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The latest articles from The Annals of Probability on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTWed, 16 Mar 2011 09:23 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Uniform convergence of Vapnik–Chervonenkis classes under ergodic sampling
http://projecteuclid.org/euclid.aop/1278593952
<strong>Terrence M. Adams</strong>, <strong>Andrew B. Nobel</strong><p><strong>Source: </strong>Ann. Probab., Volume 38, Number 4, 1345--1367.</p><p><strong>Abstract:</strong><br/>
We show that if $\mathcal{X}$ is a complete separable metric space and $\mathcal{C}$ is a countable family of Borel subsets of $\mathcal{X}$ with finite VC dimension, then, for every stationary ergodic process with values in $\mathcal{X}$ , the relative frequencies of sets $C\in\mathcal{C}$ converge uniformly to their limiting probabilities. Beyond ergodicity, no assumptions are imposed on the sampling process, and no regularity conditions are imposed on the elements of $\mathcal{C}$ . The result extends existing work of Vapnik and Chervonenkis, among others, who have studied uniform convergence for i.i.d. and strongly mixing processes. Our method of proof is new and direct: it does not rely on symmetrization techniques, probability inequalities or mixing conditions. The uniform convergence of relative frequencies for VC-major and VC-graph classes of functions under ergodic sampling is established as a corollary of the basic result for sets.
</p>projecteuclid.org/euclid.aop/1278593952_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTLargest entries of sample correlation matrices from equi-correlated normal populationshttps://projecteuclid.org/euclid.aop/1571731453<strong>Jianqing Fan</strong>, <strong>Tiefeng Jiang</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 5, 3321--3374.</p><p><strong>Abstract:</strong><br/>
The paper studies the limiting distribution of the largest off-diagonal entry of the sample correlation matrices of high-dimensional Gaussian populations with equi-correlation structure. Assume the entries of the population distribution have a common correlation coefficient $\rho >0$ and both the population dimension $p$ and the sample size $n$ tend to infinity with $\log p=o(n^{\frac{1}{3}})$. As $0<\rho <1$, we prove that the largest off-diagonal entry of the sample correlation matrix converges to a Gaussian distribution, and the same is true for the sample covariance matrix as $0<\rho <1/2$. This differs substantially from a well-known result for the independent case where $\rho =0$, in which the above limiting distribution is an extreme-value distribution. We then study the phase transition between these two limiting distributions and identify the regime of $\rho $ where the transition occurs. If $\rho $ is less than, larger than or is equal to the threshold, the corresponding limiting distribution is the extreme-value distribution, the Gaussian distribution and a convolution of the two distributions, respectively. The proofs rely on a subtle use of the Chen–Stein Poisson approximation method, conditioning, a coupling to create independence and a special property of sample correlation matrices. An application is given for a statistical testing problem.
</p>projecteuclid.org/euclid.aop/1571731453_20191022040429Tue, 22 Oct 2019 04:04 EDTDynkin isomorphism and Mermin–Wagner theorems for hyperbolic sigma models and recurrence of the two-dimensional vertex-reinforced jump processhttps://projecteuclid.org/euclid.aop/1571731454<strong>Roland Bauerschmidt</strong>, <strong>Tyler Helmuth</strong>, <strong>Andrew Swan</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 5, 3375--3396.</p><p><strong>Abstract:</strong><br/>
We prove the vertex-reinforced jump process (VRJP) is recurrent in two dimensions for any translation invariant finite-range initial rates. Our proof has two main ingredients. The first is a direct connection between the VRJP and sigma models whose target space is a hyperbolic space $\mathbb{H}^{n}$ or its supersymmetric counterpart $\mathbb{H}^{2|2}$. These results are analogues of well-known relations between the Gaussian free field and the local times of simple random walk. The second ingredient is a Mermin–Wagner theorem for these sigma models. This result is of intrinsic interest for the sigma models and also implies our main theorem on the VRJP. Surprisingly, our Mermin–Wagner theorem applies even though the symmetry groups of $\mathbb{H}^{n}$ and $\mathbb{H}^{2|2}$ are nonamenable.
</p>projecteuclid.org/euclid.aop/1571731454_20191022040429Tue, 22 Oct 2019 04:04 EDTClassification of scaling limits of uniform quadrangulations with a boundaryhttps://projecteuclid.org/euclid.aop/1575277335<strong>Erich Baur</strong>, <strong>Grégory Miermont</strong>, <strong>Gourab Ray</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 6, 3397--3477.</p><p><strong>Abstract:</strong><br/>
We study noncompact scaling limits of uniform random planar quadrangulations with a boundary when their size tends to infinity. Depending on the asymptotic behavior of the boundary size and the choice of the scaling factor, we observe different limiting metric spaces. Among well-known objects like the Brownian plane or the self-similar continuum random tree, we construct two new one-parameter families of metric spaces that appear as scaling limits: the Brownian half-plane with skewness parameter $\theta$ and the infinite-volume Brownian disk of perimeter $\sigma$. We also obtain various coupling and limit results clarifying the relation between these objects.
</p>projecteuclid.org/euclid.aop/1575277335_20191202040243Mon, 02 Dec 2019 04:02 ESTLarge scale limit of interface fluctuation modelshttps://projecteuclid.org/euclid.aop/1575277336<strong>Martin Hairer</strong>, <strong>Weijun Xu</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 6, 3478--3550.</p><p><strong>Abstract:</strong><br/>
We extend the weak universality of KPZ in [An analytic BPHZ theorem for regularity structures (2016)] to weakly asymmetric interface models with general growth mechanisms beyond polynomials. A key new ingredient is a pointwise bound on correlations of trigonometric functions of Gaussians in terms of their polynomial counterparts. This enables us to reduce the problem of a general nonlinearity with sufficient regularity to that of a polynomial.
</p>projecteuclid.org/euclid.aop/1575277336_20191202040243Mon, 02 Dec 2019 04:02 ESTSample path large deviations for Lévy processes and random walks with regularly varying incrementshttps://projecteuclid.org/euclid.aop/1575277337<strong>Chang-Han Rhee</strong>, <strong>Jose Blanchet</strong>, <strong>Bert Zwart</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 6, 3551--3605.</p><p><strong>Abstract:</strong><br/>
Let $X$ be a Lévy process with regularly varying Lévy measure $\nu$. We obtain sample-path large deviations for scaled processes $\bar{X}_{n}(t)\triangleq X(nt)/n$ and obtain a similar result for random walks with regularly varying increments. Our results yield detailed asymptotic estimates in scenarios where multiple big jumps in the increment are required to make a rare event happen; we illustrate this through detailed conditional limit theorems. In addition, we investigate connections with the classical large deviations framework. In that setting, we show that a weak large deviation principle (with logarithmic speed) holds, but a full large deviation principle does not hold.
</p>projecteuclid.org/euclid.aop/1575277337_20191202040243Mon, 02 Dec 2019 04:02 ESTGaussian free field light cones and $\mathrm{SLE}_{\kappa }(\rho )$https://projecteuclid.org/euclid.aop/1575277338<strong>Jason Miller</strong>, <strong>Scott Sheffield</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 6, 3606--3648.</p><p><strong>Abstract:</strong><br/>
Let $h$ be an instance of the GFF. Fix $\kappa \in (0,4)$ and $\chi =2/\sqrt{\kappa }-\sqrt{\kappa }/2$. Recall that an imaginary geometry ray is a flow line of $e^{i(h/\chi +\theta)}$ that looks locally like $\mathrm{SLE}_{\kappa }$. The light cone with parameter $\theta \in [0,\pi ]$ is the set of points reachable from the origin by a sequence of rays with angles in $[-\theta /2,\theta /2]$. It is known that when $\theta =0$, the light cone looks like $\mathrm{SLE}_{\kappa }$, and when $\theta =\pi $ it looks like the range of an $\mathrm{SLE}_{16/\kappa }$ counterflow line . We find that when $\theta \in (0,\pi )$ the light cones are either fractal carpets with a dense set of holes or space-filling regions with no holes. We show that every non-space-filling light cone agrees in law with the range of an $\mathrm{SLE}_{\kappa }(\rho )$ process with $\rho \in ((-2-\kappa /2)\vee (\kappa /2-4),-2)$. Conversely, the range of any such $\mathrm{SLE}_{\kappa }(\rho )$ process agrees in law with a non-space-filling light cone. As a consequence of our analysis, we obtain the first proof that these $\mathrm{SLE}_{\kappa }(\rho )$ processes are a.s. continuous curves and show that they can be constructed as natural path-valued functions of the GFF.
</p>projecteuclid.org/euclid.aop/1575277338_20191202040243Mon, 02 Dec 2019 04:02 ESTFormation of large-scale random structure by competitive erosionhttps://projecteuclid.org/euclid.aop/1575277339<strong>Shirshendu Ganguly</strong>, <strong>Lionel Levine</strong>, <strong>Sourav Sarkar</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 6, 3649--3704.</p><p><strong>Abstract:</strong><br/>
We study the following one-dimensional model of annihilating particles. Beginning with all sites of $\mathbb{Z}$ uncolored, a blue particle performs simple random walk from $0$ until it reaches a nonzero red or uncolored site, and turns that site blue; then a red particle performs simple random walk from $0$ until it reaches a nonzero blue or uncolored site, and turns that site red. We prove that after $n$ blue and $n$ red particles alternately perform such walks, the total number of colored sites is of order $n^{1/4}$. The resulting random color configuration, after rescaling by $n^{1/4}$ and taking $n\to \infty $, has an explicit description in terms of alternating extrema of Brownian motion (the global maximum on a certain interval, the global minimum attained after that maximum, etc.).
</p>projecteuclid.org/euclid.aop/1575277339_20191202040243Mon, 02 Dec 2019 04:02 ESTCutoff for the Swendsen–Wang dynamics on the latticehttps://projecteuclid.org/euclid.aop/1575277340<strong>Danny Nam</strong>, <strong>Allan Sly</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 6, 3705--3761.</p><p><strong>Abstract:</strong><br/>
We study the Swendsen–Wang dynamics for the $q$-state Potts model on the lattice. Introduced as an alternative algorithm of the classical single-site Glauber dynamics, the Swendsen–Wang dynamics is a nonlocal Markov chain that recolors many vertices at once based on the random-cluster representation of the Potts model. In this work, we establish cutoff phenomenon for the Swendsen–Wang dynamics on the lattice at sufficiently high temperatures, proving that it exhibits a sharp transition from “unmixed” to “well mixed.” In particular, we show that at high enough temperatures the Swendsen–Wang dynamics on the torus $(\mathbb{Z}/n\mathbb{Z})^{d}$ has cutoff at time $\frac{d}{2}(-\log (1-\gamma ))^{-1}\log n$, where $\gamma (\beta )$ is the spectral gap of the infinite-volume dynamics.
