expansion which makes it clearer which terms determine the leading order fluctuations.
In Sect. 2.2 we decomposed \(\Delta _n(z)=\Phi _n(z)-\Phi _{n-1}(e^cz)\) as a sum of a martingale difference \(B_n(z)\) and drift \(A_n(z)\), and in the previous section we justified writing
In view of (23), it is convenient to split the martingale difference \(B_n\) as a sum of analytic functions
where
and
We will see that \(M_n\) is the main term: its explicit form allows for precise estimates, and it determines the Gaussian fluctuations. On the other hand, \(W_n\) is accessible less directly, but is of smaller order, so can also be handled adequately. Then, using (26),
so we obtain the linear evolution equation
where P is as in (14). Note that P acts diagonally on the Laurent coefficients, with multipliers
In the case \(\eta \in [0,1]\), we factorize P by writing
It is straightforward to check then that, for all k,
In particular, we can define a multiplier operator \(P_0\) acting on analytic functions f on \(\{|z|>1\}\), bounded at \(\infty \), such that
Note that, by the factorization above,
Being able to “push-out” the point at which f is evaluated in this way will allow us to exploit that the derivative of a conformal map becomes more regular away from the boundary. Losing this push-out in the \(\eta =1\) case is the reason that the bounds in Sect. 5 are larger when \(\eta =1\).
In order to adapt our argument to the case \(\eta \in (-\infty ,0)\), we would modify the equation defining \(p_0(k)\) to
The subsequent argument is very similar so we will not give further details for this case.
We iterate (28) to obtain
where
Then, on differentiating,
where
We will focus initially on bounding the terms in the decomposition (32) of the differentiated fluctuations \(\Psi _n\). We will refer to \({{\mathcal {M}}}_n\), \({{\mathcal {W}}}_n\) and \({{\mathcal {R}}}_n\) as the principal martingale term, the second martingale term and the remainder term respectively. Later, we will return also to the undifferentiated decomposition (31).
4.1 Norms
We conclude this section by describing the normed spaces on which we will obtain our bounds.
Recall from (7) the definition of \(\Vert f\Vert _{p,r}\) for a measurable function f on \(\{|z|>1\}\). For a measurable function \(\Phi : {\Omega }\times \{|z| > 1\} \rightarrow {\mathbb {C}}\), we will write
Then, by Fubini,
where \(\Vert \cdot \Vert _p\) denotes the \(L^p({\mathbb {P}})\)-norm on the probability space.
Note that, for all \(n\ge 0\), the boundedness and monotonicity seen in (30) allows an application of the Marcinkiewicz multiplier theorem (see Appendix B.2), with \(m_k=p_0(k)^n\) and \(M=1\) to see that for all \(p\in (1,\infty )\) and all \(r>1\), there is a constant \(C=C(p)<\infty \) such that
Some further operator estimates which will be used in the subsequent analysis are stated in Appendix B.2.
5 Estimation of terms in the decomposition of the differentiated fluctuations
In this section we collect estimates for the principal martingale term, the second martingale term and remainder term.
We first estimate the principal martingale term \({{\mathcal {M}}}_n(z)\) in the decomposition (32) of the differentiated fluctuation process, which is given by
Lemma 5.1
For all \(p\in [2,\infty )\), there is a constant \(C=C(p)<\infty \) such that
where \(r_n=re^{c(1-\eta )n}\).
It follows that if \(r \ge 1 + c^{1/2 - {\varepsilon }}\) for some \({\varepsilon }\in (0, 1/2)\),
Proof
By Burkholder’s inequality (cf. Theorem B.1), for all \(p\in [2,\infty )\), there is a constant \(C=C(p)<\infty \) such that
where
So, on taking the \(\Vert \cdot \Vert _{p/2,r}\)-norm,
Recall from (22) that
Observe that
We have, almost surely,
and, by (36), for \(|z|=r\),
For \(j\le N_0\), by (17), we have \(h_{j}({\theta })\le 3/2\), so we obtain, for \(|z|=r\), almost surely,
where we have used
Hence, for \(|z|=r\), almost surely,
Moreover,
and
Hence, using that ,
Finally, we take the \(\Vert .\Vert _{p/2,r}\)-norm and substitute into (35) to obtain (34).
Now suppose \(r\ge 1+c^{1/2 - {\varepsilon }}\) for some \({\varepsilon }\in (0,2)\) and \(p \ge 1 + 1/(2 {\varepsilon })\). If \(\eta <1\), by using an integral comparison in (34) we obtain
where we used the assumption on p in the second inequality, and absorbed a factor of \(2+1/(2-2\eta )\) in the final constant C. Hence
If \(\eta =1\), we now have \(r_n = r\), so
and then, using that \(p\ge 2\),
\(\square \)
We now state the estimate of the second martingale term \({{\mathcal {W}}}_n(z)\) in the decomposition (32) of the differentiated fluctuation process, which is given by
The proof is deferred to Appendix D.1.
Lemma 5.2
For all \({\varepsilon }\in (0, 1/2)\) and \(p\in [2,\infty )\), there is a constant \(C=C(\Lambda , \eta , \epsilon , p)<\infty \) such that, for all \(r\ge 1+ 2c^{1/2 - {\varepsilon }}\),
where \(r_n=re^{c(1-\eta )n}\) and \(\rho _n = (1 + r_n)/2\).
It follows that, setting \(\rho = (1+r)/2\), for \(p \ge 1 + 1/(2 {\varepsilon })\),
We finish this section with the estimate of the remainder term \({{\mathcal {R}}}_n(z)\) in the decomposition (32) of the differentiated fluctuation process, which is given by
The proof is deferred to Appendix D.2.
Lemma 5.3
For all \(p\in [2,\infty )\), there is a constant \(C=C(\Lambda , \eta , p)<\infty \) such that, for all \(r\ge 1+ 2\sqrt{c}\),
where we have used the same notation as in Lemma 5.2.
Now suppose that for some constant \(T>0\). Then there is a constant \(C=C(\Lambda , \eta , p, T)<\infty \) such that
when \(\eta < 1\) and
when \(\eta = 1\).
6 Convergence to a disk for ALE\((\eta )\)
In this section we derive our main disk theorem. Recall that
First we show that \(|||\Psi _n1_{\{n\le N_0\}}|||_{p,r}\) is small, provided \({\delta }_0\) is appropriately chosen. Then we deduce estimates on the random norms \(\Vert \Psi _n1_{\{n\le N_0\}}\Vert _{\infty ,r}\), valid with high probability, and use them to dispense with the restriction that \(n\le N_0\). Finally, we apply these results to show that \(\Phi _n(z)\) is close to \(e^{cn}z\).
