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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 EDTLower bounds for the smallest singular value of structured random matriceshttps://projecteuclid.org/euclid.aop/1537862438<strong>Nicholas Cook</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 6, 3442--3500.</p><p><strong>Abstract:</strong><br/>
We obtain lower tail estimates for the smallest singular value of random matrices with independent but nonidentically distributed entries. Specifically, we consider $n\times n$ matrices with complex entries of the form
\[M=A\circ X+B=(a_{ij}\xi_{ij}+b_{ij}),\] where $X=(\xi_{ij})$ has i.i.d. centered entries of unit variance and $A$ and $B$ are fixed matrices. In our main result, we obtain polynomial bounds on the smallest singular value of $M$ for the case that $A$ has bounded (possibly zero) entries, and $B=Z\sqrt{n}$ where $Z$ is a diagonal matrix with entries bounded away from zero. As a byproduct of our methods we can also handle general perturbations $B$ under additional hypotheses on $A$, which translate to connectivity hypotheses on an associated graph. In particular, we extend a result of Rudelson and Zeitouni for Gaussian matrices to allow for general entry distributions satisfying some moment hypotheses. Our proofs make use of tools which (to our knowledge) were previously unexploited in random matrix theory, in particular Szemerédi’s regularity lemma, and a version of the restricted invertibility theorem due to Spielman and Srivastava.
</p>projecteuclid.org/euclid.aop/1537862438_20180925040105Tue, 25 Sep 2018 04:01 EDTThe scaling limits of the Minimal Spanning Tree and Invasion Percolation in the planehttps://projecteuclid.org/euclid.aop/1537862439<strong>Christophe Garban</strong>, <strong>Gábor Pete</strong>, <strong>Oded Schramm</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 6, 3501--3557.</p><p><strong>Abstract:</strong><br/>
We prove that the Minimal Spanning Tree and the Invasion Percolation Tree on a version of the triangular lattice in the complex plane have unique scaling limits, which are invariant under rotations, scalings, and, in the case of the $\mathsf{MST}$, also under translations. However, they are not expected to be conformally invariant. We also prove some geometric properties of the limiting $\mathsf{MST}$. The topology of convergence is the space of spanning trees introduced by Aizenman et al. [ Random Structures Algorithms 15 (1999) 319–365], and the proof relies on the existence and conformal covariance of the scaling limit of the near-critical percolation ensemble, established in our earlier works.
</p>projecteuclid.org/euclid.aop/1537862439_20180925040105Tue, 25 Sep 2018 04:01 EDTQuenched central limit theorem for random walks in doubly stochastic random environmenthttps://projecteuclid.org/euclid.aop/1537862440<strong>Bálint Tóth</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 6, 3558--3577.</p><p><strong>Abstract:</strong><br/>
We prove the quenched version of the central limit theorem for the displacement of a random walk in doubly stochastic random environment, under the $H_{-1}$-condition, with slightly stronger, $\mathcal{L}^{2+\varepsilon}$ (rather than $\mathcal{L}^{2}$) integrability condition on the stream tensor. On the way we extend Nash’s moment bound to the nonreversible, divergence-free drift case, with unbounded ($\mathcal{L}^{2+\varepsilon}$) stream tensor. This paper is a sequel of [ Ann. Probab. 45 (2017) 4307–4347] and relies on technical results quoted from there.
</p>projecteuclid.org/euclid.aop/1537862440_20180925040105Tue, 25 Sep 2018 04:01 EDTThe KLS isoperimetric conjecture for generalized Orlicz ballshttps://projecteuclid.org/euclid.aop/1537862441<strong>Alexander V. Kolesnikov</strong>, <strong>Emanuel Milman</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 6, 3578--3615.</p><p><strong>Abstract:</strong><br/>
What is the optimal way to cut a convex bounded domain $K$ in Euclidean space $(\mathbb{R}^{n},|\cdot|)$ into two halves of equal volume, so that the interface between the two halves has least surface area? A conjecture of Kannan, Lovász and Simonovits asserts that, if one does not mind gaining a universal numerical factor (independent of $n$) in the surface area, one might as well dissect $K$ using a hyperplane. This conjectured essential equivalence between the former nonlinear isoperimetric inequality and its latter linear relaxation, has been shown over the last two decades to be of fundamental importance to the understanding of volume-concentration and spectral properties of convex domains. In this work, we address the conjecture for the subclass of generalized Orlicz balls
\[K=\{x\in\mathbb{R}^{n};\sum_{i=1}^{n}V_{i}(x_{i})\leq E\},\] confirming its validity for certain levels $E\in\mathbb{R}$ under a mild technical assumption on the growth of the convex functions $V_{i}$ at infinity [without which we confirm the conjecture up to a $\log(1+n)$ factor]. In sharp contrast to previous approaches for tackling the KLS conjecture, we emphasize that no symmetry is required from $K$. This significantly enlarges the subclass of convex bodies for which the conjecture is confirmed.
</p>projecteuclid.org/euclid.aop/1537862441_20180925040105Tue, 25 Sep 2018 04:01 EDTA Stratonovich–Skorohod integral formula for Gaussian rough pathshttps://projecteuclid.org/euclid.aop/1544691617<strong>Thomas Cass</strong>, <strong>Nengli Lim</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 1, 1--60.</p><p><strong>Abstract:</strong><br/>
Given a Gaussian process $X$, its canonical geometric rough path lift $\mathbf{X}$, and a solution $Y$ to the rough differential equation (RDE) $\mathrm{d}Y_{t}=V(Y_{t})\circ\mathrm{d}\mathbf{X}_{t}$, we present a closed-form correction formula for $\int Y\circ\mathrm{d}\mathbf{X}-\int Y\,\mathrm{d}X$, that is, the difference between the rough and Skorohod integrals of $Y$ with respect to $X$. When $X$ is standard Brownian motion, we recover the classical Stratonovich-to-Itô conversion formula, which we generalize to Gaussian rough paths with finite $p$-variation, $p<3$, and satisfying an additional natural condition. This encompasses many familiar examples, including fractional Brownian motion with $H>\frac{1}{3}$. To prove the formula, we first show that the Riemann-sum approximants of the Skorohod integral converge in $L^{2}(\Omega)$ by using a novel characterization of the Cameron–Martin norm in terms of higher-dimensional Young–Stieltjes integrals. Next, we append the approximants of the Skorohod integral with a suitable compensation term without altering the limit, and the formula is finally obtained after a rebalancing of terms.
</p>projecteuclid.org/euclid.aop/1544691617_20181213040100Thu, 13 Dec 2018 04:01 ESTBerry–Esseen bounds of normal and nonnormal approximation for unbounded exchangeable pairshttps://projecteuclid.org/euclid.aop/1544691618<strong>Qi-Man Shao</strong>, <strong>Zhuo-Song Zhang</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 1, 61--108.</p><p><strong>Abstract:</strong><br/>
An exchangeable pair approach is commonly taken in the normal and nonnormal approximation using Stein’s method. It has been successfully used to identify the limiting distribution and provide an error of approximation. However, when the difference of the exchangeable pair is not bounded by a small deterministic constant, the error bound is often not optimal. In this paper, using the exchangeable pair approach of Stein’s method, a new Berry–Esseen bound for an arbitrary random variable is established without a bound on the difference of the exchangeable pair. An optimal convergence rate for normal and nonnormal approximation is achieved when the result is applied to various examples including the quadratic forms, general Curie–Weiss model, mean field Heisenberg model and colored graph model.
</p>projecteuclid.org/euclid.aop/1544691618_20181213040100Thu, 13 Dec 2018 04:01 ESTStructure of optimal martingale transport plans in general dimensionshttps://projecteuclid.org/euclid.aop/1544691619<strong>Nassif Ghoussoub</strong>, <strong>Young-Heon Kim</strong>, <strong>Tongseok Lim</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 1, 109--164.</p><p><strong>Abstract:</strong><br/>
Given two probability measures $\mu$ and $\nu$ in “convex order” on $\mathbb{R}^{d}$, we study the profile of one-step martingale plans $\pi$ on $\mathbb{R}^{d}\times\mathbb{R}^{d}$ that optimize the expected value of the modulus of their increment among all martingales having $\mu$ and $\nu$ as marginals. While there is a great deal of results for the real line (i.e., when $d=1$), much less is known in the richer and more delicate higher-dimensional case that we tackle in this paper. We show that many structural results can be obtained, provided the initial measure $\mu$ is absolutely continuous with respect to the Lebesgue measure. One such a property is that $\mu$-almost every $x$ in $\mathbb{R}^{d}$ is transported by the optimal martingale plan into a probability measure $\pi_{x}$ concentrated on the extreme points of the closed convex hull of its support. This will be established for the distance cost $c(x,y)=\vert x-y\vert $ in the two-dimensional case, and also for any $d\geq3$ as long as the marginals are in “subharmonic order.” In some cases, $\pi_{x}$ is supported on the vertices of a $k(x)$-dimensional polytope, such as when the target measure is discrete. Duality plays a crucial role in our approach, even though, in contrast to standard optimal transports, the dual extremal problem may not be attained in general. We show however that “martingale supporting” Borel subsets of $\mathbb{R}^{d}\times\mathbb{R}^{d}$ can be decomposed into a collection of mutually disjoint components by means of a “convex paving” of the source space, in such a way that when the martingale is optimal for a general cost function, each of the components then supports a restricted optimal martingale transport whose dual problem is attained. This decomposition is used to obtain structural results in cases where global duality is not attained. On the other hand, it shows that certain “optimal martingale supporting” Borel sets can be viewed as higher-dimensional versions of Nikodym-type sets. The paper focuses on the distance cost, but much of the results hold for general Lipschitz cost functions.