</p>projecteuclid.org/euclid.aop/1575277340_20191202040243Mon, 02 Dec 2019 04:02 ESTTotal variation distance between stochastic polynomials and invariance principleshttps://projecteuclid.org/euclid.aop/1575277341<strong>Vlad Bally</strong>, <strong>Lucia Caramellino</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 6, 3762--3811.</p><p><strong>Abstract:</strong><br/>
The goal of this paper is to estimate the total variation distance between two general stochastic polynomials. As a consequence, one obtains an invariance principle for such polynomials. This generalizes known results concerning the total variation distance between two multiple stochastic integrals on one hand, and invariance principles in Kolmogorov distance for multilinear stochastic polynomials on the other hand. As an application, we first discuss the asymptotic behavior of U-statistics associated to polynomial kernels. Moreover, we also give an example of CLT associated to quadratic forms.
</p>projecteuclid.org/euclid.aop/1575277341_20191202040243Mon, 02 Dec 2019 04:02 ESTRandom gluing of metric spaceshttps://projecteuclid.org/euclid.aop/1575277342<strong>Delphin Sénizergues</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 6, 3812--3865.</p><p><strong>Abstract:</strong><br/>
We construct random metric spaces by gluing together an infinite sequence of pointed metric spaces that we call blocks . At each step, we glue the next block to the structure constructed so far by randomly choosing a point on the structure and then identifying it with the distinguished point of the block. The random object that we study is the completion of the structure that we obtain after an infinite number of steps. In ( Ann. Inst. Fourier (Grenoble) 67 (2017) 1963–2001), Curien and Haas study the case of segments, where the sequence of lengths is deterministic and typically behaves like $n^{-\alpha}$. They proved that for $\alpha>0$, the resulting tree is compact and that the Hausdorff dimension of its set of leaves is $\alpha^{-1}$. The aim of this paper is to handle a much more general case in which the blocks are i.i.d. copies of the same random metric space, scaled by deterministic factors that we call $(\lambda_{n})_{n\geq1}$. We work under some conditions on the distribution of the blocks ensuring that their Hausdorff dimension is almost surely $d$, for some $d\geq0$. We also introduce a sequence $(w_{n})_{n\geq1}$ that we call the weights of the blocks. At each step, the probability that the next block is glued onto any of the preceding blocks is proportional to its weight. The main contribution of this paper is the computation of the Hausdorff dimension of the set $\mathcal{L}$ of points which appear during the completion procedure when the sequences $(\lambda_{n})_{n\geq1}$ and $(w_{n})_{n\geq1}$ typically behave like a power of $n$, say $n^{-\alpha}$ for the scaling factors and $n^{-\beta}$ for the weights, with $\alpha>0$ and $\beta\in\mathbb{R}$. For a large domain of $\alpha$ and $\beta$, we have the same behaviour as the one observed in ( Ann. Inst. Fourier (Grenoble) 67 (2017) 1963–2001), which is that $\operatorname{dim}_{\mathrm{H}}(\mathcal{L})=\alpha^{-1}$. However, for $\beta>1$ and $\alpha<1/d$, our results reveal an interesting phenomenon: the dimension has a nontrivial dependence in $\alpha$, $\beta$ and $d$, namely \begin{equation*}\operatorname{dim}_{\mathrm{H}}(\mathcal{L})=\frac{2\beta-1-2\sqrt{(\beta-1)(\beta-\alpha d)}}{\alpha}.\end{equation*} The computation of the dimension in the latter case involves new tools, which are specific to our model.
</p>projecteuclid.org/euclid.aop/1575277342_20191202040243Mon, 02 Dec 2019 04:02 ESTFour-dimensional loop-erased random walkhttps://projecteuclid.org/euclid.aop/1575277343<strong>Gregory Lawler</strong>, <strong>Xin Sun</strong>, <strong>Wei Wu</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 6, 3866--3910.</p><p><strong>Abstract:</strong><br/>
The loop-erased random walk (LERW) in $\mathbb{Z}^{4}$ is the process obtained by erasing loops chronologically for a simple random walk. We prove that the escape probability of the LERW renormalized by $(\log n)^{\frac{1}{3}}$ converges almost surely and in $L^{p}$ for all $p>0$. Along the way, we extend previous results by the first author building on slowly recurrent sets. We provide two applications for the escape probability. We construct the two-sided LERW, and we construct a $\pm 1$ spin model coupled with the wired spanning forests on $\mathbb{Z}^{4}$ with the bi-Laplacian Gaussian field on $\mathbb{R}^{4}$ as its scaling limit.
</p>projecteuclid.org/euclid.aop/1575277343_20191202040243Mon, 02 Dec 2019 04:02 ESTModulus of continuity of polymer weight profiles in Brownian last passage percolationhttps://projecteuclid.org/euclid.aop/1575277344<strong>Alan Hammond</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 6, 3911--3962.</p><p><strong>Abstract:</strong><br/>
In last passage percolation models lying in the KPZ universality class, the energy of long energy-maximizing paths may be studied as a function of the paths’ pair of endpoint locations. Scaled coordinates may be introduced, so that these maximizing paths, or polymers, now cross unit distances with unit-order fluctuations, and have scaled energy, or weight, of unit order. In this article, we consider Brownian last passage percolation in these scaled coordinates. In the narrow wedge case, one endpoint of such polymers is fixed, say at $(0,0)\in \mathbb{R}^{2}$, and the other is varied horizontally, over $(z,1)$, $z\in \mathbb{R}$, so that the polymer weight profile is a function of $z\in \mathbb{R}$. This profile is known to manifest a one-half power law, having $1/2$-Hölder continuity. The polymer weight profile may be defined beginning from a much more general initial condition. In this article, we present a more general assertion of this one-half power law, as well as a bound on the polylogarithmic correction. The polymer weight profile admits a modulus of continuity of order $x^{1/2}(\log x^{-1})^{2/3}$, with a high degree of uniformity in the scaling parameter and over a very broad class of initial data.
</p>projecteuclid.org/euclid.aop/1575277344_20191202040243Mon, 02 Dec 2019 04:02 ESTThe scaling limit of the membrane modelhttps://projecteuclid.org/euclid.aop/1575277345<strong>Alessandra Cipriani</strong>, <strong>Biltu Dan</strong>, <strong>Rajat Subhra Hazra</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 6, 3963--4001.</p><p><strong>Abstract:</strong><br/>
On the integer lattice, we consider the discrete membrane model, a random interface in which the field has Laplacian interaction. We prove that, under appropriate rescaling, the discrete membrane model converges to the continuum membrane model in $d\ge2$. Namely, it is shown that the scaling limit in $d=2,3$ is a Hölder continuous random field, while in $d\ge4$ the membrane model converges to a random distribution. As a by-product of the proof in $d=2,3$, we obtain the scaling limit of the maximum. This work complements the analogous results of Caravenna and Deuschel ( Ann. Probab. 37 (2009) 903–945) in $d=1$.
</p>projecteuclid.org/euclid.aop/1575277345_20191202040243Mon, 02 Dec 2019 04:02 ESTThe structure of low-complexity Gibbs measures on product spaceshttps://projecteuclid.org/euclid.aop/1575277346<strong>Tim Austin</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 6, 4002--4023.</p><p><strong>Abstract:</strong><br/>
Let $K_{1},\ldots,K_{n}$ be bounded, complete, separable metric spaces. Let $\lambda_{i}$ be a Borel probability measure on $K_{i}$ for each $i$. Let $f:\prod_{i}K_{i}\longrightarrow \mathbb{R}$ be a bounded and continuous potential function, and let \begin{equation*}\mu (\mathrm{d}\boldsymbol{x})\ \propto \ \mathrm{e}^{f(\boldsymbol{x})}\lambda_{1}(\mathrm{d}x_{1})\cdots \lambda_{n}(\mathrm{d}x_{n})\end{equation*} be the associated Gibbs distribution.
At each point $\boldsymbol{{x}\in \prod_{i}K_{i}}$, one can define a ‘discrete gradient’ $\nabla f(\boldsymbol{x},\cdot )$ by comparing the values of $f$ at all points which differ from $\boldsymbol{{x}}$ in at most one coordinate. In case $\prod_{i}K_{i}=\{-1,1\}^{n}\subset \mathbb{R}^{n}$, the discrete gradient $\nabla f(\boldsymbol{x},\cdot )$ is naturally identified with a vector in $\mathbb{R}^{n}$.
This paper shows that a ‘low-complexity’ assumption on $\nabla f$ implies that $\mu $ can be approximated by a mixture of other measures, relatively few in number, and most of them close to product measures in the sense of optimal transport. This implies also an approximation to the partition function of $f$ in terms of product measures, along the lines of Chatterjee and Dembo’s theory of ‘nonlinear large deviations’.
An important precedent for this work is a result of Eldan in the case $\prod_{i}K_{i}=\{-1,1\}^{n}$. Eldan’s assumption is that the discrete gradients $\nabla f(\boldsymbol{x},\cdot )$ all lie in a subset of $\mathbb{R}^{n}$ that has small Gaussian width. His proof is based on the careful construction of a diffusion in $\mathbb{R}^{n}$ which starts at the origin and ends with the desired distribution on the subset $\{-1,1\}^{n}$. Here our assumption is a more naive covering-number bound on the set of gradients $\{\nabla f(\boldsymbol{x},\cdot ):\boldsymbol{x}\in \prod_{i}K_{i}\}$, and our proof relies only on basic inequalities of information theory. As a result, it is shorter, and applies to Gibbs measures on arbitrary product spaces.
</p>projecteuclid.org/euclid.aop/1575277346_20191202040243Mon, 02 Dec 2019 04:02 ESTDirected polymers in heavy-tail random environmenthttps://projecteuclid.org/euclid.aop/1575277347<strong>Quentin Berger</strong>, <strong>Niccolò Torri</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 6, 4024--4076.</p><p><strong>Abstract:</strong><br/>
We study the directed polymer model in dimension ${1+1}$ when the environment is heavy-tailed, with a decay exponent $\alpha \in (0,2)$. We give all possible scaling limits of the model in the weak-coupling regime, that is, when the inverse temperature temperature $\beta =\beta_{n}$ vanishes as the size of the system $n$ goes to infinity. When $\alpha \in (1/2,2)$, we show that all possible transversal fluctuations $\sqrt{n}\leq h_{n}\leq n$ can be achieved by tuning properly $\beta_{n}$, allowing to interpolate between all superdiffusive scales. Moreover, we determine the scaling limit of the model, answering a conjecture by Dey and Zygouras [ Ann. Probab. 44 (2016) 4006–4048]—we actually identify five different regimes. On the other hand, when $\alpha <1/2$, we show that there are only two regimes: the transversal fluctuations are either $\sqrt{n}$ or $n$. As a key ingredient, we use the Entropy-controlled Last-Passage Percolation (E-LPP), introduced in a companion paper [ Ann. Appl. Probab. 29 (2019) 1878–1903].