6.1 \(L^p\)-estimates on the differentiated fluctuations
The proposition below shows that, for an appropriately chosen \({\delta }_0\), the \(|||\cdot |||_{p,r}\) norm of the differentiated fluctuation process \(\Psi _n1_{\{n\le N_0\}}\) is of order \(\sqrt{c}\), with quantitative control of the singularity as \(r\rightarrow 1\) and the decay as \(r\rightarrow \infty \). The dependence of the estimate on \({\sigma }\) is also explicit, allowing one to consider limits in which \({\sigma }\rightarrow 0\) as \(c\rightarrow 0\). For small c, the estimates are strongest when \({\varepsilon }\) and \(\nu \) are taken to be small. A second argument, given in the next subsection, will show that the event \(\{n\le N_0\}\) appearing in (41) and (42) below is of high probability in the limit \(c\rightarrow 0\).
Proposition 6.1
For all \(\eta \in [0,1)\), \(T\in (0,\infty )\), \({\varepsilon }\in (0,1/2)\), \(\nu \in (0,{\varepsilon }/2)\) and \(p \in [2,\infty ) \), there is a constant \(C=C({\Lambda },\eta ,T,{\varepsilon },\nu ,p)\in [1,\infty )\) with the following property. For all \(c\in (0,1]\), all \(r,e^{\sigma }\ge 1+c^{1/2-{\varepsilon }}\) and all \(n\le T/c\), we have
where \(N_0\) is given by (40) with \({\delta }_0=c^{1/2-\nu }/(e^{\sigma }-1)\).
Moreover, in the case \(\eta =1\), for all \(T\in (0,\infty )\), \({\varepsilon }\in (0,1/5)\), \(\nu \in (0,3{\varepsilon }/2)\) and \(p \in [2,\infty ) \), there is a constant \(C=C({\Lambda },T,{\varepsilon },\nu ,p)\in [1,\infty )\) with the following property. For all \(c\in (0,1]\), all \(r,e^{\sigma }\ge 1+c^{1/5-{\varepsilon }}\) and all \(n\le T/c\), we have
where \(N_0\) is given by (40) with \({\delta }_0=c^{1/2-\nu }/(e^{\sigma }-1)^{3/2}\).
Proof
As before, constants referred to in the proof by the letter C may change from line to line and are all assumed to lie in \([1,\infty )\). They may depend on \({\Lambda }\), \(\eta \), T, \({\varepsilon }\), \(\nu \) and p but they do not depend on c, n, \({\sigma }\) and r.
We begin with a crude estimate which allows us to restrict further consideration to small values of c. The function \(e^{-cn}\Phi _n(z)\) is univalent on \(\{|z|>1\}\), with \(e^{-cn}\Phi _n(z)\sim z\) as \(z\rightarrow \infty \). By same distortion estimate used in Sect. 2.1, for all \(|z|=r>1\),
and so
It is straightforward to check that this implies the claimed estimates in the case where \(c>1/C\), for any given constant C of the allowed dependence. Hence it will suffice to consider the case where \(c\le 1/C\).
Consider first the case where \(\eta <1\). Fix T, \({\varepsilon }\), p and \(\nu \) as in the statement, and assume that \(c\le 1/e\) and \(r\ge 1+c^{1/2-{\varepsilon }/2}\) and \(e^{\sigma }\ge 1+c^{1/2-{\varepsilon }}\) and \(n\le T/c\). Set \(\rho =(r+1)/2\). It will suffice to prove the result for p large enough, so assume \(p > 1+1/(2{\varepsilon })\).
By the triangle inequality,
where, by Lemmas 5.1, 5.2 and 5.3,
and
On combining the estimates above and substituting the chosen value of \({\delta }_0\), we obtain, for all \(r\ge 1+c^{1/2-{\varepsilon }/2}\),
where
and
Note that, for all \(r\ge 1+c^{1/2-{\varepsilon }}\), we have
for all sufficiently small c. Similarly, for \(r\ge 1+c^{1/2-{\varepsilon }/2}\), we have \({\bar{{\delta }}}(r)\le 1\) for all sufficiently small c. As noted above, it suffices to deal with the case where c is sufficiently small.
A complication in the analysis is that the right hand side of the inequality (44) requires estimates of \(\Psi _{j-1}(z)\) when \(|z| = \rho \), but the left hand side only gives information about \(\Psi _n(z)\) when \(|z| = r > \rho \). Our approach is therefore to use the universal distortion estimate (43) to obtain an initial (very weak) bound and then recursively feed the bounds through the inequality. This generates stronger and stronger estimates, but at the cost of moving r further away from 1.
Set \(C_0=1\) and for \(k\ge 0\) define recursively \(C_{k+1}=2^{k+1}C_k+1\). We will show that, for all \(k\ge 0\), all \(r\ge 1+2^kc^{1/2-{\varepsilon }/2}\) and all \(n\le T/c\),
The case \(k=0\) is implied by (43). Suppose inductively that (45) holds for k and that \(r\ge 1+2^{k+1}c^{1/2-{\varepsilon }/2}\) and \(n\le T/c\). Then \(\rho =(r+1)/2\ge 1+2^k c^{1/2-{\varepsilon }/2}\) so, for all \(j\le n\),
where we used the inequalities \({\delta }(\rho )\le 2{\delta }(r)\) and \({\bar{{\delta }}}(\rho )\le 2{\bar{{\delta }}}(r)\). Since \(r\ge 1+c^{1/2-{\varepsilon }/2}\), we can substitute into (44) to obtain
Hence (45) holds for \(k+1\) and the induction proceeds.
Choose now \(k=\lceil 1/{\varepsilon }\rceil \). Then
For c sufficiently small, we have \(c^{{\varepsilon }/2}\le 2^{-k/2}\) so, for all \(r\ge 1+c^{1/2-{\varepsilon }}\), we have \(r\ge 1+2^kc^{1/2-{\varepsilon }/2}\) and so
giving a bound of the desired form (41).
We turn to the case where \(\eta =1\). Fix T, \({\varepsilon }\), p and \(\nu \) as in the statement for \(\eta =1\). Assume that \(c\le 1/e\) and \(n\le T/c\), and assume now that \(r\ge 1+c^{1/5}\) and \(e^{\sigma }\ge 1+c^{1/5-{\varepsilon }}\). It will suffice to prove the result for p sufficiently large. The argument follows the same pattern as the case where \(\eta <1\), except for modifications necessary because of the different estimates in Lemmas 5.1, 5.2 and 5.3 (and different choice of \({\delta }_0\)), which arose because \(r_n=re^{c(1-\eta )n}=r\).
We obtain for \(r\ge 1+c^{1/5}\),
where now
and
Note that, for \(r\ge 1+c^{1/5-{\varepsilon }}\), we have, for all sufficiently small c
Similarly, we have \({\delta }(r)\le 1\) whenever \(r\ge 1+c^{1/5}\), for all sufficiently small c. We restrict to such c. For \(\rho =(r+1)/2\), we now have modified inequalities \({\delta }(\rho )\le 2^{3/2}{\delta }(r)\) and \({\bar{{\delta }}}(\rho )\le 2^{3/2}{\bar{{\delta }}}(r)\). Set \(C_0=1\) and for \(k\ge 0\) define now recursively \(C_{k+1}=2^{3k/2+1}C_k+1\). Then, by an analogous inductive argument, we obtain, for all \(k\ge 0\), all \(n\le T/c\) and all \(r\ge 1+2^kc^{1/5}\),
Choose now \(k=\lceil 1/{\varepsilon }\rceil \) and assume that \(r\ge 1+c^{1/5-{\varepsilon }}\). Then
and, for c sufficiently small, we have \(c^{\varepsilon }\le 2^{-k}\), so \(r\ge 1+2^kc^{1/5}\) and so
which is a bound of the required form (42). \(\square \)
6.2 Spatially-uniform high-probability estimates on the differentiated fluctuations
We now use the results from the previous section to obtain uniform estimates on \(\Psi _n(z)\).