</p>projecteuclid.org/euclid.aop/1544691619_20181213040100Thu, 13 Dec 2018 04:01 ESTRegularization by noise and flows of solutions for a stochastic heat equationhttps://projecteuclid.org/euclid.aop/1544691620<strong>Oleg Butkovsky</strong>, <strong>Leonid Mytnik</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 1, 165--212.</p><p><strong>Abstract:</strong><br/>
Motivated by the regularization by noise phenomenon for SDEs, we prove existence and uniqueness of the flow of solutions for the non-Lipschitz stochastic heat equation
\[\frac{\partial u}{\partial t}=\frac{1}{2}\frac{\partial^{2}u}{\partial z^{2}}+b\bigl(u(t,z)\bigr)+\dot{W}(t,z),\] where $\dot{W}$ is a space-time white noise on $\mathbb{R}_{+}\times\mathbb{R}$ and $b$ is a bounded measurable function on $\mathbb{R}$. As a byproduct of our proof, we also establish the so-called path-by-path uniqueness for any initial condition in a certain class on the same set of probability one. To obtain these results, we develop a new approach that extends Davie’s method (2007) to the context of stochastic partial differential equations.
</p>projecteuclid.org/euclid.aop/1544691620_20181213040100Thu, 13 Dec 2018 04:01 ESTBrownian motion on some spaces with varying dimensionhttps://projecteuclid.org/euclid.aop/1544691621<strong>Zhen-Qing Chen</strong>, <strong>Shuwen Lou</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 1, 213--269.</p><p><strong>Abstract:</strong><br/>
In this paper, we introduce and study Brownian motion on a class of state spaces with varying dimension. Starting with a concrete case of such state spaces that models a big square with a flag pole, we construct a Brownian motion on it and study how heat propagates on such a space. We derive sharp two-sided global estimates on its transition density function (also called heat kernel). These two-sided estimates are of Gaussian type, but the measure on the underlying state space does not satisfy volume doubling property. Parabolic Harnack inequality fails for such a process. Nevertheless, we show Hölder regularity holds for its parabolic functions. We also derive the Green function estimates for this process on bounded smooth domains. Brownian motion on some other state spaces with varying dimension are also constructed and studied in this paper.
</p>projecteuclid.org/euclid.aop/1544691621_20181213040100Thu, 13 Dec 2018 04:01 ESTRényi divergence and the central limit theoremhttps://projecteuclid.org/euclid.aop/1544691622<strong>S. G. Bobkov</strong>, <strong>G. P. Chistyakov</strong>, <strong>F. Götze</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 1, 270--323.</p><p><strong>Abstract:</strong><br/>
We explore properties of the $\chi^{2}$ and Rényi distances to the normal law and in particular propose necessary and sufficient conditions under which these distances tend to zero in the central limit theorem (with exact rates with respect to the increasing number of summands).
</p>projecteuclid.org/euclid.aop/1544691622_20181213040100Thu, 13 Dec 2018 04:01 ESTTowards a universality picture for the relaxation to equilibrium of kinetically constrained modelshttps://projecteuclid.org/euclid.aop/1544691623<strong>Fabio Martinelli</strong>, <strong>Cristina Toninelli</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 1, 324--361.</p><p><strong>Abstract:</strong><br/>
Recent years have seen a great deal of progress in our understanding of bootstrap percolation models, a particular class of monotone cellular automata . In the two-dimensional lattice $\mathbb{Z}^{2}$, there is now a quite satisfactory understanding of their evolution starting from a random initial condition, with a strikingly beautiful universality picture for their critical behavior. Much less is known for their nonmonotone stochastic counterpart, namely kinetically constrained models (KCM). In KCM, each vertex is resampled (independently) at rate one by tossing a $p$-coin iff it can be infected in the next step by the bootstrap model. In particular, an infection can also heal, hence the nonmonotonicity. Besides the connection with bootstrap percolation, KCM have an interest in their own as they feature some of the most striking features of the liquid/glass transition, a major and still largely open problem in condensed matter physics. In this paper, we pave the way towards proving universality results for the characteristic time scales of KCM. Our novel and general approach gives the right tools to establish a close connection between the critical scaling of characteristic time scales for KCM and the scaling of the critical length in critical bootstrap models. When applied to the Fredrickson–Andersen $k$-facilitated models in dimension $d\ge2$, among the most studied KCM, and to the Gravner–Griffeath model, our results are close to optimal.
</p>projecteuclid.org/euclid.aop/1544691623_20181213040100Thu, 13 Dec 2018 04:01 ESTThe spectral gap of dense random regular graphshttps://projecteuclid.org/euclid.aop/1544691624<strong>Konstantin Tikhomirov</strong>, <strong>Pierre Youssef</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 1, 362--419.</p><p><strong>Abstract:</strong><br/>
For any $\alpha\in(0,1)$ and any $n^{\alpha}\leq d\leq n/2$, we show that $\lambda({\mathbf{G}})\leq C_{\alpha}\sqrt{d}$ with probability at least $1-\frac{1}{n}$, where ${\mathbf{G}}$ is the uniform random undirected $d$-regular graph on $n$ vertices, $\lambda({\mathbf{G}})$ denotes its second largest eigenvalue (in absolute value) and $C_{\alpha}$ is a constant depending only on $\alpha$. Combined with earlier results in this direction covering the case of sparse random graphs, this completely settles the problem of estimating the magnitude of $\lambda({\mathbf{G}})$, up to a multiplicative constant, for all values of $n$ and $d$, confirming a conjecture of Vu. The result is obtained as a consequence of an estimate for the second largest singular value of adjacency matrices of random directed graphs with predefined degree sequences. As the main technical tool, we prove a concentration inequality for arbitrary linear forms on the space of matrices, where the probability measure is induced by the adjacency matrix of a random directed graph with prescribed degree sequences. The proof is a nontrivial application of the Freedman inequality for martingales, combined with self-bounding and tensorization arguments. Our method bears considerable differences compared to the approach used by Broder et al. [ SIAM J. Comput. 28 (1999) 541–573] who established the upper bound for $\lambda({\mathbf{G}})$ for $d=o(\sqrt{n})$, and to the argument of Cook, Goldstein and Johnson [ Ann. Probab. 46 (2018) 72–125] who derived a concentration inequality for linear forms and estimated $\lambda({\mathbf{G}})$ in the range $d=O(n^{2/3})$ using size-biased couplings.
</p>projecteuclid.org/euclid.aop/1544691624_20181213040100Thu, 13 Dec 2018 04:01 ESTCanonical RDEs and general semimartingales as rough pathshttps://projecteuclid.org/euclid.aop/1544691625<strong>Ilya Chevyrev</strong>, <strong>Peter K. Friz</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 1, 420--463.</p><p><strong>Abstract:</strong><br/>
In the spirit of Marcus canonical stochastic differential equations, we study a similar notion of rough differential equations (RDEs), notably dropping the assumption of continuity prevalent in the rough path literature. A new metric is exhibited in which the solution map is a continuous function of the driving rough path and a so-called path function, which directly models the effect of the jump on the system. In a second part, we show that general multidimensional semimartingales admit canonically defined rough path lifts. An extension of Lépingle’s BDG inequality to this setting is given, and in turn leads to a number of novel limit theorems for semimartingale driven differential equations, both in law and in probability, conveniently phrased a uniformly-controlled-variations (UCV) condition (Kurtz–Protter, Jakubowski–Mémin–Pagès). A number of examples illustrate the scope of our results.