</p>projecteuclid.org/euclid.aop/1575277347_20191202040243Mon, 02 Dec 2019 04:02 ESTHeavy Bernoulli-percolation clusters are indistinguishablehttps://projecteuclid.org/euclid.aop/1575277348<strong>Pengfei Tang</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 6, 4077--4115.</p><p><strong>Abstract:</strong><br/>
We prove that the heavy clusters are indistinguishable for Bernoulli percolation on quasi-transitive nonunimodular graphs. As an application, we show that the uniqueness threshold of any quasi-transitive graph is also the threshold for connectivity decay. This resolves a question of Lyons and Schramm ( Ann. Probab. 27 (1999) 1809–1836) in the Bernoulli percolation case and confirms a conjecture of Schonmann ( Comm. Math. Phys. 219 (2001) 271–322). We also prove that every infinite cluster of Bernoulli percolation on a nonamenable quasi-transitive graph is transient almost surely.
</p>projecteuclid.org/euclid.aop/1575277348_20191202040243Mon, 02 Dec 2019 04:02 ESTStrict monotonicity of percolation thresholds under covering mapshttps://projecteuclid.org/euclid.aop/1575277349<strong>Sébastien Martineau</strong>, <strong>Franco Severo</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 6, 4116--4136.</p><p><strong>Abstract:</strong><br/>
We answer a question of Benjamini and Schramm by proving that under reasonable conditions, quotienting a graph strictly increases the value of its percolation critical parameter $p_{c}$. More precisely, let $\mathcal{G}=(V,E)$ be a quasi-transitive graph with $p_{c}(\mathcal{G})<1$, and let $G$ be a nontrivial group that acts freely on $V$ by graph automorphisms. Assume that $\mathcal{H}:=\mathcal{G}/G$ is quasi-transitive. Then one has $p_{c}(\mathcal{G})<p_{c}(\mathcal{H})$.
We provide results beyond this setting: we treat the case of general covering maps and provide a similar result for the uniqueness parameter $p_{u}$, under an additional assumption of boundedness of the fibres. The proof makes use of a coupling built by lifting the exploration of the cluster, and an exploratory counterpart of Aizenman–Grimmett’s essential enhancements.
</p>projecteuclid.org/euclid.aop/1575277349_20191202040243Mon, 02 Dec 2019 04:02 ESTA stochastic telegraph equation from the six-vertex modelhttps://projecteuclid.org/euclid.aop/1575277350<strong>Alexei Borodin</strong>, <strong>Vadim Gorin</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 6, 4137--4194.</p><p><strong>Abstract:</strong><br/>
A stochastic telegraph equation is defined by adding a random inhomogeneity to the classical (second-order linear hyperbolic) telegraph differential equation. The inhomogeneities we consider are proportional to the two-dimensional white noise, and solutions to our equation are two-dimensional random Gaussian fields. We show that such fields arise naturally as asymptotic fluctuations of the height function in a certain limit regime of the stochastic six-vertex model in a quadrant. The corresponding law of large numbers—the limit shape of the height function—is described by the (deterministic) homogeneous telegraph equation.
</p>projecteuclid.org/euclid.aop/1575277350_20191202040243Mon, 02 Dec 2019 04:02 ESTDimers and imaginary geometryhttps://projecteuclid.org/euclid.aop/1585123322<strong>Nathanaël Berestycki</strong>, <strong>Benoȋt Laslier</strong>, <strong>Gourab Ray</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 48, Number 1, 1--52.</p><p><strong>Abstract:</strong><br/>
We show that the winding of the branches in a uniform spanning tree on a planar graph converge in the limit of fine mesh size to a Gaussian free field. The result holds assuming only convergence of simple random walk to Brownian motion and a Russo–Seymour–Welsh type crossing estimate, thereby establishing a strong form of universality. As an application, we prove universality of the fluctuations of the height function associated to the dimer model, in several situations.
The proof relies on a connection to imaginary geometry, where the scaling limit of a uniform spanning tree is viewed as a set of flow lines associated to a Gaussian free field. In particular, we obtain an explicit construction of the a.s. unique Gaussian free field coupled to a continuum uniform spanning tree in this way, which is of independent interest.
</p>projecteuclid.org/euclid.aop/1585123322_20200325040214Wed, 25 Mar 2020 04:02 EDTOn a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with nonglobally monotone coefficientshttps://projecteuclid.org/euclid.aop/1585123323<strong>Martin Hutzenthaler</strong>, <strong>Arnulf Jentzen</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 48, Number 1, 53--93.</p><p><strong>Abstract:</strong><br/>
We develop a perturbation theory for stochastic differential equations (SDEs) by which we mean both stochastic ordinary differential equations (SODEs) and stochastic partial differential equations (SPDEs). In particular, we estimate the $L^{p}$-distance between the solution process of an SDE and an arbitrary Itô process, which we view as a perturbation of the solution process of the SDE, by the $L^{q}$-distances of the differences of the local characteristics for suitable $p,q>0$. As one application of the developed perturbation theory, we establish strong convergence rates for numerical approximations of a class of SODEs with nonglobally monotone coefficients. As another application of the developed perturbation theory, we prove strong convergence rates for spatial spectral Galerkin approximations of solutions of semilinear SPDEs with nonglobally monotone nonlinearities including Cahn–Hilliard–Cook-type equations and stochastic Burgers equations. Further applications of the developed perturbation theory include regularity analyses of solutions of SDEs with respect to their initial values as well as small-noise analyses for ordinary and partial differential equations.
</p>projecteuclid.org/euclid.aop/1585123323_20200325040214Wed, 25 Mar 2020 04:02 EDTQuenched invariance principles for the maximal particle in branching random walk in random environment and the parabolic Anderson modelhttps://projecteuclid.org/euclid.aop/1585123324<strong>Jiří Černý</strong>, <strong>Alexander Drewitz</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 48, Number 1, 94--146.</p><p><strong>Abstract:</strong><br/>
We consider branching random walk in spatial random branching environment (BRWRE) in dimension one, as well as related differential equations: the Fisher–KPP equation with random branching and its linearized version, the parabolic Anderson model (PAM). When the random environment is bounded, we show that after recentering and scaling, the position of the maximal particle of the BRWRE, the front of the solution of the PAM, as well as the front of the solution of the randomized Fisher–KPP equation fulfill quenched invariance principles. In addition, we prove that at time $t$ the distance between the median of the maximal particle of the BRWRE and the front of the solution of the PAM is in $O(\ln t)$. This partially transfers classical results of Bramson ( Comm. Pure Appl. Math. 31 (1978) 531–581) to the setting of BRWRE.
</p>projecteuclid.org/euclid.aop/1585123324_20200325040214Wed, 25 Mar 2020 04:02 EDTContinuous Breuer–Major theorem: Tightness and nonstationarityhttps://projecteuclid.org/euclid.aop/1585123325<strong>Simon Campese</strong>, <strong>Ivan Nourdin</strong>, <strong>David Nualart</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 48, Number 1, 147--177.</p><p><strong>Abstract:</strong><br/>
Let $Y=(Y(t))_{t\geq 0}$ be a zero-mean Gaussian stationary process with covariance function $\rho :\mathbb{R}\to \mathbb{R}$ satisfying $\rho (0)=1$. Let $f:\mathbb{R}\to \mathbb{R}$ be a square-integrable function with respect to the standard Gaussian measure, and suppose the Hermite rank of $f$ is $d\geq 1$. If $\int_{\mathbb{R}}|\rho (s)|^{d}\,ds<\infty $, then the celebrated Breuer–Major theorem (in its continuous version) asserts that the finite-dimensional distributions of $Z_{\varepsilon }:=\sqrt{\varepsilon }\int_{0}^{\cdot /\varepsilon }f(Y(s))\,ds$ converge to those of $\sigma W$ as $\varepsilon \to 0$, where $W$ is a standard Brownian motion and $\sigma $ is some explicit constant. Since its first appearance in 1983, this theorem has become a crucial probabilistic tool in different areas, for instance in signal processing or in statistical inference for fractional Gaussian processes.
The goal of this paper is twofold. First, we investigate the tightness in the Breuer–Major theorem. Surprisingly, this problem did not receive a lot of attention until now, and the best available condition due to Ben Hariz [ J. Multivariate Anal. 80 (2002) 191–216] is neither arguably very natural, nor easy-to-check in practice. In contrast, our condition very simple, as it only requires that $|f|^{p}$ must be integrable with respect to the standard Gaussian measure for some $p$ strictly bigger than 2. It is obtained by means of the Malliavin calculus, in particular Meyer inequalities.
Second, and motivated by a problem of geometrical nature, we extend the continuous Breuer–Major theorem to the notoriously difficult case of self-similar Gaussian processes which are not necessarily stationary. An application to the fluctuations associated with the length process of a regularized version of the bifractional Brownian motion concludes the paper.
</p>projecteuclid.org/euclid.aop/1585123325_20200325040214Wed, 25 Mar 2020 04:02 EDTStrong existence and uniqueness for stable stochastic differential equations with distributional drifthttps://projecteuclid.org/euclid.aop/1585123326<strong>Siva Athreya</strong>, <strong>Oleg Butkovsky</strong>, <strong>Leonid Mytnik</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 48, Number 1, 178--210.</p><p><strong>Abstract:</strong><br/>
We consider the stochastic differential equation \begin{equation*}dX_{t}=b(X_{t})\,dt+dL_{t},\end{equation*} where the drift $b$ is a generalized function and $L$ is a symmetric one dimensional $\alpha $-stable Lévy processes, $\alpha \in (1,2)$. We define the notion of solution to this equation and establish strong existence and uniqueness whenever $b$ belongs to the Besov–Hölder space $\mathcal{C}^{\beta }$ for $\beta >1/2-\alpha /2$.
</p>projecteuclid.org/euclid.aop/1585123326_20200325040214Wed, 25 Mar 2020 04:02 EDTFrom the master equation to mean field game limit theory: Large deviations and concentration of measurehttps://projecteuclid.org/euclid.aop/1585123327<strong>François Delarue</strong>, <strong>Daniel Lacker</strong>, <strong>Kavita Ramanan</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 48, Number 1, 211--263.</p><p><strong>Abstract:</strong><br/>
We study a sequence of symmetric $n$-player stochastic differential games driven by both idiosyncratic and common sources of noise, in which players interact with each other through their empirical distribution. The unique Nash equilibrium empirical measure of the $n$-player game is known to converge, as $n$ goes to infinity, to the unique equilibrium of an associated mean field game. Under suitable regularity conditions, in the absence of common noise, we complement this law of large numbers result with nonasymptotic concentration bounds for the Wasserstein distance between the $n$-player Nash equilibrium empirical measure and the mean field equilibrium. We also show that the sequence of Nash equilibrium empirical measures satisfies a weak large deviation principle, which can be strengthened to a full large deviation principle only in the absence of common noise. For both sets of results, we first use the master equation, an infinite-dimensional partial differential equation that characterizes the value function of the mean field game, to construct an associated McKean–Vlasov interacting $n$-particle system that is exponentially close to the Nash equilibrium dynamics of the $n$-player game for large $n$, by refining estimates obtained in our companion paper. Then we establish a weak large deviation principle for McKean–Vlasov systems in the presence of common noise. In the absence of common noise, we upgrade this to a full large deviation principle and obtain new concentration estimates for McKean–Vlasov systems. Finally, in two specific examples that do not satisfy the assumptions of our main theorems, we show how to adapt our methodology to establish large deviations and concentration results.