Proposition 6.2
For all \(\eta \in [0,1)\), \({\varepsilon }\in (0,1/2)\), \(\nu \in (0,{\varepsilon }/4)\), \(m\in {{\mathbb {N}}}\) and \(T\in (0,\infty )\), there is a constant \(C=C({\Lambda },\eta ,{\varepsilon },\nu ,m,T)<\infty \) with the following properties. For all \(c\in (0,1]\) and all \(e^{\sigma }\ge 1+c^{1/2-{\varepsilon }}\), there is an event \({\Omega }_0\) of probability exceeding \(1-c^m\) on which, for all \(n\le T/c\) and all \(|z|=r\ge 1+c^{1/2-{\varepsilon }}\),
Moreover, for \(c\le 1/C\), we have \({\Omega }_0\subseteq \{n\le N_0\}\) for all \(n\le T/c\), where \(N_0\) is given by (40) with \({\delta }_0=c^{1/2-\nu }/(e^{\sigma }-1)\).
For \(\eta =1\), \({\varepsilon }\in (0,1/5)\), \(\nu \in (0,{\varepsilon }/2)\), \(m\in {{\mathbb {N}}}\) and \(T\in (0,\infty )\), there is a constant \(C=C({\Lambda },{\varepsilon },\nu ,m,T)<\infty \) with the following property. For all \(c\in (0,1]\) and all \(e^{\sigma }\ge 1+c^{1/5-{\varepsilon }}\), there is an event \({\Omega }_0\) of probability exceeding \(1-c^m\) on which, for all \(n\le T/c\) and all \(|z|=r\ge 1+c^{1/5-{\varepsilon }}\),
Morover, for \(c\le 1/C\), we have \({\Omega }_0\subseteq \{n\le N_0\}\) for all \(n\le T/c\), where \(N_0\) is given by (40) with \({\delta }_0=c^{1/2-\nu }/(e^{\sigma }-1)^{3/2}\).
Proof
We will give details for the case \(\eta \in [0,1)\). The minor modifications needed for the case \(\eta =1\) are left to the reader. Fix \(\eta ,{\varepsilon },\nu ,m\) and T as in the statement. It will suffice to consider the case where \(e^{\sigma }\ge 1+2c^{1/2-{\varepsilon }}\), and to find an event \({\Omega }_0\) of probability exceeding \(1-c^m\) on which (46) holds whenever \(r\ge 1+2c^{1/2-{\varepsilon }}\) and \(n\le T/c\). Set
Then \(K\le \lfloor \log (1/c)\rfloor +1\). For \(k=1,\dots ,K\), set
Then \(\rho (k)\ge 1+c^{1/2-{\varepsilon }}\) and \(r(K)\in [2,4]\). Choose \(p\ge \max \{ 1+1/(2{\varepsilon }) ; (m+2)/\nu \}\) even integer, and set
By Proposition 6.1, there is a constant \(C=C({\Lambda },\eta ,{\varepsilon },\nu ,p,T)<\infty \) such that, for all \(n\le T/c\),
where \(N_0\) is defined as in the statement and
Set \({\lambda }=Rc^{-1/p}\) and consider the event
By Chebyshev’s inequality,
so
Fix \(r\ge 1+2c^{1/2-{\varepsilon }}\). Then \(r(k)\le r<r(k+1)\) for some \(k\in \{1,\dots ,K\}\), where we set \(r(K+1)=\infty \). Note that \(z\Psi _n(z)\) is a bounded analytic function on \(\{|z|>\rho (1)\}\). We use the inequality (8) to see that, on the event \({\Omega }_0\), for \(n\le N_0\wedge N\),
so, using that \(r(k) \ge (r+1) /2\), we get
where
By our choice of p, we have \({\gamma }_k\le 1\) for all sufficiently small c. We can restrict to such c, since the desired estimate follows from the distortion inequality (43) otherwise. Then, on the event \({\Omega }_0\), for \(n\le N_0\wedge N\),
and in particular, since \(e^{\sigma }\ge 1+c^{1/2-{\varepsilon }}\) and \(\nu < {\varepsilon }/4\), we have
which forces \(N_0>N\) on \({\Omega }_0\). \(\square \)
6.3 \(L^p\)-estimates on the fluctuations
In this section we prove a result analogous to Proposition 6.1 for the undifferentiated dynamics. This allows us to prove Theorem 1.1.
Proposition 6.3
For all \(\eta \in [0,1)\), \(T\in (0,\infty )\), \({\varepsilon }\in (0,1/2)\), \(\nu \in (0,{\varepsilon }/4)\) and \(p \in [2,\infty ) \), there is a constant \(C=C({\Lambda },\eta ,T,{\varepsilon },\nu ,p)\in [1,\infty )\) with the following property. For all \(c\in (0,1]\), all \(r,e^{\sigma }\ge 1+c^{1/2-{\varepsilon }}\) and all \(n\le T/c\), we have
and
where \(N_0\) is given by (40) with \({\delta }_0=c^{1/2-\nu }/(e^{\sigma }-1)\).
Moreover, in the case \(\eta =1\), for all \(T\in (0,\infty )\), \({\varepsilon }\in (0,1/5)\), \(\nu \in (0,{\varepsilon }/2)\) and \(p \in [2,\infty ) \), there is a constant \(C=C({\Lambda },T,{\varepsilon },\nu ,p)\in [1,\infty )\) with the following property. For all \(c\in (0,1]\), all \(r,e^{\sigma }\ge 1+c^{1/5-{\varepsilon }}\) and all \(n\le T/c\), we have
and
where \(N_0\) is given by (40) with \({\delta }_0=c^{1/2-\nu }/(e^{\sigma }-1)^{3/2}\).
Proof
Let us first consider \(\eta \in [0,1)\). It suffices to prove the result for \(p> 1+1/(2{\varepsilon })\). The argument follows along almost exactly the same lines as that used to establish (34). The only difference is that we delete the D operator, which has the effect of removing the \(k^2\) factor from some of the summations, and it is necessary to consider separately the constant term of the Laurent expansion. This gives
and
Similarly, for the second martingale term we obtain
where we used the bound on \(\Psi _n\) from Proposition 6.1. Finally, for the remainder term, we find
and
On assembling these bounds, and simplifying using our constraints on r and \({\sigma }\), we obtain (47) and (48).