</p>projecteuclid.org/euclid.aop/1544691625_20181213040100Thu, 13 Dec 2018 04:01 ESTRate of convergence to equilibrium of fractional driven stochastic differential equations with rough multiplicative noisehttps://projecteuclid.org/euclid.aop/1544691626<strong>Aurélien Deya</strong>, <strong>Fabien Panloup</strong>, <strong>Samy Tindel</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 1, 464--518.</p><p><strong>Abstract:</strong><br/>
We investigate the problem of the rate of convergence to equilibrium for ergodic stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H\in(1/3,1)$ and multiplicative noise component $\sigma$. When $\sigma$ is constant and for every $H\in(0,1)$, it was proved in [ Ann. Probab. 33 (2005) 703–758] that, under some mean-reverting assumptions, such a process converges to its equilibrium at a rate of order $t^{-\alpha}$ where $\alpha\in(0,1)$ (depending on $H$). In [ Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017) 503–538], this result has been extended to the multiplicative case when $H>1/2$. In this paper, we obtain these types of results in the rough setting $H\in(1/3,1/2)$. Once again, we retrieve the rate orders of the additive setting. Our methods also extend the multiplicative results of [ Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017) 503–538] by deleting the gradient assumption on the noise coefficient $\sigma$. The main theorems include some existence and uniqueness results for the invariant distribution.
</p>projecteuclid.org/euclid.aop/1544691626_20181213040100Thu, 13 Dec 2018 04:01 ESTGlobal solutions to stochastic reaction–diffusion equations with super-linear drift and multiplicative noisehttps://projecteuclid.org/euclid.aop/1544691627<strong>Robert C. Dalang</strong>, <strong>Davar Khoshnevisan</strong>, <strong>Tusheng Zhang</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 1, 519--559.</p><p><strong>Abstract:</strong><br/>
Let $\xi (t,x)$ denote space–time white noise and consider a reaction–diffusion equation of the form \[\dot{u}(t,x)=\frac{1}{2}u"(t,x)+b\big(u(t,x)\big)+\sigma \big(u(t,x)\big)\xi (t,x),\] on $\mathbb{R}_{+}\times [0,1]$, with homogeneous Dirichlet boundary conditions and suitable initial data, in the case that there exists $\varepsilon >0$ such that $\vert b(z)\vert \ge \vert z\vert (\log \vert z\vert )^{1+\varepsilon }$ for all sufficiently-large values of $\vert z\vert $. When $\sigma \equiv 0$, it is well known that such PDEs frequently have nontrivial stationary solutions. By contrast, Bonder and Groisman [ Phys. D 238 (2009) 209–215] have recently shown that there is finite-time blowup when $\sigma $ is a nonzero constant. In this paper, we prove that the Bonder–Groisman condition is unimprovable by showing that the reaction–diffusion equation with noise is “typically” well posed when $\vert b(z)\vert =O(\vert z\vert \log_{+}\vert z\vert )$ as $\vert z\vert \to \infty $. We interpret the word “typically” in two essentially-different ways without altering the conclusions of our assertions.
</p>projecteuclid.org/euclid.aop/1544691627_20181213040100Thu, 13 Dec 2018 04:01 ESTSharp interface limit for stochastically perturbed mass conserving Allen–Cahn equationhttps://projecteuclid.org/euclid.aop/1544691628<strong>Tadahisa Funaki</strong>, <strong>Satoshi Yokoyama</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 1, 560--612.</p><p><strong>Abstract:</strong><br/>
This paper studies the sharp interface limit for a mass conserving Allen–Cahn equation, added an external noise and derives a stochastically perturbed mass conserving mean curvature flow in the limit. The stochastic term destroys the precise conservation law, instead the total mass changes like a Brownian motion in time. For our equation, the comparison argument does not work, so that to study the limit we adopt the asymptotic expansion method, which extends that for deterministic equations used originally in de Mottoni and Schatzman [ Interfaces Free Bound. 12 (2010) 527–549] for the nonconservative case and then in Chen et al. [ Trans. Amer. Math. Soc. 347 (1995) 1533–1589] for the conservative case. Differently from the deterministic case, each term except the leading term appearing in the expansion of the solution in a small parameter $\varepsilon$ diverges as $\varepsilon$ tends to $0$, since our equation contains the noise which converges to a white noise and the products or the powers of the white noise diverge. To derive the error estimate for our asymptotic expansion, we need to establish the Schauder estimate for a diffusion operator with coefficients determined from higher order derivatives of the noise and their powers. We show that one can choose the noise sufficiently mild in such a manner that it converges to the white noise and at the same time its diverging speed is slow enough for establishing a necessary error estimate.
</p>projecteuclid.org/euclid.aop/1544691628_20181213040100Thu, 13 Dec 2018 04:01 ESTPhase transitions in the ASEP and stochastic six-vertex modelhttps://projecteuclid.org/euclid.aop/1551171634<strong>Amol Aggarwal</strong>, <strong>Alexei Borodin</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 2, 613--689.</p><p><strong>Abstract:</strong><br/>
In this paper, we consider two models in the Kardar–Parisi–Zhang (KPZ) universality class, the asymmetric simple exclusion process (ASEP) and the stochastic six-vertex model. We introduce a new class of initial data (which we call shape generalized step Bernoulli initial data ) for both of these models that generalizes the step Bernoulli initial data studied in a number of recent works on the ASEP. Under this class of initial data, we analyze the current fluctuations of both the ASEP and stochastic six-vertex model and establish the existence of a phase transition along a characteristic line, across which the fluctuation exponent changes from $1/2$ to $1/3$. On the characteristic line, the current fluctuations converge to the general (rank $k$) Baik–Ben–Arous–Péché distribution for the law of the largest eigenvalue of a critically spiked covariance matrix. For $k=1$, this was established for the ASEP by Tracy and Widom; for $k>1$ (and also $k=1$, for the stochastic six-vertex model), the appearance of these distributions in both models is new.
</p>projecteuclid.org/euclid.aop/1551171634_20190226040136Tue, 26 Feb 2019 04:01 ESTLiouville first-passage percolation: Subsequential scaling limits at high temperaturehttps://projecteuclid.org/euclid.aop/1551171635<strong>Jian Ding</strong>, <strong>Alexander Dunlap</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 2, 690--742.</p><p><strong>Abstract:</strong><br/>
Let $\{Y_{\mathfrak{B}}(x):x\in\mathfrak{B}\}$ be a discrete Gaussian free field in a two-dimensional box $\mathfrak{B}$ of side length $S$ with Dirichlet boundary conditions. We study Liouville first-passage percolation: the shortest-path metric in which each vertex $x$ is given a weight of $e^{\gamma Y_{\mathfrak{B}}(x)}$ for some $\gamma>0$. We show that for sufficiently small but fixed $\gamma>0$, for any sequence of scales $\{S_{k}\}$ there exists a subsequence along which the appropriately scaled and interpolated Liouville FPP metric converges in the Gromov–Hausdorff sense to a random metric on the unit square in $\mathbf{R}^{2}$. In addition, all possible (conjecturally unique) scaling limits are homeomorphic by bi-Hölder-continuous homeomorphisms to the unit square with the Euclidean metric.
</p>projecteuclid.org/euclid.aop/1551171635_20190226040136Tue, 26 Feb 2019 04:01 ESTDerivative and divergence formulae for diffusion semigroupshttps://projecteuclid.org/euclid.aop/1551171636<strong>Anton Thalmaier</strong>, <strong>James Thompson</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 2, 743--773.</p><p><strong>Abstract:</strong><br/>
For a semigroup $P_{t}$ generated by an elliptic operator on a smooth manifold $M$, we use straightforward martingale arguments to derive probabilistic formulae for $P_{t}(V(f))$, not involving derivatives of $f$, where $V$ is a vector field on $M$. For nonsymmetric generators, such formulae correspond to the derivative of the heat kernel in the forward variable. As an application, these formulae can be used to derive various shift-Harnack inequalities.
</p>projecteuclid.org/euclid.aop/1551171636_20190226040136Tue, 26 Feb 2019 04:01 ESTLow-dimensional lonely branching random walks die outhttps://projecteuclid.org/euclid.aop/1551171637<strong>Matthias Birkner</strong>, <strong>Rongfeng Sun</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 2, 774--803.</p><p><strong>Abstract:</strong><br/>
The lonely branching random walks on $\mathbb{Z}^{d}$ is an interacting particle system where each particle moves as an independent random walk and undergoes critical binary branching when it is alone. We show that if the symmetrized walk is recurrent, lonely branching random walks die out locally. Furthermore, the same result holds if additional branching is allowed when the walk is not alone.
</p>projecteuclid.org/euclid.aop/1551171637_20190226040136Tue, 26 Feb 2019 04:01 ESTBoundary regularity of stochastic PDEshttps://projecteuclid.org/euclid.aop/1551171638<strong>Máté Gerencsér</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 2, 804--834.</p><p><strong>Abstract:</strong><br/>
The boundary behaviour of solutions of stochastic PDEs with Dirichlet boundary conditions can be surprisingly—and in a sense, arbitrarily—bad: as shown by Krylov [ SIAM J. Math. Anal. 34 (2003) 1167–1182], for any $\alpha>0$ one can find a simple $1$-dimensional constant coefficient linear equation whose solution at the boundary is not $\alpha$-Hölder continuous.