</p>projecteuclid.org/euclid.aop/1585123327_20200325040214Wed, 25 Mar 2020 04:02 EDTConvergence of transport noise to Ornstein–Uhlenbeck for 2D Euler equations under the enstrophy measurehttps://projecteuclid.org/euclid.aop/1585123328<strong>Franco Flandoli</strong>, <strong>Dejun Luo</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 48, Number 1, 264--295.</p><p><strong>Abstract:</strong><br/>
We consider the vorticity form of the 2D Euler equations which is perturbed by a suitable transport type noise and has white noise initial condition. It is shown that stationary solutions of this equation converge to the unique stationary solution of the 2D Navier–Stokes equation driven by the space-time white noise.
</p>projecteuclid.org/euclid.aop/1585123328_20200325040214Wed, 25 Mar 2020 04:02 EDTQuenched invariance principle for random walks among random degenerate conductanceshttps://projecteuclid.org/euclid.aop/1585123329<strong>Peter Bella</strong>, <strong>Mathias Schäffner</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 48, Number 1, 296--316.</p><p><strong>Abstract:</strong><br/>
We consider the random conductance model in a stationary and ergodic environment. Under suitable moment conditions on the conductances and their inverse, we prove a quenched invariance principle for the random walk among the random conductances. The moment conditions improve earlier results of Andres, Deuschel and Slowik ( Ann. Probab. 43 (2015) 1866–1891) and are the minimal requirement to ensure that the corrector is sublinear everywhere. The key ingredient is an essentially optimal deterministic local boundedness result for finite difference equations in divergence form.
</p>projecteuclid.org/euclid.aop/1585123329_20200325040214Wed, 25 Mar 2020 04:02 EDTExact asymptotics for Duarte and supercritical rooted kinetically constrained modelshttps://projecteuclid.org/euclid.aop/1585123330<strong>Laure Marêché</strong>, <strong>Fabio Martinelli</strong>, <strong>Cristina Toninelli</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 48, Number 1, 317--342.</p><p><strong>Abstract:</strong><br/>
Kinetically constrained models (KCM) are a class of interacting particle systems which represent a natural stochastic (and nonmonotone) counterpart of the family of cellular automata known as $\mathcal{U}$-bootstrap percolation. A key issue for KCM is to identify the divergence of the characteristic time scales when the equilibrium density of empty sites, $q$, goes to zero. In ( Ann. Probab. 47 (2019) 324–361; Comm. Math. Phys. 369 (2019) 761–809), a general scheme was devised to determine a sharp upper bound for these time scales. Our paper is devoted to developing a (very different) technique which allows to prove matching lower bounds. We analyse the class of two-dimensional supercritical rooted KCM and the Duarte KCM . We prove that the relaxation time and the mean infection time diverge for supercritical rooted KCM as $e^{\Theta ((\log q)^{2})}$ and for Duarte KCM as $e^{\Theta ((\log q)^{4}/q^{2})}$ when $q\downarrow 0$. These results prove the conjectures put forward in ( European J. Combin. 66 (2017) 250–263; Comm. Math. Phys. 369 (2019) 761–809) for these models, and establish that the time scales for these KCM diverge much faster than for the corresponding $\mathcal{U}$-bootstrap processes, the main reason being the occurrence of energy barriers which determine the dominant behaviour for KCM, but which do not matter for the bootstrap dynamics.
</p>projecteuclid.org/euclid.aop/1585123330_20200325040214Wed, 25 Mar 2020 04:02 EDTMallows permutations and finite dependencehttps://projecteuclid.org/euclid.aop/1585123331<strong>Alexander E. Holroyd</strong>, <strong>Tom Hutchcroft</strong>, <strong>Avi Levy</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 48, Number 1, 343--379.</p><p><strong>Abstract:</strong><br/>
We use the Mallows permutation model to construct a new family of stationary finitely dependent proper colorings of the integers. We prove that these colorings can be expressed as finitary factors of i.i.d. processes with finite mean coding radii. They are the first colorings known to have these properties. Moreover, we prove that the coding radii have exponential tails, and that the colorings can also be expressed as functions of countable-state Markov chains. We deduce analogous existence statements concerning shifts of finite type and higher-dimensional colorings.
</p>projecteuclid.org/euclid.aop/1585123331_20200325040214Wed, 25 Mar 2020 04:02 EDTGeometric ergodicity in a weighted Sobolev spacehttps://projecteuclid.org/euclid.aop/1585123332<strong>Adithya Devraj</strong>, <strong>Ioannis Kontoyiannis</strong>, <strong>Sean Meyn</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 48, Number 1, 380--403.</p><p><strong>Abstract:</strong><br/>
For a discrete-time Markov chain $\boldsymbol{X}=\{X(t)\}$ evolving on $\mathbb{R}^{\ell}$ with transition kernel $P$, natural, general conditions are developed under which the following are established:
(i) The transition kernel $P$ has a purely discrete spectrum, when viewed as a linear operator on a weighted Sobolev space $L_{\infty}^{v,1}$ of functions with norm, \begin{equation*}\Vert f\Vert_{v,1}=\mathop{\mathrm{sup}}_{x\in\mathbb{R}^{\ell}}\frac{1}{v(x)}\max\{\vert f(x)\vert ,\vert \partial_{1}f(x)\vert ,\ldots,\vert \partial_{\ell}f(x)\vert \},\end{equation*} where $v\colon\mathbb{R}^{\ell}\to[1,\infty)$ is a Lyapunov function and $\partial_{i}:=\partial/\partial x_{i}$.
(ii) The Markov chain is geometrically ergodic in $L_{\infty}^{v,1}$: There is a unique invariant probability measure $\pi$ and constants $B<\infty$ and $\delta>0$ such that, for each $f\in L_{\infty}^{v,1}$, any initial condition $X(0)=x$, and all $t\geq0$: \begin{eqnarray*}\vert \mathsf{E}_{x}[f(X(t))]-\pi(f)\vert &\le&B\Vert f\Vert_{v,1}e^{-\delta t}v(x),\\\Vert \nabla\mathsf{E}_{x}[f(X(t))]\Vert_{2}&\le&B\Vert f\Vert_{v,1}e^{-\delta t}v(x),\end{eqnarray*} where $\pi(f)=\int f\,d\pi$.
(iii) For any function $f\in L_{\infty}^{v,1}$ there is a function $h\in L_{\infty}^{v,1}$ solving Poisson’s equation: \begin{equation*}h-Ph=f-\pi(f).\end{equation*}
Part of the analysis is based on an operator-theoretic treatment of the sensitivity process that appears in the theory of Lyapunov exponents. Relationships with topological coupling, in terms of the Wasserstein metric, are also explored.
</p>projecteuclid.org/euclid.aop/1585123332_20200325040214Wed, 25 Mar 2020 04:02 EDTHydrodynamics in a condensation regime: The disordered asymmetric zero-range processhttps://projecteuclid.org/euclid.aop/1585123333<strong>C. Bahadoran</strong>, <strong>T. Mountford</strong>, <strong>K. Ravishankar</strong>, <strong>E. Saada</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 48, Number 1, 404--444.</p><p><strong>Abstract:</strong><br/>
We study asymmetric zero-range processes on $\mathbb{Z}$ with nearest-neighbour jumps and site disorder. The jump rate of particles is an arbitrary but bounded nondecreasing function of the number of particles. For any given environment satisfying suitable averaging properties, we establish a hydrodynamic limit given by a scalar conservation law including the domain above critical density, where the flux is shown to be constant.
</p>projecteuclid.org/euclid.aop/1585123333_20200325040214Wed, 25 Mar 2020 04:02 EDTA simple proof of the DPRZ theorem for 2d cover timeshttps://projecteuclid.org/euclid.aop/1585123334<strong>Marius A. Schmidt</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 48, Number 1, 445--457.</p><p><strong>Abstract:</strong><br/>
We give a simple proof of the theorem by Dembo, Peres, Rosen and Zeitouni (DPRZ) regarding the time Brownian motion needs to cover every $\varepsilon$ ball on the two-dimensional unit torus in the $\varepsilon\searrow 0$ limit.
</p>projecteuclid.org/euclid.aop/1585123334_20200325040214Wed, 25 Mar 2020 04:02 EDTItô’s formula for Gaussian processes with stochastic discontinuitieshttps://projecteuclid.org/euclid.aop/1585123335<strong>Christian Bender</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 48, Number 1, 458--492.</p><p><strong>Abstract:</strong><br/>
We introduce a Skorokhod type integral and prove an Itô formula for a wide class of Gaussian processes which may exhibit stochastic discontinuities. Our Itô formula unifies and extends the classical one for general (i.e., possibly discontinuous) Gaussian martingales in the sense of Itô integration and the one for stochastically continuous Gaussian non-martingales in the Skorokhod sense, which was first derived in Alòs et al. ( Ann. Probab. 29 (2001) 766–801).
</p>projecteuclid.org/euclid.aop/1585123335_20200325040214Wed, 25 Mar 2020 04:02 EDTOn the probability of nonexistence in binomial subsetshttps://projecteuclid.org/euclid.aop/1585123336<strong>Frank Mousset</strong>, <strong>Andreas Noever</strong>, <strong>Konstantinos Panagiotou</strong>, <strong>Wojciech Samotij</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 48, Number 1, 493--525.</p><p><strong>Abstract:</strong><br/>
Given a hypergraph $\Gamma =(\Omega ,\mathcal{X})$ and a sequence $\mathbf{p}=(p_{\omega })_{\omega \in \Omega }$ of values in $(0,1)$, let $\Omega_{\mathbf{p}}$ be the random subset of $\Omega $ obtained by keeping every vertex $\omega $ independently with probability $p_{\omega }$. We investigate the general question of deriving fine (asymptotic) estimates for the probability that $\Omega_{\mathbf{p}}$ is an independent set in $\Gamma $, which is an omnipresent problem in probabilistic combinatorics. Our main result provides a sequence of upper and lower bounds on this probability, each of which can be evaluated explicitly in terms of the joint cumulants of small sets of edge indicator random variables. Under certain natural conditions, these upper and lower bounds coincide asymptotically, thus giving the precise asymptotics of the probability in question. We demonstrate the applicability of our results with two concrete examples: subgraph containment in random (hyper)graphs and arithmetic progressions in random subsets of the integers.