As in the proof of Proposition 6.1, in the case \(\eta =1\), we do not benefit from the push-out of \(r_n=re^{c(1-\eta )n}\), and the bound on \(\Psi _n\) is weaker. After some straightforward modifications, for p sufficiently large we obtain
On assembling these bounds, and simplifying using our constraints on r and \({\sigma }\), we obtain (49). Similarly
giving (50). \(\square \)
Proof of Theorem 1.1
The argument is a variation of that for Proposition 6.2. We do it when \(\eta <1\); the \(\eta =1\) case is similar. Let \({\Omega }_0\), N, K, r(k), \(\rho (k)\) and \({\lambda }\) be as in the proof of Proposition 6.2. Define
where
and C is the larger of the constant in (47) and that in (48). Then \({\mathbb {P}}({\Omega }_1)\le 2c^m\) and the desired uniform estimate on \(\Phi _n\) holds on \({\Omega }_1\), by the argument used in the proof of Proposition 6.2. In arriving at this estimate we use the fact that for \(r\ge 1+c^{1/2}\) we have \((1+\log (r/(r-1)))^{1/2}\le c^{-{\varepsilon }}\) for all sufficiently small \(c>0\), for all \({\varepsilon }>0\). \(\square \)
7 Fluctuation scaling limit for ALE\((\eta )\)
In this section, we show that the fluctuations of ALE\((\eta )\) for \(\eta \in (-\infty ,1]\) are of order \(\sqrt{c}\), and we determine the distribution of the rescaled fluctuations.
Let \((\Phi _n)_{n\ge 0}\) be an ALE\((\eta )\) process with basic map F and regularization parameter \({\sigma }\). Assume that F has capacity \(c\in (0,1]\) and regularity bound \({\Lambda }\in [0,\infty )\). We consider the limit \(c\rightarrow 0\) with \({\sigma }\rightarrow 0\), and will show weak limits which are otherwise uniform in F, subject to the given regularity bound. We embed in continuous time by setting \(n(t)=\lfloor t/c\rfloor \) and defining
We will show that the process of analytic functions \(({{\tilde{\Phi }}}(t,.)/\sqrt{c})_{t\ge 0}\) converges weakly to a Gaussian limit.
Let us define the metric spaces our processes will live in. To start with, let \(D[0,\infty )\) denote the space of complex-valued càdlàg processes equipped with the Skorohod metric \({\mathbf {d}}\). To discuss weak convergence of sequences of Laurent coefficients, it is convenient to introduce the product space \(D[0,\infty )^{{{\mathbb {Z}}}^+}\) of sequences of complex-valued càdlàg processes, with the metric of coordinate-wise convergence, given by
Finally, to talk about convergence of functions, let \({{\mathcal {H}}}\) denote the space of analytic functions on \(D_0=\{|z|>1\}\) with limits at \(\infty \), equipped with the metric of uniform convergence on compacts in \(D_0\cup \{\infty \}\), given by
We let \(D_{{\mathcal {H}}}[0,\infty )\) denote the space of \({{\mathcal {H}}}\)-valued càdlàg processes equipped with the associated Skorohod metric \({\mathbf {d}}_{{\mathcal {H}}}\). Then all the above spaces are complete separable metric spaces [3], and \(({{\tilde{\Phi }}}(t,.)/\sqrt{c})_{t\ge 0}\) lies in \(D_{{\mathcal {H}}}[0,\infty )\).
To state our main fluctuation result, we now define the limiting fluctuation field on \(C_{{\mathcal {H}}}[0,\infty )\), the space of continuous processes with values in \({{\mathcal {H}}}\). Let \((A(\cdot ,k))_{k\ge 0}\) denote a sequence of independent complex Ornstein–Uhlenbeck processes, solutions to
where \((B_k)_{k\ge 0}\) are independent complex Brownian motions. Thus \((A(\cdot ,k))_{k\ge 0}\) is a zero-mean Gaussian process, with covariance given for \(s,t\in [0,\infty )\) by
Here, on the left, we use the tensor product from \({{\mathbb {R}}}^2\). Thus
By standard estimates, the following series both converge almost surely, uniformly on compacts in \((t,z)\in [0,\infty )\times (D_0\cup \{\infty \})\)
Hence \({{\mathcal {F}}}=({{\mathcal {F}}}(t,.):t\ge 0)\) and \(\xi =(\xi (t,.):t\ge 0)\) are continuous random processes in \({{\mathcal {H}}}\). It is straightforward to check that
and \(\xi \) is the analytic extension in \(D_0\) of space-time white noise on the unit circle, so \({{\mathcal {F}}}\) satisfies the stochastic PDE (6). In this section we prove Theorem 1.2 by showing that \({{\tilde{\Phi }}}/\sqrt{c}\rightarrow {{\mathcal {F}}}\) in distribution on \(D_{{\mathcal {H}}}[0,\infty )\).
7.1 Discarding lower order fluctuations
Our analysis is based on the decomposition (31), which we rewrite in continuous time, with obvious notation as
Define \({\tilde{{{\mathcal {M}}}}}^0(t,z) = {\beta }^{-1} {\tilde{{{\mathcal {M}}}}}(t,z)\), where \({\beta }\) is defined in Proposition A.1, and recall that . In a first step, we will show that \({\tilde{{{\mathcal {M}}}}}^0\) is the only term that matters in the limiting fluctuations.
Lemma 7.1
Under the hypotheses of Theorem 1.2, for all \(t\ge 0\), we have
in probability as \(c\rightarrow 0\), uniformly in \({\sigma }\) and F.
Proof
Fix \({\varepsilon }\in (0,1/6)\) as in the statement of Theorem 1.2 and set
We first consider the case \(\eta \in (-\infty ,1)\). Recall that in the proof of Proposition 6.3 we showed that, for all \(T\in [0,\infty )\), \(p > 1+1/(2{\varepsilon })\) and \(r>1\), there is a constant \(C=C({\Lambda },\eta ,T,{\varepsilon },p,r)<\infty \) such that for all \(c\le 1/C\), \(e^{\sigma }\ge 1+c^{1/2-{\varepsilon }}\) and \(n\le T/c\), we have
Here we have used that \(|{\beta }-1|\le {\Lambda }\sqrt{c}\). Note that under the further restriction \({\sigma }\ge c^{1/4-{\varepsilon }}\),
By arguments from the proof of Proposition 6.2, it follows that
in probability as \(c\rightarrow 0\), uniformly on compacts in \((t,z)\in [0,T]\times (D_0\cup \{\infty \})\), and uniformly in \({\sigma }\) and F subject to the given constraints. On the other hand, by Proposition 6.2, we know that \({\mathbb {P}}(N_0<T/c)\rightarrow 0\) in the same limiting regime. The claim of the lemma follows.