We obtain a positive counterpart of this: under some mild regularity assumptions on the coefficients, solutions of semilinear SPDEs on $\mathcal{C}^{1}$ domains are proved to be $\alpha$-Hölder continuous up to the boundary with some $\alpha>0$.
</p>projecteuclid.org/euclid.aop/1551171638_20190226040136Tue, 26 Feb 2019 04:01 ESTLimit theory for geometric statistics of point processes having fast decay of correlationshttps://projecteuclid.org/euclid.aop/1551171639<strong>B. Błaszczyszyn</strong>, <strong>D. Yogeshwaran</strong>, <strong>J. E. Yukich</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 2, 835--895.</p><p><strong>Abstract:</strong><br/>
Let $\mathcal{P}$ be a simple, stationary point process on $\mathbb{R}^{d}$ having fast decay of correlations, that is, its correlation functions factorize up to an additive error decaying faster than any power of the separation distance. Let $\mathcal{P}_{n}:=\mathcal{P}\cap W_{n}$ be its restriction to windows $W_{n}:=[-{\frac{1}{2}}n^{1/d},{\frac{1}{2}}n^{1/d}]^{d}\subset\mathbb{R}^{d}$. We consider the statistic $H_{n}^{\xi}:=\sum_{x\in\mathcal{P}_{n}}\xi(x,\mathcal{P}_{n})$ where $\xi(x,\mathcal{P}_{n})$ denotes a score function representing the interaction of $x$ with respect to $\mathcal{P}_{n}$. When $\xi$ depends on local data in the sense that its radius of stabilization has an exponential tail, we establish expectation asymptotics, variance asymptotics and central limit theorems for $H_{n}^{\xi}$ and, more generally, for statistics of the re-scaled, possibly signed, $\xi$-weighted point measures $\mu_{n}^{\xi}:=\sum_{x\in\mathcal{P}_{n}}\xi(x,\mathcal{P}_{n})\delta_{n^{-1/d}x}$, as $W_{n}\uparrow\mathbb{R}^{d}$. This gives the limit theory for nonlinear geometric statistics (such as clique counts, the number of Morse critical points, intrinsic volumes of the Boolean model and total edge length of the $k$-nearest neighbors graph) of $\alpha$-determinantal point processes (for $-1/\alpha\in\mathbb{N}$) having fast decreasing kernels, including the $\beta$-Ginibre ensembles, extending the Gaussian fluctuation results of Soshnikov [ Ann. Probab. 30 (2002) 171–187] to nonlinear statistics. It also gives the limit theory for geometric $U$-statistics of $\alpha$-permanental point processes (for $1/\alpha\in\mathbb{N}$) as well as the zero set of Gaussian entire functions, extending the central limit theorems of Nazarov and Sodin [ Comm. Math. Phys. 310 (2012) 75–98] and Shirai and Takahashi [ J. Funct. Anal. 205 (2003) 414–463], which are also confined to linear statistics. The proof of the central limit theorem relies on a factorial moment expansion originating in [ Stochastic Process. Appl. 56 (1995) 321–335; Statist. Probab. Lett. 36 (1997) 299–306] to show the fast decay of the correlations of $\xi$-weighted point measures. The latter property is shown to imply a condition equivalent to Brillinger mixing, and consequently yields the asymptotic normality of $\mu_{n}^{\xi}$ via an extension of the cumulant method.
</p>projecteuclid.org/euclid.aop/1551171639_20190226040136Tue, 26 Feb 2019 04:01 ESTDifferential subordination under change of lawhttps://projecteuclid.org/euclid.aop/1551171640<strong>Komla Domelevo</strong>, <strong>Stefanie Petermichl</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 2, 896--925.</p><p><strong>Abstract:</strong><br/>
We prove optimal $L^{2}$ bounds for a pair of Hilbert space valued differentially subordinate martingales under a change of law. The change of law is given by a process called a weight and sharpness, and in this context refers to the optimal growth with respect to the characteristic of the weight. The pair of martingales is adapted, uniformly integrable and càdlàg. Differential subordination is in the sense of Burkholder, defined through the use of the square bracket. In the scalar dyadic setting with underlying Lebesgue measure, this was proved by Wittwer [ Math. Res. Lett. 7 (2000) 1–12], where homogeneity was heavily used. Recent progress by Thiele–Treil–Volberg [ Adv. Math. 285 (2015) 1155–1188] and Lacey [ Israel J. Math. 217 (2017) 181–195] independently resolved the so-called nonhomogenous case using discrete in time filtrations, where one martingale is a predictable multiplier of the other. The general case for continuous-in-time filtrations and pairs of martingales that are not necessarily predictable multipliers, remained open and is addressed here. As a very useful second main result, we give the explicit expression of a Bellman function of four variables for the weighted estimate of subordinate martingales with jumps. This construction includes an analysis of the regularity of this function as well as a very precise convexity, needed to deal with the jump part.
</p>projecteuclid.org/euclid.aop/1551171640_20190226040136Tue, 26 Feb 2019 04:01 ESTCentral limit theorems for empirical transportation cost in general dimensionhttps://projecteuclid.org/euclid.aop/1551171641<strong>Eustasio del Barrio</strong>, <strong>Jean-Michel Loubes</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 2, 926--951.</p><p><strong>Abstract:</strong><br/>
We consider the problem of optimal transportation with quadratic cost between a empirical measure and a general target probability on $\mathbb{R}^{d}$, with $d\geq1$. We provide new results on the uniqueness and stability of the associated optimal transportation potentials, namely, the minimizers in the dual formulation of the optimal transportation problem. As a consequence, we show that a CLT holds for the empirical transportation cost under mild moment and smoothness requirements. The limiting distributions are Gaussian and admit a simple description in terms of the optimal transportation potentials.
</p>projecteuclid.org/euclid.aop/1551171641_20190226040136Tue, 26 Feb 2019 04:01 ESTDeterminantal spanning forests on planar graphshttps://projecteuclid.org/euclid.aop/1551171642<strong>Richard Kenyon</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 2, 952--988.</p><p><strong>Abstract:</strong><br/>
We generalize the uniform spanning tree to construct a family of determinantal measures on essential spanning forests on periodic planar graphs in which every component tree is bi-infinite. Like the uniform spanning tree, these measures arise naturally from the Laplacian on the graph.
More generally, these results hold for the “massive” Laplacian determinant which counts rooted spanning forests with weight $M$ per finite component. These measures typically have a form of conformal invariance, unlike the usual rooted spanning tree measure. We show that the spectral curve for these models is always a simple Harnack curve; this fact controls the decay of edge-edge correlations in these models.
We construct a limit shape theory in these settings, where the limit shapes are defined by measured foliations of fixed isotopy type.
</p>projecteuclid.org/euclid.aop/1551171642_20190226040136Tue, 26 Feb 2019 04:01 ESTComparison principle for stochastic heat equation on $\mathbb{R}^{d}$https://projecteuclid.org/euclid.aop/1551171643<strong>Le Chen</strong>, <strong>Jingyu Huang</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 2, 989--1035.</p><p><strong>Abstract:</strong><br/>
We establish the strong comparison principle and strict positivity of solutions to the following nonlinear stochastic heat equation on $\mathbb{R}^{d}$
\[\biggl(\frac{\partial}{\partial t}-\frac{1}{2}\Delta \biggr)u(t,x)=\rho\bigl(u(t,x)\bigr)\dot{M}(t,x),\] for measure-valued initial data, where $\dot{M}$ is a spatially homogeneous Gaussian noise that is white in time and $\rho$ is Lipschitz continuous. These results are obtained under the condition that $\int_{\mathbb{R}^{d}}(1+|\xi|^{2})^{\alpha-1}\hat{f}(\text{d}\xi)<\infty$ for some $\alpha\in(0,1]$, where $\hat{f}$ is the spectral measure of the noise. The weak comparison principle and nonnegativity of solutions to the same equation are obtained under Dalang’s condition, that is, $\alpha=0$. As some intermediate results, we obtain handy upper bounds for $L^{p}(\Omega)$-moments of $u(t,x)$ for all $p\ge2$, and also prove that $u$ is a.s. Hölder continuous with order $\alpha-\varepsilon$ in space and $\alpha/2-\varepsilon$ in time for any small $\varepsilon>0$.
</p>projecteuclid.org/euclid.aop/1551171643_20190226040136Tue, 26 Feb 2019 04:01 ESTKirillov–Frenkel character formula for loop groups, radial part and Brownian sheethttps://projecteuclid.org/euclid.aop/1551171644<strong>Manon Defosseux</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 2, 1036--1055.</p><p><strong>Abstract:</strong><br/>
We consider the coadjoint action of a Loop group of a compact group on the dual of the corresponding centrally extended Loop algebra and prove that a Brownian motion in a Cartan subalgebra conditioned to remain in an affine Weyl chamber—which can be seen as a space time conditioned Brownian motion—is distributed as the radial part process of a Brownian sheet on the underlying Lie algebra.