</p>projecteuclid.org/euclid.aop/1585123336_20200325040214Wed, 25 Mar 2020 04:02 EDTAn almost sure KPZ relation for SLE and Brownian motionhttps://projecteuclid.org/euclid.aop/1587542672<strong>Ewain Gwynne</strong>, <strong>Nina Holden</strong>, <strong>Jason Miller</strong>. <p><strong>Source: </strong>Annals of Probability, Volume 48, Number 2, 527--573.</p><p><strong>Abstract:</strong><br/>
The peanosphere construction of Duplantier, Miller and Sheffield provides a means of representing a $\gamma $-Liouville quantum gravity (LQG) surface, $\gamma \in (0,2)$, decorated with a space-filling form of Schramm’s $\mathrm{SLE}_{\kappa }$, $\kappa =16/\gamma^{2}\in (4,\infty)$, $\eta $ as a gluing of a pair of trees which are encoded by a correlated two-dimensional Brownian motion $Z$. We prove a KPZ-type formula which relates the Hausdorff dimension of any Borel subset $A$ of the range of $\eta $, which can be defined as a function of $\eta $ (modulo time parameterization) to the Hausdorff dimension of the corresponding time set $\eta^{-1}(A)$. This result serves to reduce the problem of computing the Hausdorff dimension of any set associated with an $\mathrm{SLE}$, $\mathrm{CLE}$ or related processes in the interior of a domain to the problem of computing the Hausdorff dimension of a certain set associated with a Brownian motion. For many natural examples, the associated Brownian motion set is well known. As corollaries, we obtain new proofs of the Hausdorff dimensions of the $\mathrm{SLE}_{\kappa}$ curve for $\kappa \neq4$; the double points and cut points of $\mathrm{SLE}_{\kappa }$ for $\kappa >4$; and the intersection of two flow lines of a Gaussian free field. We obtain the Hausdorff dimension of the set of $m$-tuple points of space-filling $\mathrm{SLE}_{\kappa }$ for $\kappa >4$ and $m\geq 3$ by computing the Hausdorff dimension of the so-called $(m-2)$-tuple $\pi /2$-cone times of a correlated planar Brownian motion.
</p>projecteuclid.org/euclid.aop/1587542672_20200422040440Wed, 22 Apr 2020 04:04 EDTLocalization in random geometric graphs with too many edgeshttps://projecteuclid.org/euclid.aop/1587542673<strong>Sourav Chatterjee</strong>, <strong>Matan Harel</strong>. <p><strong>Source: </strong>Annals of Probability, Volume 48, Number 2, 574--621.</p><p><strong>Abstract:</strong><br/>
We consider a random geometric graph $G(\chi_{n},r_{n})$, given by connecting two vertices of a Poisson point process $\chi_{n}$ of intensity $n$ on the $d$-dimensional unit torus whenever their distance is smaller than the parameter $r_{n}$. The model is conditioned on the rare event that the number of edges observed, $|E|$, is greater than $(1+\delta )\mathbb{E}(|E|)$, for some fixed $\delta >0$. This article proves that upon conditioning, with high probability there exists a ball of diameter $r_{n}$ which contains a clique of at least $\sqrt{2\delta \mathbb{E}(|E|)}(1-\varepsilon )$ vertices, for any given $\varepsilon >0$. Intuitively, this region contains all the “excess” edges the graph is forced to contain by the conditioning event, up to lower order corrections. As a consequence of this result, we prove a large deviations principle for the upper tail of the edge count of the random geometric graph. The rate function of this large deviation principle turns out to be nonconvex.
</p>projecteuclid.org/euclid.aop/1587542673_20200422040440Wed, 22 Apr 2020 04:04 EDTThe maximal flow from a compact convex subset to infinity in first passage percolation on $\mathbb{Z}^{d}$https://projecteuclid.org/euclid.aop/1587542674<strong>Barbara Dembin</strong>. <p><strong>Source: </strong>Annals of Probability, Volume 48, Number 2, 622--645.</p><p><strong>Abstract:</strong><br/>
We consider the standard first passage percolation model on $\mathbb{Z}^{d}$ with a distribution $G$ on $\mathbb{R}^{+}$ that admits an exponential moment. We study the maximal flow between a compact convex subset $A$ of $\mathbb{R}^{d}$ and infinity. The study of maximal flow is associated with the study of sets of edges of minimal capacity that cut $A$ from infinity. We prove that the rescaled maximal flow between $nA$ and infinity $\phi (nA)/n^{d-1}$ almost surely converges toward a deterministic constant depending on $A$. This constant corresponds to the capacity of the boundary $\partial A$ of $A$ and is the integral of a deterministic function over $\partial A$. This result was shown in dimension $2$ and conjectured for higher dimensions by Garet in ( Annals of Applied Probability 19 (2009) 641–660).
</p>projecteuclid.org/euclid.aop/1587542674_20200422040440Wed, 22 Apr 2020 04:04 EDTHitting probabilities of a Brownian flow with radial drifthttps://projecteuclid.org/euclid.aop/1587542675<strong>Jong Jun Lee</strong>, <strong>Carl Mueller</strong>, <strong>Eyal Neuman</strong>. <p><strong>Source: </strong>Annals of Probability, Volume 48, Number 2, 646--671.</p><p><strong>Abstract:</strong><br/>
We consider a stochastic flow $\phi_{t}(x,\omega )$ in $\mathbb{R}^{n}$ with initial point $\phi_{0}(x,\omega )=x$, driven by a single $n$-dimensional Brownian motion, and with an outward radial drift of magnitude $\frac{F(\|\phi_{t}(x)\|)}{\|\phi_{t}(x)\|}$, with $F$ nonnegative, bounded and Lipschitz. We consider initial points $x$ lying in a set of positive distance from the origin. We show that there exist constants $C^{*},c^{*}>0$ not depending on $n$, such that if $F>C^{*}n$ then the image of the initial set under the flow has probability 0 of hitting the origin. If $0\leq F\leq c^{*}n^{3/4}$, and if the initial set has a nonempty interior, then the image of the set has positive probability of hitting the origin.
</p>projecteuclid.org/euclid.aop/1587542675_20200422040440Wed, 22 Apr 2020 04:04 EDTRandom moment problems under constraintshttps://projecteuclid.org/euclid.aop/1587542676<strong>Holger Dette</strong>, <strong>Dominik Tomecki</strong>, <strong>Martin Venker</strong>. <p><strong>Source: </strong>Annals of Probability, Volume 48, Number 2, 672--713.</p><p><strong>Abstract:</strong><br/>
We investigate moment sequences of probability measures on subsets of the real line under constraints of certain moments being fixed. This corresponds to studying sections of $n$th moment spaces, that is, the spaces of moment sequences of order $n$. By equipping these sections with the uniform or more general probability distributions, we manage to give for large $n$ precise results on the (probabilistic) barycenters of moment space sections and the fluctuations of random moments around these barycenters. The measures associated to the barycenters belong to the Bernstein–Szegő class and show strong universal behavior. We prove Gaussian fluctuations and moderate and large deviations principles. Furthermore, we demonstrate how fixing moments by a constraint leads to breaking the connection between random moments and random matrices.
</p>projecteuclid.org/euclid.aop/1587542676_20200422040440Wed, 22 Apr 2020 04:04 EDTThe maximum of the four-dimensional membrane modelhttps://projecteuclid.org/euclid.aop/1587542677<strong>Florian Schweiger</strong>. <p><strong>Source: </strong>Annals of Probability, Volume 48, Number 2, 714--741.</p><p><strong>Abstract:</strong><br/>
We show that the centred maximum of the four-dimensional membrane model on a box of sidelength $N$ converges in distribution. To do so, we use a criterion of Ding, Roy and Zeitouni ( Ann. Probab. 45 (2017) 3886–3928) and prove sharp estimates for the Green’s function of the discrete Bilaplacian. These estimates are the main contribution of this work and might also be of independent interest. To derive them, we use estimates for the approximation quality of finite difference schemes as well as results for the Green’s function of the continuous Bilaplacian.
</p>projecteuclid.org/euclid.aop/1587542677_20200422040440Wed, 22 Apr 2020 04:04 EDTCutoff for the mean-field zero-range process with bounded monotone rateshttps://projecteuclid.org/euclid.aop/1587542678<strong>Jonathan Hermon</strong>, <strong>Justin Salez</strong>. <p><strong>Source: </strong>Annals of Probability, Volume 48, Number 2, 742--759.</p><p><strong>Abstract:</strong><br/>
We consider the zero-range process with arbitrary bounded monotone rates on the complete graph, in the regime where the number of sites diverges while the density of particles per site converges. We determine the asymptotics of the mixing time from any initial configuration, and establish the cutoff phenomenon. The intuitive picture is that the system separates into a slowly evolving solid phase and a quickly relaxing liquid phase: as time passes, the solid phase dissolves into the liquid phase, and the mixing time is essentially the time at which the system becomes completely liquid. Our proof uses the path coupling technique of Bubley and Dyer, and the analysis of a suitable hydrodynamic limit. To the best of our knowledge, even the order of magnitude of the mixing time was unknown, except in the special case of constant rates.
</p>projecteuclid.org/euclid.aop/1587542678_20200422040440Wed, 22 Apr 2020 04:04 EDTTranslation-invariant Gibbs states of the Ising model: General settinghttps://projecteuclid.org/euclid.aop/1587542679<strong>Aran Raoufi</strong>. <p><strong>Source: </strong>Annals of Probability, Volume 48, Number 2, 760--777.</p><p><strong>Abstract:</strong><br/>
We prove that at any inverse temperature $\beta $ and on any transitive amenable graph, the automorphism-invariant Gibbs states of the ferromagnetic Ising model are convex combinations of the plus and minus states. The theorem is equivalent with the differentiability of the free energy with respect to the temperature at any temperature. This is obtained for a general class of interactions, that is automorphism-invariant and irreducible coupling constants. The proof uses the random current representation of the Ising model. The result is novel when the graph is not $\mathbb{Z}^{d}$, or when the graph is $\mathbb{Z}^{d}$ but endowed with infinite-range interactions, or even $\mathbb{Z}^{2}$ with finite-range interactions.
Among the other corollaries of this result, we can list continuity of the magnetization at any noncritical temperature and the uniqueness of FK-Ising infinite-volume measures at any temperature.
</p>projecteuclid.org/euclid.aop/1587542679_20200422040440Wed, 22 Apr 2020 04:04 EDTBusemann functions and Gibbs measures in directed polymer models on $\mathbb{Z}^{2}$https://projecteuclid.org/euclid.aop/1587542680<strong>Christopher Janjigian</strong>, <strong>Firas Rassoul-Agha</strong>. <p><strong>Source: </strong>Annals of Probability, Volume 48, Number 2, 778--816.</p><p><strong>Abstract:</strong><br/>
We consider random walk in a space-time random potential, also known as directed random polymer measures, on the planar square lattice with nearest-neighbor steps and general i.i.d. weights on the vertices. We construct covariant cocycles and use them to prove new results on existence, uniqueness/nonuniqueness, and asymptotic directions of semi-infinite polymer measures (solutions to the Dobrushin–Lanford–Ruelle equations). We also prove nonexistence of covariant or deterministically directed bi-infinite polymer measures. Along the way, we prove almost sure existence of Busemann function limits in directions where the limiting free energy has some regularity.