The case \(\eta =1\) is handled by the same argument with straightforward modifications. \(\square \)
7.2 Covariance structure
We now focus on the leading order fluctuations, coming from the martingale term
where
Let \((\Theta _n^u)_{n\ge 1}\) be a sequence of independent uniform random variables in \([0,2\pi )\). Define for \(|z|>1\)
where \({{\mathcal {F}}}_{n-1}^u \) is the \(\sigma \)-algebra generated by . Expanding in Laurent series, we find
where
Recalling that the operator P acts diagonally on Laurent coefficients, set
where
and define for \(t\ge 0\)
Let \({{\tilde{{{\mathcal {M}}}}}}^u(t,z)\) be defined as in (52) with \(M^0_j\) replaced by \(M_j^u\). Then we have
By an elementary calculation, we obtain
from which
Recall that for \(\eta \in [0,1]\)
By some straightforward estimation, recalling that \({\sigma }\rightarrow 0\), we have
Note that if for some \(t>0\), and k is fixed, then the right hand side converges to 0 as \(c\rightarrow 0\). In the case \(\eta <0\), define \(p_0(k)\) exactly as above (note that this differs from the definition in (29)). Provided c is taken sufficiently small that \({\sigma }-c-c|\eta |>0\), we have
and hence \(p_0(k) \ge 1\). A straightforward estimation therefore gives
Hence
Now, for any \(k,k'\ge 0\) and \(s,t\in [0,\infty )\) with \(s\le t\), the following limit holds in probability as \(c\rightarrow 0\), uniformly in \({\sigma }\) and F,
To see this, recall that by Proposition 6.1 for all \(m\in {{\mathbb {N}}}\) there exists a constant \(C=C({\Lambda },\eta , {\varepsilon },m,T)<\infty \) such that, for \(c\le 1/C\) and \({\delta }_0\) defined as in the proof of Lemma 7.1, there exists an event \({\Omega }_0\) of probability at least \(1-c^m\) on which, for all \(n\le T/c\) and all \({\theta }\in [0,2\pi )\),
and hence, by (17), \(|h_n({\theta })-1|\le 63{\delta }_0\). Then, on \({\Omega }_0\), for \(c\le 1/C\) and \(t\le T\),
Since \(c{\delta }_0n(s)\rightarrow 0\) as \(c\rightarrow 0\), this shows the claimed limit in probability.
7.3 Convergence of Laurent coefficients
We now show that the processes of rescaled Laurent coefficients \(({\tilde{A}}(\cdot ,k))_{k\ge 0}\) of \({{\tilde{{{\mathcal {M}}}}}}^0(t,z)\) converge weakly to those of the limiting process \({{\mathcal {F}}}\).
Theorem 7.2
Under the hypotheses of Theorem 1.2, in the limit \(c\rightarrow 0\) and \({\sigma }\rightarrow 0\) and uniformly in the basic map F, we have
in distribution in \((D[0,\infty )^{{{\mathbb {Z}}}^+},{\mathbf {d}}^{{{\mathbb {Z}}}^+})\).
Proof
It will suffice to show that the finite-dimensional distributions of \(({\tilde{A}}(\cdot ,k))_{k\ge 0}\) converge to those of \((A(\cdot ,k))_{k\ge 0}\), and that for each fixed k the processes \({\tilde{A}}(\cdot ,k)\) are tight in \((D[0,\infty ),{\mathbf {d}})\).
We start by proving convergence of finite-dimensional distributions. Fix positive integers K and m and pick arbitrary \(0\le t_1<t_2<\dots <t_m\). We aim to show the following convergence in distribution
Write \(n_i\) in place of \(n(t_i)\) for brevity. Fix real-linear maps \({\alpha }_{k,l}:{{\mathbb {C}}}\rightarrow {{\mathbb {R}}}\), for \(k=1,\dots ,K\) and \(l=1,\dots ,m\) and consider the real-valued random variables given by
Then
It is readily verified that \((X_{j,n_m}:j=1,\dots ,n_m)\) is a martingale difference sequence with respect to the filtration \(({{\mathcal {F}}}_j:j=1,\dots ,n_m)\). Set
and note that \(\Sigma \) is the variance of
We will use the following martingale central limit theorem [3, Theorem 18.1].
Theorem 7.3
Suppose given, for each \(n\in {{\mathbb {N}}}\), a martingale difference array \((X_{j,n}:j=1,\dots ,n)\) with filtration \(({{\mathcal {F}}}_{j,n}:j=1,\dots ,n)\). Assume that, for some \(\Sigma \in [0,\infty )\) and for all \({\varepsilon }>0\), the following two conditions hold in the limit \(n\rightarrow \infty \):
-
(i)
\(\displaystyle \sum _{j=1}^n{{\mathbb {E}}}\left( X_{j,n}^2|{{\mathcal {F}}}_{j-1,n}\right) \rightarrow \Sigma \) in probability,
-
(ii)
\(\displaystyle \sum _{j=1}^n{{\mathbb {E}}}(|X_{j,n}|^2\,1_{\{|X_{j,n}|>{\varepsilon }\}})\rightarrow 0\).
Then \(\displaystyle \sum _{j=1}^nX_{j,n}\rightarrow {{\mathcal {N}}}(0,\Sigma )\) in distribution as \(n\rightarrow \infty \).
We can apply this theorem to the limit \(c\rightarrow 0\) and the martingale difference array \((X_{j,n_m}:j=1,\dots ,n_m)\), with \(n_m=n(t_m)=\lfloor t_m/c\rfloor \). We have
in probability as \(c\rightarrow 0\) by (54) and (53), which proves (i). To see (ii) note that
from which, for arbitrary \({\varepsilon }>0\) and a constant C allowed to depend on the constants \({\alpha }_{k,l}\), K and m, for all sufficiently small c,
Since the linear maps \({\alpha }_{k,l}\) were arbitrary, this shows convergence of the finite-dimensional distributions of \(({\tilde{A}}(t,k))_{k\ge 0}\) to those of \((A(t,k))_{k\ge 0}\).
It remains to prove tightness. We will show that, for all \(p\in [2,\infty )\), all \(k\ge 0\) and all \(T\in [0,\infty )\), there is a constant \(C=C(p,\eta ,k,T)<\infty \) such that, for all \(s,t\in [0,T]\),
Since we may choose \(p>2\), this implies tightness, by a standard criterion.
Recall that
and that \(({\hat{M}}^0_j(k):j\ge 0)\) is a martingale difference sequence with \(|{\hat{M}}^0_j(k)|\le 2ce^{cj}\). Also \(0\le p(k)\le 1\) and, estimating as above,
Fix \(s,t\in [0,T]\) with \(s\le t\) and note that \(n(t)-n(s)\le 1+(t-s)/c\). Then
Using that
and combining
with (56), by Burkholder’s inequality we find that, for some constant \(C=C(p,\eta ,k,T)<\infty \),
The asymptotic Hölder condition (55) follows. \(\square \)
7.4 Convergence as an analytic function
In this section we deduce the convergence of \({{\tilde{{{\mathcal {M}}}}}}^0(t,z)\) from that of the Laurent coefficients, thus concluding the proof of Theorem 1.2. To this end, set
These define processes in \(D_{{\mathcal {H}}}[0,\infty )\). For any \(T>0\) let \(D_{{\mathcal {H}}}[0,T]\) denote the space of \({{\mathcal {H}}}\)-valued càdlàg processes on [0, T]. Then \({{\tilde{{{\mathcal {F}}}}}},{{\mathcal {F}}}\) define processes in \(D_{{\mathcal {H}}}[0,T]\) by restriction, for all \(T>0\). For any \(r>1\) let \({{\mathcal {H}}}_r\) denote the space of analytic functions on \(\{|z|\ge r\}\) with limits at \(\infty \), equipped with the metric
We let \(D_{{{\mathcal {H}}}_r}[0,T]\) denote the space of càdlàg processes with values in \({{\mathcal {H}}}_r\) equipped with the associated Skorohod metric \({\mathbf {d}}_{T,r}\). To show that \({{\tilde{{{\mathcal {F}}}}}}\) converges to \({{\mathcal {F}}}\) in distribution on \((D_{{\mathcal {H}}}[0,\infty ),{\mathbf {d}}_{{\mathcal {H}}})\), it suffices to show that, for any \(T>0\) and \(r>1\), the process \({{\tilde{{{\mathcal {F}}}}}}\) converge to \({{\mathcal {F}}}\) in distribution on \((D_{{{\mathcal {H}}}_r}[0,T],{\mathbf {d}}_{T,r})\) as \(c\rightarrow 0\) (see Billingsley [3]). This in turn follows from the lemma below.