</p>projecteuclid.org/euclid.aop/1551171644_20190226040136Tue, 26 Feb 2019 04:01 ESTHeat kernel upper bounds for interacting particle systemshttps://projecteuclid.org/euclid.aop/1551171645<strong>Arianna Giunti</strong>, <strong>Yu Gu</strong>, <strong>Jean-Christophe Mourrat</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 2, 1056--1095.</p><p><strong>Abstract:</strong><br/>
We show a diffusive upper bound on the transition probability of a tagged particle in the symmetric simple exclusion process. The proof relies on optimal spectral gap estimates for the dynamics in finite volume, which are of independent interest. We also show off-diagonal estimates of Carne–Varopoulos type.
</p>projecteuclid.org/euclid.aop/1551171645_20190226040136Tue, 26 Feb 2019 04:01 ESTParacontrolled quasilinear SPDEshttps://projecteuclid.org/euclid.aop/1551171646<strong>Marco Furlan</strong>, <strong>Massimiliano Gubinelli</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 2, 1096--1135.</p><p><strong>Abstract:</strong><br/>
We introduce a nonlinear paracontrolled calculus and use it to renormalise a class of singular SPDEs including certain quasilinear variants of the periodic two-dimensional parabolic Anderson model.
</p>projecteuclid.org/euclid.aop/1551171646_20190226040136Tue, 26 Feb 2019 04:01 ESTErdős–Feller–Kolmogorov–Petrowsky law of the iterated logarithm for self-normalized martingales: A game-theoretic approachhttps://projecteuclid.org/euclid.aop/1551171647<strong>Takeyuki Sasai</strong>, <strong>Kenshi Miyabe</strong>, <strong>Akimichi Takemura</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 2, 1136--1161.</p><p><strong>Abstract:</strong><br/>
We prove an Erdős–Feller–Kolmogorov–Petrowsky law of the iterated logarithm for self-normalized martingales. Our proof is given in the framework of the game-theoretic probability of Shafer and Vovk. Like many other game-theoretic proofs, our proof is self-contained and explicit.
</p>projecteuclid.org/euclid.aop/1551171647_20190226040136Tue, 26 Feb 2019 04:01 ESTCritical radius and supremum of random spherical harmonicshttps://projecteuclid.org/euclid.aop/1551171648<strong>Renjie Feng</strong>, <strong>Robert J. Adler</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 2, 1162--1184.</p><p><strong>Abstract:</strong><br/>
We first consider deterministic immersions of the $d$-dimensional sphere into high dimensional Euclidean spaces, where the immersion is via spherical harmonics of level $n$. The main result of the article is the, a priori unexpected, fact that there is a uniform lower bound to the critical radius of the immersions as $n\to\infty$. This fact has immediate implications for random spherical harmonics with fixed $L^{2}$-norm. In particular, it leads to an exact and explicit formulae for the tail probability of their (large deviation) suprema by the tube formula, and also relates this to the expected Euler characteristic of their upper level sets.
</p>projecteuclid.org/euclid.aop/1551171648_20190226040136Tue, 26 Feb 2019 04:01 ESTComponent sizes for large quantum Erdős–Rényi graph near criticalityhttps://projecteuclid.org/euclid.aop/1551171650<strong>Amir Dembo</strong>, <strong>Anna Levit</strong>, <strong>Sreekar Vadlamani</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 2, 1185--1219.</p><p><strong>Abstract:</strong><br/>
The $N$ vertices of a quantum random graph are each a circle independently punctured at Poisson points of arrivals, with parallel connections derived through for each pair of these punctured circles by yet another independent Poisson process. Considering these graphs at their critical parameters, we show that the joint law of the rescaled by $N^{2/3}$ and ordered sizes of their connected components, converges to that of the ordered lengths of excursions above zero for a reflected Brownian motion with drift. Thereby, this work forms the first example of an inhomogeneous random graph, beyond the case of effectively rank-1 models, which is rigorously shown to be in the Erdős–Rényi graphs universality class in terms of Aldous’s results.
</p>projecteuclid.org/euclid.aop/1551171650_20190226040136Tue, 26 Feb 2019 04:01 ESTThe Wiener condition and the conjectures of Embrechts and Goldiehttps://projecteuclid.org/euclid.aop/1556784018<strong>Toshiro Watanabe</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 3, 1221--1239.</p><p><strong>Abstract:</strong><br/>
We show that the class of convolution equivalent distributions and the class of locally subexponential distributions are not closed under convolution roots. It gives a negative answer to the classical conjectures of Embrechts and Goldie. Moreover, we establish two sufficient conditions in order that the class of convolution equivalent distributions is closed under convolution roots.
</p>projecteuclid.org/euclid.aop/1556784018_20190502040034Thu, 02 May 2019 04:00 EDTBipolar orientations on planar maps and $\mathrm{SLE}_{12}$https://projecteuclid.org/euclid.aop/1556784019<strong>Richard Kenyon</strong>, <strong>Jason Miller</strong>, <strong>Scott Sheffield</strong>, <strong>David B. Wilson</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 3, 1240--1269.</p><p><strong>Abstract:</strong><br/>
We give bijections between bipolar-oriented (acyclic with unique source and sink) planar maps and certain random walks, which show that the uniformly random bipolar-oriented planar map, decorated by the “peano curve” surrounding the tree of left-most paths to the sink, converges in law with respect to the peanosphere topology to a $\sqrt{4/3}$-Liouville quantum gravity surface decorated by an independent Schramm–Loewner evolution with parameter $\kappa=12$ (i.e., $\mathrm{SLE}_{12}$). This result is universal in the sense that it holds for bipolar-oriented triangulations, quadrangulations, $k$-angulations and maps in which face sizes are mixed.
</p>projecteuclid.org/euclid.aop/1556784019_20190502040034Thu, 02 May 2019 04:00 EDTLocal single ring theorem on optimal scalehttps://projecteuclid.org/euclid.aop/1556784020<strong>Zhigang Bao</strong>, <strong>László Erdős</strong>, <strong>Kevin Schnelli</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 3, 1270--1334.</p><p><strong>Abstract:</strong><br/>
Let $U$ and $V$ be two independent $N$ by $N$ random matrices that are distributed according to Haar measure on $U(N)$. Let $\Sigma$ be a nonnegative deterministic $N$ by $N$ matrix. The single ring theorem [ Ann. of Math. (2) 174 (2011) 1189–1217] asserts that the empirical eigenvalue distribution of the matrix $X:=U\Sigma V^{*}$ converges weakly, in the limit of large $N$, to a deterministic measure which is supported on a single ring centered at the origin in $\mathbb{C}$. Within the bulk regime, that is, in the interior of the single ring, we establish the convergence of the empirical eigenvalue distribution on the optimal local scale of order $N^{-1/2+\varepsilon}$ and establish the optimal convergence rate. The same results hold true when $U$ and $V$ are Haar distributed on $O(N)$.
</p>projecteuclid.org/euclid.aop/1556784020_20190502040034Thu, 02 May 2019 04:00 EDTLarge deviation principle for random matrix productshttps://projecteuclid.org/euclid.aop/1556784021<strong>Cagri Sert</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 3, 1335--1377.</p><p><strong>Abstract:</strong><br/>
Under a Zariski density assumption, we extend the classical theorem of Cramér on large deviations of sums of i.i.d. real random variables to random matrix products.
</p>projecteuclid.org/euclid.aop/1556784021_20190502040034Thu, 02 May 2019 04:00 EDTRegenerative random permutations of integershttps://projecteuclid.org/euclid.aop/1556784022<strong>Jim Pitman</strong>, <strong>Wenpin Tang</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 3, 1378--1416.</p><p><strong>Abstract:</strong><br/>
Motivated by recent studies of large $\operatorname{Mallows}(q)$ permutations, we propose a class of random permutations of $\mathbb{N}_{+}$ and of $\mathbb{Z}$, called regenerative permutations . Many previous results of the limiting $\operatorname{Mallows}(q)$ permutations are recovered and extended. Three special examples: blocked permutations, $p$-shifted permutations and $p$-biased permutations are studied.