</p>projecteuclid.org/euclid.aop/1587542680_20200422040440Wed, 22 Apr 2020 04:04 EDTThe endpoint distribution of directed polymershttps://projecteuclid.org/euclid.aop/1587542681<strong>Erik Bates</strong>, <strong>Sourav Chatterjee</strong>. <p><strong>Source: </strong>Annals of Probability, Volume 48, Number 2, 817--871.</p><p><strong>Abstract:</strong><br/>
Probabilistic models of directed polymers in random environment have received considerable attention in recent years. Much of this attention has focused on integrable models. In this paper, we introduce some new computational tools that do not require integrability. We begin by defining a new kind of abstract limit object, called “partitioned subprobability measure,” to describe the limits of endpoint distributions of directed polymers. Inspired by a recent work of Mukherjee and Varadhan on large deviations of the occupation measure of Brownian motion, we define a suitable topology on the space of partitioned subprobability measures and prove that this topology is compact. Then using a variant of the cavity method from the theory of spin glasses, we show that any limit law of a sequence of endpoint distributions must satisfy a fixed point equation on this abstract space, and that the limiting free energy of the model can be expressed as the solution of a variational problem over the set of fixed points. As a first application of the theory, we prove that in an environment with finite exponential moment, the endpoint distribution is asymptotically purely atomic if and only if the system is in the low temperature phase. The analogous result for a heavy-tailed environment was proved by Vargas in 2007. As a second application, we prove a subsequential version of the longstanding conjecture that in the low temperature phase, the endpoint distribution is asymptotically localized in a region of stochastically bounded diameter. All our results hold in arbitrary dimensions, and make no use of integrability.
</p>projecteuclid.org/euclid.aop/1587542681_20200422040440Wed, 22 Apr 2020 04:04 EDTThe distribution of Gaussian multiplicative chaos on the unit intervalhttps://projecteuclid.org/euclid.aop/1587542682<strong>Guillaume Remy</strong>, <strong>Tunan Zhu</strong>. <p><strong>Source: </strong>Annals of Probability, Volume 48, Number 2, 872--915.</p><p><strong>Abstract:</strong><br/>
We consider a subcritical Gaussian multiplicative chaos (GMC) measure defined on the unit interval $[0,1]$ and prove an exact formula for the fractional moments of the total mass of this measure. Our formula includes the case where log-singularities (also called insertion points) are added in $0$ and $1$, the most general case predicted by the Selberg integral. The idea to perform this computation is to introduce certain auxiliary functions resembling holomorphic observables of conformal field theory that will be solutions of hypergeometric equations. Solving these equations then provides nontrivial relations that completely determine the moments we wish to compute. We also include a detailed discussion of the so-called reflection coefficients appearing in tail expansions of GMC measures and in Liouville theory. Our theorem provides an exact value for one of these coefficients. Lastly, we mention some additional applications to small deviations for GMC measures, to the behavior of the maximum of the log-correlated field on the interval and to random hermitian matrices.
</p>projecteuclid.org/euclid.aop/1587542682_20200422040440Wed, 22 Apr 2020 04:04 EDTTransition from Tracy–Widom to Gaussian fluctuations of extremal eigenvalues of sparse Erdős–Rényi graphshttps://projecteuclid.org/euclid.aop/1587542683<strong>Jiaoyang Huang</strong>, <strong>Benjamin Landon</strong>, <strong>Horng-Tzer Yau</strong>. <p><strong>Source: </strong>Annals of Probability, Volume 48, Number 2, 916--962.</p><p><strong>Abstract:</strong><br/>
We consider the statistics of the extreme eigenvalues of sparse random matrices, a class of random matrices that includes the normalized adjacency matrices of the Erdős–Rényi graph $G(N,p)$. Tracy–Widom fluctuations of the extreme eigenvalues for $p\gg N^{-2/3}$ was proved in ( Probab. Theory Related Fields 171 (2018) 543–616; Comm. Math. Phys. 314 (2012) 587–640). We prove that there is a crossover in the behavior of the extreme eigenvalues at $p\sim N^{-2/3}$. In the case that $N^{-7/9}\ll p\ll N^{-2/3}$, we prove that the extreme eigenvalues have asymptotically Gaussian fluctuations. Under a mean zero condition and when $p=CN^{-2/3}$, we find that the fluctuations of the extreme eigenvalues are given by a combination of the Gaussian and the Tracy–Widom distribution. These results show that the eigenvalues at the edge of the spectrum of sparse Erdős–Rényi graphs are less rigid than those of random $d$-regular graphs (Bauerschmidt et al. (2019)) of the same average degree.
</p>projecteuclid.org/euclid.aop/1587542683_20200422040440Wed, 22 Apr 2020 04:04 EDTCorrelated random matrices: Band rigidity and edge universalityhttps://projecteuclid.org/euclid.aop/1587542684<strong>Johannes Alt</strong>, <strong>László Erdős</strong>, <strong>Torben Krüger</strong>, <strong>Dominik Schröder</strong>. <p><strong>Source: </strong>Annals of Probability, Volume 48, Number 2, 963--1001.</p><p><strong>Abstract:</strong><br/>
We prove edge universality for a general class of correlated real symmetric or complex Hermitian Wigner matrices with arbitrary expectation. Our theorem also applies to internal edges of the self-consistent density of states. In particular, we establish a strong form of band rigidity which excludes mismatches between location and label of eigenvalues close to internal edges in these general models.
</p>projecteuclid.org/euclid.aop/1587542684_20200422040440Wed, 22 Apr 2020 04:04 EDTOn the nature of the Swiss cheese in dimension $3$https://projecteuclid.org/euclid.aop/1587542685<strong>Amine Asselah</strong>, <strong>Bruno Schapira</strong>. <p><strong>Source: </strong>Annals of Probability, Volume 48, Number 2, 1002--1013.</p><p><strong>Abstract:</strong><br/>
We study scenarii linked with the Swiss cheese picture in dimension 3 obtained when two random walks are forced to meet often, or when one random walk is forced to squeeze its range. In the case of two random walks, we show that they most likely meet in a region of optimal density . In the case of one random walk, we show that a small range is reached by a strategy uniform in time. Both results rely on an original inequality estimating the cost of visiting sparse sites, and in the case of one random walk on the precise large deviation principle of van den Berg, Bolthausen and den Hollander ( Ann. of Math. (2) 153 (2001) 355–406), including their sharp estimates of the rate functions in the neighborhood of the origin.
</p>projecteuclid.org/euclid.aop/1587542685_20200422040440Wed, 22 Apr 2020 04:04 EDTConstructing a solution of the $(2+1)$-dimensional KPZ equationhttps://projecteuclid.org/euclid.aop/1587542686<strong>Sourav Chatterjee</strong>, <strong>Alexander Dunlap</strong>. <p><strong>Source: </strong>Annals of Probability, Volume 48, Number 2, 1014--1055.</p><p><strong>Abstract:</strong><br/>
The $(d+1)$-dimensional KPZ equation is the canonical model for the growth of rough $d$-dimensional random surfaces. A deep mathematical understanding of the KPZ equation for $d=1$ has been achieved in recent years, and the case $d\ge 3$ has also seen some progress. The most physically relevant case of $d=2$, however, is not very well understood mathematically, largely due to the renormalization that is required: in the language of renormalization group analysis, the $d=2$ case is neither ultraviolet superrenormalizable like the $d=1$ case nor infrared superrenormalizable like the $d\ge 3$ case. Moreover, unlike in $d=1$, the Cole–Hopf transform is not directly usable in $d=2$ because solutions to the multiplicative stochastic heat equation are distributions rather than functions. In this article, we show the existence of subsequential scaling limits as $\varepsilon \to 0$ of Cole–Hopf solutions of the $(2+1)$-dimensional KPZ equation with white noise mollified to spatial scale $\varepsilon $ and nonlinearity multiplied by the vanishing factor $|\log \varepsilon |^{-\frac{1}{2}}$. We also show that the scaling limits obtained in this way do not coincide with solutions to the linearized equation, meaning that the nonlinearity has a nonvanishing effect. We thus propose our scaling limit as a notion of KPZ evolution in $2+1$ dimensions.
</p>projecteuclid.org/euclid.aop/1587542686_20200422040440Wed, 22 Apr 2020 04:04 EDTHitting times of interacting drifted Brownian motions and the vertex reinforced jump processhttps://projecteuclid.org/euclid.aop/1592359222<strong>Christophe Sabot</strong>, <strong>Xiaolin Zeng</strong>. <p><strong>Source: </strong>Annals of Probability, Volume 48, Number 3, 1057--1085.</p><p><strong>Abstract:</strong><br/>
Consider a negatively drifted one-dimensional Brownian motion starting at positive initial position, its first hitting time to 0 has the inverse Gaussian law. Moreover, conditionally on this hitting time, the Brownian motion up to that time has the law of a three-dimensional Bessel bridge. In this paper, we give a generalization of this result to a family of Brownian motions with interacting drifts, indexed by the vertices of a conductance network. The hitting times are equal in law to the inverse of a random potential that appears in the analysis of a self-interacting process called the vertex reinforced jump process ( Ann. Probab. 45 (2017) 3967–3986; J. Amer. Math. Soc. 32 (2019) 311–349). These Brownian motions with interacting drifts have remarkable properties with respect to restriction and conditioning, showing hidden Markov properties. This family of processes are closely related to the martingale that plays a crucial role in the analysis of the vertex reinforced jump process and edge reinforced random walk ( J. Amer. Math. Soc. 32 (2019) 311–349) on infinite graphs.
</p>projecteuclid.org/euclid.aop/1592359222_20200616220034Tue, 16 Jun 2020 22:00 EDTThe two-dimensional KPZ equation in the entire subcritical regimehttps://projecteuclid.org/euclid.aop/1592359223<strong>Francesco Caravenna</strong>, <strong>Rongfeng Sun</strong>, <strong>Nikos Zygouras</strong>. <p><strong>Source: </strong>Annals of Probability, Volume 48, Number 3, 1086--1127.</p><p><strong>Abstract:</strong><br/>
We consider the KPZ equation in space dimension $2$ driven by space-time white noise. We showed in previous work that if the noise is mollified in space on scale $\varepsilon $ and its strength is scaled as $\hat{\beta }/\sqrt{|\log \varepsilon |}$, then a transition occurs with explicit critical point $\hat{\beta }_{c}=\sqrt{2\pi }$. Recently Chatterjee and Dunlap showed that the solution admits subsequential scaling limits as $\varepsilon \downarrow 0$, for sufficiently small $\hat{\beta }$. We prove here that the limit exists in the entire subcritical regime $\hat{\beta }\in (0,\hat{\beta }_{c})$ and we identify it as the solution of an additive stochastic heat equation, establishing so-called Edwards–Wilkinson fluctuations. The same result holds for the directed polymer model in random environment in space dimension $2$.