Lemma 7.4
For any \(T>0\), \(r>1\) and \({\delta }={\delta }(r) \in [0,1]\) such that \(e^{-2{\delta }}r>1\), we have that for any \({\varepsilon }>0\)
Proof
Fix \({\varepsilon },T,r,{\delta }\) as in the statement, and partition the interval [0, T] into sub-intervals \(I_l=[(l-1){\delta },l{\delta })\) for \(1\le l\le \lceil T/{\delta }\rceil \). Then
and so
Recall that
which shows that the process \((p(k)^{-n(t)}\tilde{A}(t,k))_{t\ge 0}\) is a martingale for each \(k\ge 0\), with
Doob’s \(L^2\) inequality then gives
for some positive constant C, depending on T, changing from line to line. In the last inequality we have used that \(p(k)\le 1\) and \(cn(l{\delta })\le T+1\). Noting that \(n(l{\delta })-n((l-1){\delta })\le 1+{\delta }/c\), and that \(p(k)\ge e^{-c(k+1)}\) for \(\eta \in [0,1]\), we find
for \({\delta }\ge c\). Plugging this into (58) gives
and hence
as \(K\rightarrow \infty \) since \(e^{-2{\delta }}r>1\). If \(\eta <0\), the result follows from the same argument using that, for c small enough that \({\sigma }-c-c|\eta |>0\), we have
\(\square \)
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Acknowledgements
We would like to thank Alan Sola for helpful discussions and for suggestions on an earlier draft of this paper. Amanda Turner would like to thank the University of Geneva for a visiting position in 2019–2020 during which time this work was completed. We would like to thank the anonymous referees for their helpful suggestions, which greatly improved the presentation of this work.
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Appendices
Appendices
Particle estimates
Let \(c\in (0,\infty )\) and \({\Lambda }\in [0,\infty )\). Recall that we say a univalent function F from \(D_0=\{|z|>1\}\) into \(D_0\) has capacity c and regularity \({\Lambda }\) if it satisfies condition (4), that is to say, for all \(z\in D_0\),
We show that this in fact implies a similar condition for F but with better decay as \(z\rightarrow \infty \). Then we will give some explicit examples of suitable maps F. Finally, we will show that (4) holds whenever the corresponding particle is not too flat. Only Sect. A.1 is used in the paper.
1.1 Precise form of the particle hypothesis
Our particle hypothesis (4) can be reformulated more precisely in terms of the coefficient \(a_0\) in the Laurent expansion (1).
Proposition A.1
Suppose that F satisfies (4) and set \({\beta }=a_0/(2c)\). Then \(|{\beta }-1|\le {\Lambda }\sqrt{c}/2\) and, for all \(z\in D_0\),
Proof
Set
Then g is analytic in \(D_0\) and \(g(z)\rightarrow a_0-2c=2c({\beta }-1)\) as \(z\rightarrow \infty \). Condition (4) implies
On letting \(z\rightarrow \infty \), we see that \(2c|{\beta }-1|\le {\Lambda }c^{3/2}\) so \(|{\beta }-1|\le {\Lambda }\sqrt{c}/2\). Consider
Then h is analytic in \(D_0\) and bounded at \(\infty \). We have
so
whenever \(|z|=2\). Then, by the maximum principle, for all \(|z|\ge 2\), we have \(|h(z)|\le 6{\Lambda }c^{3/2}\) and hence
On the other hand (60) implies the same inequality for \(1<|z|<2\). \(\square \)
Note that (59) with \(|{\beta }-1|\le {\Lambda }\sqrt{c}/2\) implies (4) with \({\Lambda }\) replaced by \(7{\Lambda }\). Thus the two conditions are equivalent up to adjustment of the constant by a universal factor.
1.2 Spread out particles
Consider for \({\gamma }\in {{\mathbb {C}}}\) the map on \(D_0\) given by
It is straightforward to check that \(F_{c,{\gamma }}\) is univalent into \(D_0\) if and only if
Then \(F_{c,{\gamma }}\) has capacity c and, since
and
we see that \(F_{c,{\gamma }}\) has regularity \({\Lambda }=2|{\gamma }-1|/\sqrt{c}\). The corresponding particles \(P_{c,{\gamma }}\) are spread all around the unit circle, as illustrated in the rightmost particle in Fig. 1. When \({\gamma }={\gamma }(c)\) we find \(F'(1)=0\) so \(P_{c,{\gamma }(c)}\) has the form of a cusp with endpoint F(1). Moreover, in the limit \(c\rightarrow 0\) with \({\gamma }={\gamma }(c)\), the regularity constant \({\Lambda }\) stays bounded and \(\log F(1)\sim \sqrt{2c}\), so the endpoint lies at distance \(F(1)-1\sim \sqrt{2c}\) from the unit circle.
1.3 Small particles of a fixed shape
The following proposition shows that our condition (4) holds generically for particles attached near 1 which are not too flat. In particular, it shows that, for particles of a fixed shape, such as slits or disks, attached to the unit circle at 1, in the small diameter limit \({\delta }\rightarrow 0\), the capacity \(c\rightarrow 0\) while the regularity constant \({\Lambda }\) stays bounded, which is the regime in which our limit theorems apply.
Proposition A.2
There is a constant \(C<\infty \) with the following property. Let P be a basic particle such that, for some \({\delta }_0,{\delta }\in (0,1]\),
-
(a)
\(|z|=1+{\delta }_0\) for some \(z\in P\),
-
(b)
\(|z-1|\le {\delta }\) for all \(z\in P\).
Then P has capacity c satisfying \( {\delta }_0^2/C\le c\le C{\delta }^2. \) Moreover, if \({\delta }\le 1/C\), then P has regularity \( {\Lambda }\le C{\delta }/{\delta }_0. \)
Proof
The bounds on c are well known. The lower bound relies on Beurling’s projection theorem and a comparison with the case of a slit particle. The upper bound follows from a comparison with the case \(P_{\delta }=S_{\delta }\cap D_0\), where \(S_{\delta }\) is the closed disk whose boundary intersects the unit circle orthogonally at \(e^{\pm i{\theta }_{\delta }}\) with \({\theta }_{\delta }\in [0,\pi ]\) is determined by \(|e^{i{\theta }_{\delta }}-1|={\delta }\). See Pommerenke [16].