</p>projecteuclid.org/euclid.aop/1556784022_20190502040034Thu, 02 May 2019 04:00 EDTFour moments theorems on Markov chaoshttps://projecteuclid.org/euclid.aop/1556784023<strong>Solesne Bourguin</strong>, <strong>Simon Campese</strong>, <strong>Nikolai Leonenko</strong>, <strong>Murad S. Taqqu</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 3, 1417--1446.</p><p><strong>Abstract:</strong><br/>
We obtain quantitative four moments theorems establishing convergence of the laws of elements of a Markov chaos to a Pearson distribution, where the only assumption we make on the Pearson distribution is that it admits four moments. These results are obtained by first proving a general carré du champ bound on the distance between laws of random variables in the domain of a Markov diffusion generator and invariant measures of diffusions, which is of independent interest, and making use of the new concept of chaos grade. For the heavy-tailed Pearson distributions, this seems to be the first time that sufficient conditions in terms of (finitely many) moments are given in order to converge to a distribution that is not characterized by its moments.
</p>projecteuclid.org/euclid.aop/1556784023_20190502040034Thu, 02 May 2019 04:00 EDTCapacity of the range of random walk on $\mathbb{Z}^{4}$https://projecteuclid.org/euclid.aop/1556784024<strong>Amine Asselah</strong>, <strong>Bruno Schapira</strong>, <strong>Perla Sousi</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 3, 1447--1497.</p><p><strong>Abstract:</strong><br/>
We study the scaling limit of the capacity of the range of a random walk on the integer lattice in dimension four. We establish a strong law of large numbers and a central limit theorem with a non-Gaussian limit. The asymptotic behaviour is analogous to that found by Le Gall in ’86 [ Comm. Math. Phys. 104 (1986) 471–507] for the volume of the range in dimension two.
</p>projecteuclid.org/euclid.aop/1556784024_20190502040034Thu, 02 May 2019 04:00 EDTSeparating cycles and isoperimetric inequalities in the uniform infinite planar quadrangulationhttps://projecteuclid.org/euclid.aop/1556784025<strong>Jean-François Le Gall</strong>, <strong>Thomas Lehéricy</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 3, 1498--1540.</p><p><strong>Abstract:</strong><br/>
We study geometric properties of the infinite random lattice called the uniform infinite planar quadrangulation or UIPQ. We establish a precise form of a conjecture of Krikun stating that the minimal size of a cycle that separates the ball of radius $R$ centered at the root vertex from infinity grows linearly in $R$. As a consequence, we derive certain isoperimetric bounds showing that the boundary size of any simply connected set $A$ consisting of a finite union of faces of the UIPQ and containing the root vertex is bounded below by a (random) constant times $|A|^{1/4}(\log|A|)^{-(3/4)-\delta}$, where the volume $|A|$ is the number of faces in $A$.
</p>projecteuclid.org/euclid.aop/1556784025_20190502040034Thu, 02 May 2019 04:00 EDTCutoff phenomenon for the asymmetric simple exclusion process and the biased card shufflinghttps://projecteuclid.org/euclid.aop/1556784026<strong>Cyril Labbé</strong>, <strong>Hubert Lacoin</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 3, 1541--1586.</p><p><strong>Abstract:</strong><br/>
We consider the biased card shuffling and the Asymmetric Simple Exclusion Process (ASEP) on the segment. We obtain the asymptotic of their mixing times: our results show that these two continuous-time Markov chains display cutoff. Our analysis combines several ingredients including: a study of the hydrodynamic profile for ASEP, the use of monotonic eigenfunctions, stochastic comparisons and concentration inequalities.
</p>projecteuclid.org/euclid.aop/1556784026_20190502040034Thu, 02 May 2019 04:00 EDTSuboptimality of local algorithms for a class of max-cut problemshttps://projecteuclid.org/euclid.aop/1556784027<strong>Wei-Kuo Chen</strong>, <strong>David Gamarnik</strong>, <strong>Dmitry Panchenko</strong>, <strong>Mustazee Rahman</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 3, 1587--1618.</p><p><strong>Abstract:</strong><br/>
We show that in random $K$-uniform hypergraphs of constant average degree, for even $K\geq 4$, local algorithms defined as factors of i.i.d. can not find nearly maximal cuts, when the average degree is sufficiently large. These algorithms have been used frequently to obtain lower bounds for the max-cut problem on random graphs, but it was not known whether they could be successful in finding nearly maximal cuts. This result follows from the fact that the overlap of any two nearly maximal cuts in such hypergraphs does not take values in a certain nontrivial interval—a phenomenon referred to as the overlap gap property—which is proved by comparing diluted models with large average degree with appropriate fully connected spin glass models and showing the overlap gap property in the latter setting.
</p>projecteuclid.org/euclid.aop/1556784027_20190502040034Thu, 02 May 2019 04:00 EDTInfinitely ramified point measures and branching Lévy processeshttps://projecteuclid.org/euclid.aop/1556784028<strong>Jean Bertoin</strong>, <strong>Bastien Mallein</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 3, 1619--1652.</p><p><strong>Abstract:</strong><br/>
We call a random point measure infinitely ramified if for every $n\in\mathbb{N}$, it has the same distribution as the $n$th generation of some branching random walk. On the other hand, branching Lévy processes model the evolution of a population in continuous time, such that individuals move in space independently, according to some Lévy process, and further beget progenies according to some Poissonian dynamics, possibly on an everywhere dense set of times. Our main result connects these two classes of processes much in the same way as in the case of infinitely divisible distributions and Lévy processes: the value at time $1$ of a branching Lévy process is an infinitely ramified point measure, and conversely, any infinitely ramified point measure can be obtained as the value at time $1$ of some branching Lévy process.
</p>projecteuclid.org/euclid.aop/1556784028_20190502040034Thu, 02 May 2019 04:00 EDTLargest eigenvalues of sparse inhomogeneous Erdős–Rényi graphshttps://projecteuclid.org/euclid.aop/1556784029<strong>Florent Benaych-Georges</strong>, <strong>Charles Bordenave</strong>, <strong>Antti Knowles</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 3, 1653--1676.</p><p><strong>Abstract:</strong><br/>
We consider inhomogeneous Erdős–Rényi graphs. We suppose that the maximal mean degree $d$ satisfies $d\ll\log n$. We characterise the asymptotic behaviour of the $n^{1-o(1)}$ largest eigenvalues of the adjacency matrix and its centred version. We prove that these extreme eigenvalues are governed at first order by the largest degrees and, for the adjacency matrix, by the nonzero eigenvalues of the expectation matrix. Our results show that the extreme eigenvalues exhibit a novel behaviour which in particular rules out their convergence to a nondegenerate point process. Together with the companion paper [Benaych-Georges, Bordenave and Knowles (2017)], where we analyse the extreme eigenvalues in the complementary regime $d\gg\log n$, this establishes a crossover in the behaviour of the extreme eigenvalues around $d\sim\log n$. Our proof relies on a tail estimate for the Poisson approximation of an inhomogeneous sum of independent Bernoulli random variables, as well as on an estimate on the operator norm of a pruned graph due to Le, Levina, and Vershynin from [ Random Structures Algorithms 51 (2017) 538–561].
</p>projecteuclid.org/euclid.aop/1556784029_20190502040034Thu, 02 May 2019 04:00 EDTOn the almost eigenvectors of random regular graphshttps://projecteuclid.org/euclid.aop/1556784030<strong>Ágnes Backhausz</strong>, <strong>Balázs Szegedy</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 3, 1677--1725.</p><p><strong>Abstract:</strong><br/>
Let $d\geq3$ be fixed and $G$ be a large random $d$-regular graph on $n$ vertices. We show that if $n$ is large enough then the entry distribution of every almost eigenvector of $G$ (with entry sum 0 and normalized to have length $\sqrt{n}$) is close to some Gaussian distribution $N(0,\sigma)$ in the weak topology where $0\leq\sigma\leq1$. Our theorem holds even in the stronger sense when many entries are looked at simultaneously in small random neighborhoods of the graph. Furthermore, we also get the Gaussianity of the joint distribution of several almost eigenvectors if the corresponding eigenvalues are close. Our proof uses graph limits and information theory. Our results have consequences for factor of i.i.d. processes on the infinite regular tree.
In particular, we obtain that if an invariant eigenvector process on the infinite $d$-regular tree is in the weak closure of factor of i.i.d. processes then it has Gaussian distribution.
</p>projecteuclid.org/euclid.aop/1556784030_20190502040034Thu, 02 May 2019 04:00 EDTIrreducible convex paving for decomposition of multidimensional martingale transport planshttps://projecteuclid.org/euclid.aop/1556784031<strong>Hadrien De March</strong>, <strong>Nizar Touzi</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 3, 1726--1774.</p><p><strong>Abstract:</strong><br/>
Martingale transport plans on the line are known from Beiglböck and Juillet ( Ann. Probab. 44 (2016) 42–106) to have an irreducible decomposition on a (at most) countable union of intervals. We provide an extension of this decomposition for martingale transport plans in $\mathbb{R}^{d}$, $d\ge1$. Our decomposition is a partition of $\mathbb{R}^{d}$ consisting of a possibly uncountable family of relatively open convex components, with the required measurability so that the disintegration is well defined. We justify the relevance of our decomposition by proving the existence of a martingale transport plan filling these components. We also deduce from this decomposition a characterization of the structure of polar sets with respect to all martingale transport plans.