</p>projecteuclid.org/euclid.aop/1592359223_20200616220034Tue, 16 Jun 2020 22:00 EDTExchangeable interval hypergraphs and limits of ordered discrete structureshttps://projecteuclid.org/euclid.aop/1592359224<strong>Julian Gerstenberg</strong>. <p><strong>Source: </strong>Annals of Probability, Volume 48, Number 3, 1128--1167.</p><p><strong>Abstract:</strong><br/>
A hypergraph $(V,E)$ is called an interval hypergraph if there exists a linear order $l$ on $V$ such that every edge $e\in E$ is an interval w.r.t. $l$; we also assume that $\{j\}\in E$ for every $j\in V$. Our main result is a de Finetti-type representation of random exchangeable interval hypergraphs on $\mathbb{N}$ (EIHs): the law of every EIH can be obtained by sampling from some random compact subset $K$ of the triangle $\{(x,y):0\leq x\leq y\leq1\}$ at i.i.d. uniform positions $U_{1},U_{2},\dots$, in the sense that, restricted to the node set $[n]:=\{1,\dots,n\}$ every nonsingleton edge is of the form $e=\{i\in[n]:x<U_{i}<y\}$ for some $(x,y)\in K$. We obtain this result via the study of a related class of stochastic objects: erased-interval processes (EIPs). These are certain transient Markov chains $(I_{n},\eta_{n})_{n\in\mathbb{N}}$ such that $I_{n}$ is an interval hypergraph on $V=[n]$ w.r.t. the usual linear order (called interval system). We present an almost sure representation result for EIPs. Attached to each transient Markov chain is the notion of Martin boundary. The points in the boundary of EIPs can be seen as limits of growing interval systems. We obtain a one-to-one correspondence between these limits and compact subsets $K$ of the triangle with $(x,x)\in K$ for all $x\in[0,1]$.
Interval hypergraphs are a generalizations of hierarchies and as a consequence we obtain a representation result for exchangeable hierarchies, which is close to a result of Forman, Haulk and Pitman in ( Probab. Theory Related Fields 172 (2018) 1–29). Several ordered discrete structures can be seen as interval systems with additional properties, that is, Schröder trees (rooted, ordered, no node has outdegree one) or even more special: binary trees. We describe limits of Schröder trees as certain tree-like compact sets. These can be seen as an ordered counterpart to real trees, which are widely used to describe limits of discrete unordered trees. Considering binary trees, we thus obtain a homeomorphic description of the Martin boundary of Rémy’s tree growth chain, which has been analyzed by Evans, Grübel and Wakolbinger in ( Ann. Probab. 45 (2017) 225–277).
</p>projecteuclid.org/euclid.aop/1592359224_20200616220034Tue, 16 Jun 2020 22:00 EDTOn the topological boundary of the range of super-Brownian motionhttps://projecteuclid.org/euclid.aop/1592359225<strong>Jieliang Hong</strong>, <strong>Leonid Mytnik</strong>, <strong>Edwin Perkins</strong>. <p><strong>Source: </strong>Annals of Probability, Volume 48, Number 3, 1168--1201.</p><p><strong>Abstract:</strong><br/>
We show that if $\partial\mathcal{R}$ is the boundary of the range of super-Brownian motion and dim denotes Hausdorff dimension, then with probability one, for any open set $U$, $U\cap\partial\mathcal{R}\neq\varnothing$ implies \[\operatorname{dim}(U\cap\partial\mathcal{R})=\begin{cases}4-2\sqrt{2}\approx1.17\quad\text{if }d=2,\\\frac{9-\sqrt{17}}{2}\approx2.44\quad\text{if }d=3.\end{cases}\] This improves recent results of the last two authors by working with the actual topological boundary, rather than the boundary of the zero set of the local time, and establishing a local result for the dimension.
</p>projecteuclid.org/euclid.aop/1592359225_20200616220034Tue, 16 Jun 2020 22:00 EDTNormal approximation for weighted sums under a second-order correlation conditionhttps://projecteuclid.org/euclid.aop/1592359226<strong>S. G. Bobkov</strong>, <strong>G. P. Chistyakov</strong>, <strong>F. Götze</strong>. <p><strong>Source: </strong>Annals of Probability, Volume 48, Number 3, 1202--1219.</p><p><strong>Abstract:</strong><br/>
Under correlation-type conditions, we derive an upper bound of order $(\log n)/n$ for the average Kolmogorov distance between the distributions of weighted sums of dependent summands and the normal law. The result is based on improved concentration inequalities on high-dimensional Euclidean spheres. Applications are illustrated on the example of log-concave probability measures.
</p>projecteuclid.org/euclid.aop/1592359226_20200616220034Tue, 16 Jun 2020 22:00 EDTEntrance and exit at infinity for stable jump diffusionshttps://projecteuclid.org/euclid.aop/1592359227<strong>Leif Döring</strong>, <strong>Andreas E. Kyprianou</strong>. <p><strong>Source: </strong>Annals of Probability, Volume 48, Number 3, 1220--1265.</p><p><strong>Abstract:</strong><br/>
In his seminal work from the 1950s, William Feller classified all one-dimensional diffusions on $-\infty \leq a<b\leq \infty $ in terms of their ability to access the boundary (Feller’s test for explosions) and to enter the interior from the boundary. Feller’s technique is restricted to diffusion processes as the corresponding differential generators allow explicit computations and the use of Hille–Yosida theory. In the present article, we study exit and entrance from infinity for the most natural generalization, that is, jump diffusions of the form \begin{equation*}dZ_{t}=\sigma (Z_{t-})\,dX_{t},\end{equation*} driven by stable Lévy processes for $\alpha \in (0,2)$. Many results have been proved for jump diffusions, employing a variety of techniques developed after Feller’s work but exit and entrance from infinite boundaries has long remained open. We show that the presence of jumps implies features not seen in the diffusive setting without drift. Finite time explosion is possible for $\alpha \in (0,1)$, whereas entrance from different kinds of infinity is possible for $\alpha \in [1,2)$. Accordingly, we derive necessary and sufficient conditions on $\sigma $.
Our proofs are based on very recent developments for path transformations of stable processes via the Lamperti–Kiu representation and new Wiener–Hopf factorisations for Lévy processes that lie therein. The arguments draw together original and intricate applications of results using the Riesz–Bogdan–Żak transformation, entrance laws for self-similar Markov processes, perpetual integrals of Lévy processes and fluctuation theory, which have not been used before in the SDE setting, thereby allowing us to employ classical theory such as Hunt–Nagasawa duality and Getoor’s characterisation of transience and recurrence.
</p>projecteuclid.org/euclid.aop/1592359227_20200616220034Tue, 16 Jun 2020 22:00 EDTErgodic Poisson splittingshttps://projecteuclid.org/euclid.aop/1592359228<strong>Élise Janvresse</strong>, <strong>Emmanuel Roy</strong>, <strong>Thierry de la Rue</strong>. <p><strong>Source: </strong>Annals of Probability, Volume 48, Number 3, 1266--1285.</p><p><strong>Abstract:</strong><br/>
In this paper, we study splittings of a Poisson point process which are equivariant under a conservative transformation. We show that, if the Cartesian powers of this transformation are all ergodic, the only ergodic splitting is the obvious one, that is, a collection of independent Poisson processes. We apply this result to the case of a marked Poisson process: under the same hypothesis, the marks are necessarily independent of the point process and i.i.d. Under additional assumptions on the transformation, a further application is derived, giving a full description of the structure of a random measure invariant under the action of the transformation.
</p>projecteuclid.org/euclid.aop/1592359228_20200616220034Tue, 16 Jun 2020 22:00 EDTOperator limit of the circular beta ensemblehttps://projecteuclid.org/euclid.aop/1592359229<strong>Benedek Valkó</strong>, <strong>Bálint Virág</strong>. <p><strong>Source: </strong>Annals of Probability, Volume 48, Number 3, 1286--1316.</p><p><strong>Abstract:</strong><br/>
We provide a precise coupling of the finite circular beta ensembles and their limit process via their operator representations. We prove explicit bounds on the distance of the operators and the corresponding point processes. We also prove an estimate on the beta-dependence of the $\operatorname{Sine}_{\beta }$ process.
</p>projecteuclid.org/euclid.aop/1592359229_20200616220034Tue, 16 Jun 2020 22:00 EDTEntropic repulsion for the occupation-time field of random interlacements conditioned on disconnectionhttps://projecteuclid.org/euclid.aop/1592359230<strong>Alberto Chiarini</strong>, <strong>Maximilian Nitzschner</strong>. <p><strong>Source: </strong>Annals of Probability, Volume 48, Number 3, 1317--1351.</p><p><strong>Abstract:</strong><br/>
We investigate percolation of the vacant set of random interlacements on $\mathbb{Z}^{d}$, $d\geq 3$, in the strongly percolative regime. We consider the event that the interlacement set at level $u$ disconnects the discrete blow-up of a compact set $A\subseteq \mathbb{R}^{d}$ from the boundary of an enclosing box. We derive asymptotic large deviation upper bounds on the probability that the local averages of the occupation times deviate from a specific function depending on the harmonic potential of $A$, when disconnection occurs. If certain critical levels coincide, which is plausible but open at the moment, these bounds imply that conditionally on disconnection, the occupation-time profile undergoes an entropic push governed by a specific function depending on $A$. Similar entropic repulsion phenomena conditioned on disconnection by level-sets of the discrete Gaussian free field on $\mathbb{Z}^{d}$, $d\geq 3$, have been obtained by the authors in (Chiarini and Nitzschner (2018)). Our proofs rely crucially on the “solidification estimates” developed in (Nitzschner and Sznitman (2017)).
</p>projecteuclid.org/euclid.aop/1592359230_20200616220034Tue, 16 Jun 2020 22:00 EDTLocality of the critical probability for transitive graphs of exponential growthhttps://projecteuclid.org/euclid.aop/1592359231<strong>Tom Hutchcroft</strong>. <p><strong>Source: </strong>Annals of Probability, Volume 48, Number 3, 1352--1371.</p><p><strong>Abstract:</strong><br/>
Around 2008, Schramm conjectured that the critical probabilities for Bernoulli bond percolation satisfy the following continuity property: If $(G_{n})_{n\geq 1}$ is a sequence of transitive graphs converging locally to a transitive graph $G$ and $\limsup_{n\to \infty }p_{c}(G_{n})<1$, then $p_{c}(G_{n})\to p_{c}(G)$ as $n\to \infty $. We verify this conjecture under the additional hypothesis that there is a uniform exponential lower bound on the volume growth of the graphs in question. The result is new even in the case that the sequence of graphs is uniformly nonamenable.
In the unimodular case, this result is obtained as a corollary to the following theorem of independent interest: For every $g>1$ and $M<\infty $, there exist positive constants $C=C(g,M)$ and $\delta =\delta (g,M)$ such that if $G$ is a transitive unimodular graph with degree at most $M$ and growth $\operatorname{gr}(G):=\inf_{r\geq 1}|B(o,r)|^{1/r}\geq g$, then \[\mathbf{P}_{p_{c}}\bigl(\vert K_{o}\vert \geq n\bigr)\leq Cn^{-\delta }\] for every $n\geq 1$, where $K_{o}$ is the cluster of the root vertex $o$. The proof of this inequality makes use of new universal bounds on the probabilities of certain two-arm events, which hold for every unimodular transitive graph.