We turn to the bound on \({\Lambda }\). First we will show, for \(a=15{\delta }\le \pi \), we have
Then we will show that, if \(c\in (0,1]\) and (61) holds with \(a\in (0,\pi /2)\), then, for all \(|z|>1\),
The desired bound on \({\Lambda }\) then follows from (61) and (62) and the lower bound on c.
We can write
where u and v are harmonic functions in D with \(u(z)\rightarrow c\) and \(v(z)\rightarrow 0\) as \(z\rightarrow \infty \). Since F maps into \(D_0\), we have \(u(e^{i{\theta }})\ge 0\) for all \({\theta }\in [0,2\pi )\). We have to show that \(u(e^{i{\theta }})=0\) whenever \(|{\theta }|\in [a,\pi ]\). Set
where B is a complex Brownian motion. Consider the conformal map f of \(D_0\) to the upper half-plane \({\mathbb {H}}\) given by
Set \(b=f(e^{-i{\theta }_{\delta }})=\sin {\theta }_{\delta }/(1+\cos {\theta }_{\delta })\). Since \({\delta }\le 1\), we have \({\theta }_{\delta }\le {\delta }\pi /3\) and then \(b\le 2\pi {\delta }/9\). By conformal invariance,
Hence \(p_{\delta }\le 4b/\pi \le 8{\delta }/9\).
Now \(e^{i\pi }\) is not a limit point of P so \(e^{i\pi }=F(e^{i(\pi +{\alpha })})\) for some \({\alpha }\in {{\mathbb {R}}}\). Then \(u(e^{i(\pi +{\alpha })})=0\) and we can and do choose \({\alpha }\) so that \({\alpha }+v(e^{i(\pi +{\alpha })})=0\). Set
Then \({\theta }^-\le {\theta }^+\). It will suffice to show that \(|{\theta }^\pm |\le 15{\delta }\). For \({\theta }\in [{\theta }^-,{\theta }^+]\), we have \(F(e^{i{\theta }})\in S_{\delta }\) so \(|{\theta }+v(e^{i{\theta }})|\le {\theta }_{\delta }\). Set \(P^*=\{F(e^{i{\theta }}):{\theta }\in [{\theta }^-,{\theta }^+]\}\). Then \(P^*\subseteq S_{\delta }\) so, by conformal invariance,
On the other hand, for \({\theta },{\theta }'\in [{\theta }^+,{\theta }^-+2\pi ]\) with \({\theta }\le {\theta }'\), by conformal invariance,
so v is non-decreasing on \([{\theta }^+,{\theta }^-+2\pi ]\), and so
Hence
and similarly \({\theta }^--{\alpha }\ge -2\pi p_{\delta }-{\theta }_{\delta }\). So we obtain, for all \({\theta }\in [{\theta }^-,{\theta }^+]\),
Since v is continuous and is non-decreasing on the complementary interval, this inequality then holds for all \({\theta }\). Now v is bounded and harmonic in \(D_0\) with limit 0 at \(\infty \), so
Hence
and so \(|{\theta }^\pm |\le 3{\theta }_{\delta }+4\pi p_{\delta }\le 41\pi {\delta }/9\le 15{\delta }\), as required.
We turn to the proof of (62). Assume now that \(u(e^{i{\theta }})=0\) whenever \(|{\theta }|\in [a,\pi ]\). Since u is harmonic, we have
and, for all \(|z|>1\),
Hence, using that \(u(e^{i{\theta }}) \ge 0\) for all \({\theta }\in [0,2\pi )\),
and, since \(v(z)\rightarrow 0\) as \(z\rightarrow \infty \), a standard argument using the Cauchy–Riemann equations then shows that
This gives the claimed estimate in the case where \(|z-1|\le 2a\). It remains to consider the case where \(|z-1|>2a\). Let \(\alpha , \rho \) be defined by
Then \(|{\alpha }|\le a\) and \(\rho \in [\cos a,1)\). Now
so
The standard argument mentioned above now allows us to deduce that
Hence, by a simple calculation,
Note that
Since \(|z-1|>2a\), we have \(|e^{-i{\alpha }}z-1|\ge |z-1|/2\), so we can deduce from (63) that
\(\square \)
Preliminary estimates
In this section, we gather together some standard results which are used in our proofs.
1.1 Martingale estimates
We recall the following martingale inequality, due to Burkholder.
Theorem B.1
([4], Theorem 21.1) Let \((X_n)_{n\ge 0}\) be a martingale with respect to the filtration \(({{\mathcal {F}}}_n)_{n\ge 0}\). For \(n\ge 1 \) write \(\Delta _n = X_n - X_{n-1}\) for the increment process, and define
Then for any \(p \in [2,\infty ) \) there exists a constant \(C=C(p)\) such that for all \(n\ge 1\)
1.2 Operator estimates
We note some \(L^p\)-estimates on operators which act on the set of analytic functions f on \(\{|z|>1\}\) which are bounded at \(\infty \), and hence have a Laurent expansion
Firstly, for the operator \(Df(z)=zf'(z)\), by a standard argument using Cauchy’s integral formula, there is an absolute constant \(C<\infty \) such that, for all \(p\in {{\mathbb {N}}}\) and \(1<\rho <r\),
Secondly, let L be an operator which acts as multiplication by \(m_k\) on the the kth Laurent coefficient. Thus
Assume that there exists a finite constant \(M>0\) such that, for all \(k\ge 0\),
and, for all integers \(K\ge 0\),
The Marcinkiewicz multiplier theorem [21, Vol. II, Theorem 4.14] then asserts that, for all \(p\in (1,\infty )\), there is a constant \(C=C(p)<\infty \) such that, for all \(r>1\),
We will use also the following estimate. Write \(\Vert L\Vert _{p,\rho \rightarrow r}\) for the smallest constant K such that
for all analytic functions f on \(\{|z|>1\}\) bounded at \(\infty \).
Proposition B.2
Let f and g be analytic in \(\{|z|>1\}\) and bounded at \(\infty \). Set \(f_{\theta }(z)=f(e^{-i{\theta }}z)\). Let L be a multiplier operator and define
Then, for all \(r,\rho >1\), we have
Proof
We can write
Then
so
where \(\tau _kg(z)=z^{-k}g(z)\). Hence
\(\square \)
Computations of first order and error estimates
In this section, we provide the detailed calculations behind the estimates stated in Sect. 3. We only explicitly state estimates for \(\eta \in [0,1]\), taking advantage of the fact that certain constants can be chosen uniformly over such values of \(\eta \). Similar estimates hold for \(\eta \in (-\infty , 0)\), and we leave the necessary adjustments to the reader. Furthermore, throughout this section we assume that . This assumption can be relaxed at the cost of the absolute constants.