</p>projecteuclid.org/euclid.aop/1556784031_20190502040034Thu, 02 May 2019 04:00 EDTA nonlinear wave equation with fractional perturbationhttps://projecteuclid.org/euclid.aop/1556784032<strong>Aurélien Deya</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 3, 1775--1810.</p><p><strong>Abstract:</strong><br/>
We study a $d$-dimensional wave equation model ($2\leq d\leq4$) with quadratic nonlinearity and stochastic forcing given by a space-time fractional noise. Two different regimes are exhibited, depending on the Hurst parameter $H=(H_{0},\ldots,H_{d})\in(0,1)^{d+1}$ of the noise: If $\sum_{i=0}^{d}H_{i}>d-\frac{1}{2}$, then the equation can be treated directly, while in the case $d-\frac{3}{4}<\sum_{i=0}^{d}H_{i}\leq d-\frac{1}{2}$, the model must be interpreted in the Wick sense, through a renormalization procedure.
Our arguments essentially rely on a fractional extension of the considerations of [ Trans. Amer. Math. Soc. 370 (2017) 7335–7359] for the two-dimensional white-noise situation, and more generally follow a series of investigations related to stochastic wave models with polynomial perturbation.
</p>projecteuclid.org/euclid.aop/1556784032_20190502040034Thu, 02 May 2019 04:00 EDTWeak tail conditions for local martingaleshttps://projecteuclid.org/euclid.aop/1556784033<strong>Hardy Hulley</strong>, <strong>Johannes Ruf</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 3, 1811--1825.</p><p><strong>Abstract:</strong><br/>
The following conditions are necessary and jointly sufficient for an arbitrary càdlàg local martingale to be a uniformly integrable martingale: (A) The weak tail of the supremum of its modulus is zero; (B) its jumps at the first-exit times from compact intervals converge to zero in $L^{1}$ on the events that those times are finite; and (C) its almost sure limit is an integrable random variable.
</p>projecteuclid.org/euclid.aop/1556784033_20190502040034Thu, 02 May 2019 04:00 EDTGenealogical constructions of population modelshttps://projecteuclid.org/euclid.aop/1562205692<strong>Alison M. Etheridge</strong>, <strong>Thomas G. Kurtz</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 4, 1827--1910.</p><p><strong>Abstract:</strong><br/>
Representations of population models in terms of countable systems of particles are constructed, in which each particle has a “type,” typically recording both spatial position and genetic type, and a level. For finite intensity models, the levels are distributed on $[0,\lambda]$, whereas in the infinite intensity limit $\lambda\rightarrow\infty$, at each time $t$, the joint distribution of types and levels is conditionally Poisson, with mean measure $\Xi(t)\times\ell$ where $\ell$ denotes Lebesgue measure and $\Xi(t)$ is a measure-valued population process. The time-evolution of the levels captures the genealogies of the particles in the population.
Key forces of ecology and genetics can be captured within this common framework. Models covered incorporate both individual and event based births and deaths, one-for-one replacement, immigration, independent “thinning” and independent or exchangeable spatial motion and mutation of individuals. Since birth and death probabilities can depend on type, they also include natural selection. The primary goal of the paper is to present particle-with-level or lookdown constructions for each of these elements of a population model. Then the elements can be combined to specify the desired model. In particular, a nontrivial extension of the spatial $\Lambda$-Fleming–Viot process is constructed.
</p>projecteuclid.org/euclid.aop/1562205692_20190703220211Wed, 03 Jul 2019 22:02 EDTIntermittency for the stochastic heat equation with Lévy noisehttps://projecteuclid.org/euclid.aop/1562205693<strong>Carsten Chong</strong>, <strong>Péter Kevei</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 4, 1911--1948.</p><p><strong>Abstract:</strong><br/>
We investigate the moment asymptotics of the solution to the stochastic heat equation driven by a $(d+1)$-dimensional Lévy space-time white noise. Unlike the case of Gaussian noise, the solution typically has no finite moments of order $1+2/d$ or higher. Intermittency of order $p$, that is, the exponential growth of the $p$th moment as time tends to infinity, is established in dimension $d=1$ for all values $p\in (1,3)$, and in higher dimensions for some $p\in (1,1+2/d)$. The proof relies on a new moment lower bound for stochastic integrals against compensated Poisson measures. The behavior of the intermittency exponents when $p\to 1+2/d$ further indicates that intermittency in the presence of jumps is much stronger than in equations with Gaussian noise. The effect of other parameters like the diffusion constant or the noise intensity on intermittency will also be analyzed in detail.
</p>projecteuclid.org/euclid.aop/1562205693_20190703220211Wed, 03 Jul 2019 22:02 EDTUniqueness of Gibbs measures for continuous hardcore modelshttps://projecteuclid.org/euclid.aop/1562205694<strong>David Gamarnik</strong>, <strong>Kavita Ramanan</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 4, 1949--1981.</p><p><strong>Abstract:</strong><br/>
We formulate a continuous version of the well-known discrete hardcore (or independent set) model on a locally finite graph, parameterized by the so-called activity parameter $\lambda >0$. In this version the state or “spin value” $x_{u}$ of any node $u$ of the graph lies in the interval $[0,1]$, the hardcore constraint $x_{u}+x_{v}\leq 1$ is satisfied for every edge $(u,v)$ of the graph, and the space of feasible configurations is given by a convex polytope. When the graph is a regular tree, we show that there is a unique Gibbs measure associated to each activity parameter $\lambda >0$. Our result shows that, in contrast to the standard discrete hardcore model, the continuous hardcore model does not exhibit a phase transition on the infinite regular tree. We also consider a family of continuous models that interpolate between the discrete and continuous hardcore models on a regular tree when $\lambda =1$ and show that each member of the family has a unique Gibbs measure, even when the discrete model does not. In each case the proof entails the analysis of an associated Hamiltonian dynamical system that describes a certain limit of the marginal distribution at a node. Furthermore, given any sequence of regular graphs with fixed degree and girth diverging to infinity, we apply our results to compute the asymptotic limit of suitably normalized volumes of the corresponding sequence of convex polytopes of feasible configurations. In particular this yields an approximation for the partition function of the continuous hard core model on a regular graph with large girth in the case $\lambda =1$.
</p>projecteuclid.org/euclid.aop/1562205694_20190703220211Wed, 03 Jul 2019 22:02 EDTCouplings and quantitative contraction rates for Langevin dynamicshttps://projecteuclid.org/euclid.aop/1562205696<strong>Andreas Eberle</strong>, <strong>Arnaud Guillin</strong>, <strong>Raphael Zimmer</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 4, 1982--2010.</p><p><strong>Abstract:</strong><br/>
We introduce a new probabilistic approach to quantify convergence to equilibrium for (kinetic) Langevin processes. In contrast to previous analytic approaches that focus on the associated kinetic Fokker–Planck equation, our approach is based on a specific combination of reflection and synchronous coupling of two solutions of the Langevin equation. It yields contractions in a particular Wasserstein distance, and it provides rather precise bounds for convergence to equilibrium at the borderline between the overdamped and the underdamped regime. In particular, we are able to recover kinetic behaviour in terms of explicit lower bounds for the contraction rate. For example, for a rescaled double-well potential with local minima at distance $a$, we obtain a lower bound for the contraction rate of order $\Omega(a^{-1})$ provided the friction coefficient is of order $\Theta(a^{-1})$.
</p>projecteuclid.org/euclid.aop/1562205696_20190703220211Wed, 03 Jul 2019 22:02 EDTPoly-logarithmic localization for random walks among random obstacleshttps://projecteuclid.org/euclid.aop/1562205700<strong>Jian Ding</strong>, <strong>Changji Xu</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 4, 2011--2048.</p><p><strong>Abstract:</strong><br/>
Place an obstacle with probability $1-\mathsf{p}$ independently at each vertex of $\mathbb{Z}^{d}$, and run a simple random walk until hitting one of the obstacles. For $d\geq2$ and $\mathsf{p}$ strictly above the critical threshold for site percolation, we condition on the environment where the origin is contained in an infinite connected component free of obstacles, and we show that the following path localization holds for environments with probability tending to 1 as $n\to\infty$: conditioned on survival up to time $n$ we have that ever since $o(n)$ steps the simple random walk is localized in a region of volume poly-logarithmic in $n$ with probability tending to 1. The previous best result of this type went back to Sznitman (1996) on Brownian motion among Poisson obstacles, where a localization (only for the end point) in a region of volume $t^{o(1)}$ was derived conditioned on the survival of Brownian motion up to time $t$.