</p>projecteuclid.org/euclid.aop/1592359231_20200616220034Tue, 16 Jun 2020 22:00 EDTRandom matrix products: Universality and least singular valueshttps://projecteuclid.org/euclid.aop/1592359232<strong>Phil Kopel</strong>, <strong>Sean O’Rourke</strong>, <strong>Van Vu</strong>. <p><strong>Source: </strong>Annals of Probability, Volume 48, Number 3, 1372--1410.</p><p><strong>Abstract:</strong><br/>
We establish, under a moment matching hypothesis, the local universality of the correlation functions associated with products of $M$ independent i.i.d. random matrices, as $M$ is fixed, and the sizes of the matrices tend to infinity. This generalizes an earlier result of Tao and the third author for the case $M=1$.
We also prove Gaussian limits for the centered linear spectral statistics of products of $M$ independent i.i.d. random matrices. This is done in two steps. First, we establish the result for product random matrices with Gaussian entries, and then extend to the general case of non-Gaussian entries by another moment matching argument. Prior to our result, Gaussian limits were known only for the case $M=1$. In a similar fashion, we establish Gaussian limits for the centered linear spectral statistics of products of independent truncated random unitary matrices. In both cases, we are able to obtain explicit expressions for the limiting variances.
The main difficulty in our study is that the entries of the product matrix are no longer independent. Our key technical lemma is a lower bound on the least singular value of the translated linearization matrix associated with the product of $M$ normalized independent random matrices with independent and identically distributed sub-Gaussian entries. This lemma is of independent interest.
</p>projecteuclid.org/euclid.aop/1592359232_20200616220034Tue, 16 Jun 2020 22:00 EDTPercolation for level-sets of Gaussian free fields on metric graphshttps://projecteuclid.org/euclid.aop/1592359233<strong>Jian Ding</strong>, <strong>Mateo Wirth</strong>. <p><strong>Source: </strong>Annals of Probability, Volume 48, Number 3, 1411--1435.</p><p><strong>Abstract:</strong><br/>
We study level-set percolation for Gaussian free fields on metric graphs. In two dimensions, we give an upper bound on the chemical distance between the two boundaries of a macroscopic annulus. Our bound holds with high probability conditioned on connectivity and is sharp up to a poly-logarithmic factor with an exponent of one-quarter. This substantially improves a previous result by Li and the first author. In three dimensions and higher, we provide rather precise estimates of percolation probabilities in different regimes which altogether describe a sharp phase transition.
</p>projecteuclid.org/euclid.aop/1592359233_20200616220034Tue, 16 Jun 2020 22:00 EDTLarge deviations for the largest eigenvalue of Rademacher matriceshttps://projecteuclid.org/euclid.aop/1592359234<strong>Alice Guionnet</strong>, <strong>Jonathan Husson</strong>. <p><strong>Source: </strong>Annals of Probability, Volume 48, Number 3, 1436--1465.</p><p><strong>Abstract:</strong><br/>
In this article, we consider random Wigner matrices, that is, symmetric matrices such that the subdiagonal entries of $X_{n}$ are independent, centered and with variance one except on the diagonal where the entries have variance two. We prove that, under some suitable hypotheses on the laws of the entries, the law of the largest eigenvalue satisfies a large deviation principle with the same rate function as in the Gaussian case. The crucial assumption is that the Laplace transform of the entries must be bounded above by the Laplace transform of a centered Gaussian variable with same variance. This is satisfied by the Rademacher law and the uniform law on $[-\sqrt{3},\sqrt{3}]$. We extend our result to complex entries Wigner matrices and Wishart matrices.
</p>projecteuclid.org/euclid.aop/1592359234_20200616220034Tue, 16 Jun 2020 22:00 EDTThe almost-sure asymptotic behavior of the solution to the stochastic heat equation with Lévy noisehttps://projecteuclid.org/euclid.aop/1592359235<strong>Carsten Chong</strong>, <strong>Péter Kevei</strong>. <p><strong>Source: </strong>Annals of Probability, Volume 48, Number 3, 1466--1494.</p><p><strong>Abstract:</strong><br/>
We examine the almost-sure asymptotics of the solution to the stochastic heat equation driven by a Lévy space-time white noise. When a spatial point is fixed and time tends to infinity, we show that the solution develops unusually high peaks over short time intervals, even in the case of additive noise, which leads to a breakdown of an intuitively expected strong law of large numbers. More precisely, if we normalize the solution by an increasing nonnegative function, we either obtain convergence to $0$, or the limit superior and/or inferior will be infinite. A detailed analysis of the jumps further reveals that the strong law of large numbers can be recovered on discrete sequences of time points increasing to infinity. This leads to a necessary and sufficient condition that depends on the Lévy measure of the noise and the growth and concentration properties of the sequence at the same time. Finally, we show that our results generalize to the stochastic heat equation with a multiplicative nonlinearity that is bounded away from zero and infinity.
</p>projecteuclid.org/euclid.aop/1592359235_20200616220034Tue, 16 Jun 2020 22:00 EDTConnectivity properties of the adjacency graph of $\mathrm{SLE}_{\kappa}$ bubbles for $\kappa\in(4,8)$https://projecteuclid.org/euclid.aop/1592359236<strong>Ewain Gwynne</strong>, <strong>Joshua Pfeffer</strong>. <p><strong>Source: </strong>Annals of Probability, Volume 48, Number 3, 1495--1519.</p><p><strong>Abstract:</strong><br/>
We study the adjacency graph of bubbles, that is, complementary connected components of a $\mathrm{SLE}_{\kappa }$ curve for $\kappa \in (4,8)$, with two such bubbles considered to be adjacent if their boundaries intersect. We show that this adjacency graph is a.s. connected for $\kappa \in (4,\kappa _{0}]$, where $\kappa _{0}\approx 5.6158$ is defined explicitly. This gives a partial answer to a problem posed by Duplantier, Miller and Sheffield (2014). Our proof in fact yields a stronger connectivity result for $\kappa \in (4,\kappa _{0}]$, which says that there is a Markovian way of finding a path from any fixed bubble to $\infty$. We also show that there is a (nonexplicit) $\kappa _{1}\in (\kappa _{0},8)$ such that this stronger condition does not hold for $\kappa \in [\kappa _{1},8)$.
Our proofs are based on an encoding of $\mathrm{SLE}_{\kappa }$ in terms of a pair of independent $\kappa /4$-stable processes, which allows us to reduce our problem to a problem about stable processes. In fact, due to this encoding, our results can be rephrased as statements about the connectivity of the adjacency graph of loops when one glues together an independent pair of so-called $\kappa /4$-stable looptrees, as studied, for example, by Curien and Kortchemski (2014).
The above encoding comes from the theory of Liouville quantum gravity (LQG), but the paper can be read without any knowledge of LQG if one takes the encoding as a black box.
</p>projecteuclid.org/euclid.aop/1592359236_20200616220034Tue, 16 Jun 2020 22:00 EDTMean field systems on networks, with singular interaction through hitting timeshttps://projecteuclid.org/euclid.aop/1592359237<strong>Sergey Nadtochiy</strong>, <strong>Mykhaylo Shkolnikov</strong>. <p><strong>Source: </strong>Annals of Probability, Volume 48, Number 3, 1520--1556.</p><p><strong>Abstract:</strong><br/>
Building on the line of work ( Ann. Appl. Probab. 25 (2015) 2096–2133; Stochastic Process. Appl. 125 (2015) 2451–2492; Ann. Appl. Probab. 29 (2019) 89–129; Arch. Ration. Mech. Anal. 233 (2019) 643–699; Ann. Appl. Probab. 29 (2019) 2338–2373; Finance Stoch. 23 (2019) 535–594), we continue the study of particle systems with singular interaction through hitting times. In contrast to the previous research, we (i) consider very general driving processes and interaction functions, (ii) allow for inhomogeneous connection structures and (iii) analyze a game in which the particles determine their connections strategically. Hereby, we uncover two completely new phenomena. First, we characterize the “times of fragility” of such systems (e.g., the times when a macroscopic part of the population defaults or gets infected simultaneously, or when the neuron cells “synchronize”) explicitly in terms of the dynamics of the driving processes, the current distribution of the particles’ values and the topology of the underlying network (represented by its Perron–Frobenius eigenvalue). Second, we use such systems to describe a dynamic credit-network game and show that, in equilibrium, the system regularizes, that is, the times of fragility never occur, as the particles avoid them by adjusting their connections strategically. Two auxiliary mathematical results, useful in their own right, are uncovered during our investigation: a generalization of Schauder’s fixed-point theorem for the Skorokhod space with the M1 topology, and the application of max-plus algebra to the equilibrium version of the network flow problem.
</p>projecteuclid.org/euclid.aop/1592359237_20200616220034Tue, 16 Jun 2020 22:00 EDTFinitary codings for spatial mixing Markov random fieldshttps://projecteuclid.org/euclid.aop/1592359238<strong>Yinon Spinka</strong>. <p><strong>Source: </strong>Annals of Probability, Volume 48, Number 3, 1557--1591.</p><p><strong>Abstract:</strong><br/>
It has been shown by van den Berg and Steif ( Ann. Probab. 27 (1999) 1501–1522) that the subcritical and critical Ising model on $\mathbb{Z}^{d}$ is a finitary factor of an i.i.d. process (ffiid), whereas the super-critical model is not. In fact, they showed that the latter is a general phenomenon in that a phase transition presents an obstruction for being ffiid. The question remained whether this is the only such obstruction. We make progress on this, showing that certain spatial mixing conditions (notions of weak dependence on boundary conditions, not to be confused with other notions of mixing in ergodic theory) imply ffiid. Our main result is that weak spatial mixing implies ffiid with power-law tails for the coding radius, and that strong spatial mixing implies ffiid with exponential tails for the coding radius. The weak spatial mixing condition can be relaxed to a condition which is satisfied by some critical two-dimensional models. Using a result of the author (Spinka (2018)), we deduce that strong spatial mixing also implies ffiid with stretched-exponential tails from a finite-valued i.i.d. process.
We give several applications to models such as the Potts model, proper colorings, the hard-core model, the Widom–Rowlinson model and the beach model. For instance, for the ferromagnetic $q$-state Potts model on $\mathbb{Z}^{d}$ at inverse temperature $\beta $, we show that it is ffiid with exponential tails if $\beta $ is sufficiently small, it is ffiid if $\beta <\beta _{c}(q,d)$, it is not ffiid if $\beta >\beta_{c}(q,d)$ and, when $d=2$ and $\beta =\beta _{c}(q,d)$, it is ffiid if and only if $q\le 4$.
</p>projecteuclid.org/euclid.aop/1592359238_20200616220034Tue, 16 Jun 2020 22:00 EDTFlows, coalescence and noise. A correctionhttps://projecteuclid.org/euclid.aop/1592359239<strong>Yves Le Jan</strong>, <strong>Olivier Raimond</strong>. <p><strong>Source: </strong>Annals of Probability, Volume 48, Number 3, 1592--1595.</p>projecteuclid.org/euclid.aop/1592359239_20200616220034Tue, 16 Jun 2020 22:00 EDT