1.1 Estimates on the attachment measure \(h_n({\theta })\)
We begin by obtaining estimates on \(h_n({\theta })\), defined in (3). By an elementary calculation, for all \(\eta \in (-\infty ,1]\) and \(w\in {{\mathbb {C}}}\setminus \{0\}\), we can write
with
for some constant \(C < \infty \) depending only on \(\eta \). We will see below that for all \(\eta \in [0,1]\); the case \(\eta <0\) requires minor adjustments to take into account the dependence of C on \(\eta \), which we leave to the reader. Take
to obtain
where
(Here and throughout the remainder of this section, n is fixed and the dependence of error terms on n is suppressed in the notation). Then
where
Here we used the fact that
which holds because \({{\tilde{\Phi }}}'_{n-1}(z)\) is analytic in \(\{|z|>1\}\) and vanishes as \(z\rightarrow \infty \). Hence
where
Recall the definition of \(N_0\) from (16). Then, for all \(n\le N_0\) and all \({\theta }\in [0,2\pi )\),
so
and
Using (66) to bound \(|1+{\varepsilon }_3|\) directly,
and
This estimate, together with (67), is used to justify the bounds in (17) and (18).
1.2 Estimates on the increment \(\Delta _n({\theta },z)\)
We now move to analysing the increment \(\Delta _n({\theta },z)\), defined in (9). Recall from (23) that
where \(m_n\) is defined in (22). By (21) we can write
where
Note that \({\varepsilon }_6({\theta },\infty )={\varepsilon }_7({\theta },\infty )=0\) and
By some straightforward estimation, we obtain a constant \(C=C({\Lambda })<\infty \) such that, for all \(c\in (0,1]\) and all \(|z|>1\),
and, for \(|z|\ge 1+\sqrt{c}\),
Hence, for \(|z|\ge 1+\sqrt{c}\),
This estimate, together with (69), is used to justify the bound (24). We combine (67) and (21) to obtain
By Cauchy’s integral formula
So we obtain
where
Now suppose . Then, using (69) and (71), for \(|z|=r\) with \(r\ge 1+\sqrt{c}\), we obtain (27).
1.3 A more refined decomposition
The estimate (24), is not sufficiently tight for all our needs. In this section, we give a decomposition of \(w_n\), which can be used for more refined estimates.
Set
Then, for \(|z|=r\ge 1+\sqrt{c}\),
It follows that we can write
where
We will write
Recall the interpolation (20), which we can write as
We will use the following Taylor expansion
where \(m\in {{\mathbb {N}}}\). Hence
and so
Proofs of second order bounds
1.1 Estimation of the second martingale term
In this section, we give the proof of Lemma 5.2, which bounds the second martingale term \({{\mathcal {W}}}_n(z)\) in the decomposition (32) of the differentiated fluctuation process, which is given by
By Burkholder’s inequality, for all \(p\in [2,\infty )\), there is a constant \(C=C(p)<\infty \) such that
where
Then, on taking the \(\Vert \cdot \Vert _{p/2,r}\)-norm, we deduce that
While it is possible to use the estimate (24) to bound this expression, the bound is only sufficient to prove our final result for \({\sigma }\gg c^{1/3}\). In order to obtain a bound that works all the way down to \({\sigma }\gg c^{1/2}\), we need the refined decomposition (73), for some \(m \in {{\mathbb {N}}}\) which we will choose later. Define
where
Then \(W_n=U^0_n+U^1_n+U_n^2\) so, with obvious notation,
and so
We estimate the terms on the right. First, for \(j\le N_0\), we have
We start with \(i=0\). Then for all \(|z|=r\) and \(j\le N_0\),
Here, and in what follows, \(C<\infty \) is a constant, which may only depend on \(\Lambda \), \(\eta \), m and p, and which may change from line to line.
Next, consider \(i=1\). Note that
We use the estimates (64),(65) and (33) to see that, for \(\rho =(r+1)/2\) and \({\tilde{\rho }}=(3r+1)/4\),
It follows from (4) that, for \(|z|\ge 1+\sqrt{c}\),
Hence
where in the last line we used that \(r\ge 1+\sqrt{c}\).
Finally we turn to \(i=2\). Then, for \(|z|=r\ge 1+\sqrt{c}\) and \({{\tilde{\rho }}}=(3r+1)/4\), using Jensen’s inequality and the inequalities (64) and (33),
Note that
We use the inequalities (4), \(|F_{s,{\theta }}({{\tilde{\rho }}})|\ge {{\tilde{\rho }}}\) and (70) to see that, for \(\rho =(r+1)/2\ge 1+\sqrt{c}\),
Hence we obtain
and then
Hence
and hence
Fix \({\varepsilon }\in (0,1/2)\) and assume that \(r\ge 1+c^{1/2-{\varepsilon }}\). Then, on choosing \(m=\lceil 1/(8{\varepsilon }) \rceil \), we obtain, for all \(p\ge 2\),
where C now depends on \({\varepsilon }\), in places where before it depended on m.
On combining our estimates, we have shown that, for \(j\le N_0\), we have
We take the \(L^{p/2}({\mathbb {P}})\)-norm to deduce that
When bounding , it is sufficient to take \(m=0\) in the decomposition above. In this case \(u_n^1({\theta },z)=0\), so
By (64) and (33), now, similarly to above,
For \(i=0\),
and hence, by the same argument as in the proof of Lemma 5.1,
For \(i=2\),
Now, by the computation above,
Hence
Then, on using this estimate in (74), we obtain (38).
Now suppose \(\eta < 1\). It suffices to prove the result for p sufficiently large, so assume \(p > 1+1/(2{\varepsilon })\). We use our constraint on r, the monotonicity of norms (8), and same integral comparison as in (37) to deduce from (38) the estimate
The desired result follows, using our assumption on p.
The case when \(\eta =1\) is similar.
1.2 Estimation of the remainder term
The remainder term in the decomposition (32) of the differentiated fluctuation process is given by
In this section, we give the proof of Lemma 5.3, which bounds this quantity.
By the triangle inequality
For \(n\le N_0\) and \(|z|=r>1+\sqrt{c}\), we obtained in (27) the estimate
We bound the \(\Vert \cdot \Vert _{p,r}\)-norm of the final term on the right as follows:
We used the change of variable \({\tau }_m={\theta }_m-u\) and the identity
in the second equality. Then we used Hölder’s inequality and the fact that \(|F_{s,{\tau }_m}(r)|\ge r\) for the first inequality, and we used
for the second inequality. Hence, for all \(p\in {{\mathbb {N}}}\),
We use (64) and (33) to obtain (39).
Now suppose \(n \le T/c\) for some constant T. If \(\eta <1\), the result follows, using the integral comparison
The argument when \(\eta =1\) is similar.
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Norris, J., Silvestri, V. & Turner, A. Scaling limits for planar aggregation with subcritical fluctuations. Probab. Theory Relat. Fields 185, 185–250 (2023). https://doi.org/10.1007/s00440-022-01141-0
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DOI: https://doi.org/10.1007/s00440-022-01141-0