</p>projecteuclid.org/euclid.aop/1562205700_20190703220211Wed, 03 Jul 2019 22:02 EDTThe scaling limit of critical Ising interfaces is $\mathrm{CLE}_{3}$https://projecteuclid.org/euclid.aop/1562205701<strong>Stéphane Benoist</strong>, <strong>Clément Hongler</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 4, 2049--2086.</p><p><strong>Abstract:</strong><br/>
In this paper, we consider the set of interfaces between $+$ and $-$ spins arising for the critical planar Ising model on a domain with $+$ boundary conditions, and show that it converges to nested CLE$_{3}$.
Our proof relies on the study of the coupling between the Ising model and its random cluster (FK) representation, and of the interactions between FK and Ising interfaces. The main idea is to construct an exploration process starting from the boundary of the domain, to discover the Ising loops and to establish its convergence to a conformally invariant limit. The challenge is that Ising loops do not touch the boundary; we use the fact that FK loops touch the boundary (and hence can be explored from the boundary) and that Ising loops in turn touch FK loops, to construct a recursive exploration process that visits all the macroscopic loops.
A key ingredient in the proof is the convergence of Ising free arcs to the Free Arc Ensemble (FAE), established in [ Ann. Inst. Henri Poincaré Probab. Stat. 52 (2016) 1784–1798]. Qualitative estimates about the Ising interfaces then allow one to identify the scaling limit of Ising loops as a conformally invariant collection of simple, disjoint $\mathrm{SLE}_{3}$-like loops, and thus by the Markovian characterization of Sheffield and Werner [ Ann. of Math. (2) 176 (2012) 1827–1917] as a $\mathrm{CLE}_{3}$.
A technical point of independent interest contained in this paper is an investigation of double points of interfaces in the scaling limit of critical FK-Ising. It relies on the technology of Kemppainen and Smirnov [ Ann. Probab. 45 (2017) 698–779].
</p>projecteuclid.org/euclid.aop/1562205701_20190703220211Wed, 03 Jul 2019 22:02 EDTA Sobolev space theory for stochastic partial differential equations with time-fractional derivativeshttps://projecteuclid.org/euclid.aop/1562205704<strong>Ildoo Kim</strong>, <strong>Kyeong-hun Kim</strong>, <strong>Sungbin Lim</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 4, 2087--2139.</p><p><strong>Abstract:</strong><br/>
In this article, we present an $L_{p}$-theory ($p\geq 2$) for the semi-linear stochastic partial differential equations (SPDEs) of type \begin{equation*}\partial^{\alpha }_{t}u=L(\omega ,t,x)u+f(u)+\partial^{\beta }_{t}\sum_{k=1}^{\infty }\int^{t}_{0}(\Lambda^{k}(\omega,t,x)u+g^{k}(u))\,dw^{k}_{t},\end{equation*} where $\alpha \in (0,2)$, $\beta <\alpha +\frac{1}{2}$ and $\partial^{\alpha }_{t}$ and $\partial^{\beta }_{t}$ denote the Caputo derivatives of order $\alpha $ and $\beta $, respectively. The processes $w^{k}_{t}$, $k\in \mathbb{N}=\{1,2,\ldots \}$, are independent one-dimensional Wiener processes, $L$ is either divergence or nondivergence-type second-order operator, and $\Lambda^{k}$ are linear operators of order up to two. This class of SPDEs can be used to describe random effects on transport of particles in medium with thermal memory or particles subject to sticking and trapping.
We prove uniqueness and existence results of strong solutions in appropriate Sobolev spaces, and obtain maximal $L_{p}$-regularity of the solutions. By converting SPDEs driven by $d$-dimensional space–time white noise into the equations of above type, we also obtain an $L_{p}$-theory for SPDEs driven by space–time white noise if the space dimension $d<4-2(2\beta -1)\alpha^{-1}$. In particular, if $\beta <1/2+\alpha /4$ then we can handle space–time white noise driven SPDEs with space dimension $d=1,2,3$.
</p>projecteuclid.org/euclid.aop/1562205704_20190703220211Wed, 03 Jul 2019 22:02 EDTA general method for lower bounds on fluctuations of random variableshttps://projecteuclid.org/euclid.aop/1562205705<strong>Sourav Chatterjee</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 4, 2140--2171.</p><p><strong>Abstract:</strong><br/>
There are many ways of establishing upper bounds on fluctuations of random variables, but there is no systematic approach for lower bounds. As a result, lower bounds are unknown in many important problems. This paper introduces a general method for lower bounds on fluctuations. The method is used to obtain new results for the stochastic traveling salesman problem, the stochastic minimal matching problem, the random assignment problem, the Sherrington–Kirkpatrick model of spin glasses, first-passage percolation and random matrices. A long list of open problems is provided at the end.
</p>projecteuclid.org/euclid.aop/1562205705_20190703220211Wed, 03 Jul 2019 22:02 EDTStein kernels and moment mapshttps://projecteuclid.org/euclid.aop/1562205706<strong>Max Fathi</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 4, 2172--2185.</p><p><strong>Abstract:</strong><br/>
We describe a construction of Stein kernels using moment maps, which are solutions to a variant of the Monge–Ampère equation. As a consequence, we show how regularity bounds in certain weighted Sobolev spaces on these maps control the rate of convergence in the classical central limit theorem, and derive new rates in Kantorovitch–Wasserstein distance in the log-concave situation, with explicit polynomial dependence on the dimension.
</p>projecteuclid.org/euclid.aop/1562205706_20190703220211Wed, 03 Jul 2019 22:02 EDTLarge deviations and wandering exponent for random walk in a dynamic beta environmenthttps://projecteuclid.org/euclid.aop/1562205707<strong>Márton Balázs</strong>, <strong>Firas Rassoul-Agha</strong>, <strong>Timo Seppäläinen</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 4, 2186--2229.</p><p><strong>Abstract:</strong><br/>
Random walk in a dynamic i.i.d. beta random environment, conditioned to escape at an atypical velocity, converges to a Doob transform of the original walk. The Doob-transformed environment is correlated in time, i.i.d. in space and its marginal density function is a product of a beta density and a hypergeometric function. Under its averaged distribution, the transformed walk obeys the wandering exponent $2/3$ that agrees with Kardar–Parisi–Zhang universality. The harmonic function in the Doob transform comes from a Busemann-type limit and appears as an extremal in a variational problem for the quenched large deviation rate function.
</p>projecteuclid.org/euclid.aop/1562205707_20190703220211Wed, 03 Jul 2019 22:02 EDTThouless–Anderson–Palmer equations for generic $p$-spin glasseshttps://projecteuclid.org/euclid.aop/1562205708<strong>Antonio Auffinger</strong>, <strong>Aukosh Jagannath</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 4, 2230--2256.</p><p><strong>Abstract:</strong><br/>
We study the Thouless–Anderson–Palmer (TAP) equations for spin glasses on the hypercube. First, using a random, approximately ultrametric decomposition of the hypercube, we decompose the Gibbs measure, $\langle \cdot \rangle_{N}$, into a mixture of conditional laws, $\langle \cdot \rangle_{\alpha,N}$. We show that the TAP equations hold for the spin at any site with respect to $\langle \cdot \rangle_{\alpha,N}$ simultaneously for all $\alpha $. This result holds for generic models provided that the Parisi measure of the model has a jump at the top of its support.
</p>projecteuclid.org/euclid.aop/1562205708_20190703220211Wed, 03 Jul 2019 22:02 EDTThe structure of extreme level sets in branching Brownian motionhttps://projecteuclid.org/euclid.aop/1562205709<strong>Aser Cortines</strong>, <strong>Lisa Hartung</strong>, <strong>Oren Louidor</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 4, 2257--2302.</p><p><strong>Abstract:</strong><br/>
We study the structure of extreme level sets of a standard one-dimensional branching Brownian motion, namely the sets of particles whose height is within a fixed distance from the order of the global maximum. It is well known that such particles congregate at large times in clusters of order-one genealogical diameter around local maxima which form a Cox process in the limit. We add to these results by finding the asymptotic size of extreme level sets and the typical height of the local maxima whose clusters carry such level sets. We also find the right tail decay of the distribution of the distance between the two highest particles. These results confirm two conjectures of Brunet and Derrida ( J. Stat. Phys. 143 (2011) 420–446). The proofs rely on a careful study of the cluster distribution.
</p>projecteuclid.org/euclid.aop/1562205709_20190703220211Wed, 03 Jul 2019 22:02 EDTMetric gluing of Brownian and $\sqrt{8/3}$-Liouville quantum gravity surfaceshttps://projecteuclid.org/euclid.aop/1562205710<strong>Ewain Gwynne</strong>, <strong>Jason Miller</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 4, 2303--2358.</p><p><strong>Abstract:</strong><br/>
In a recent series of works, Miller and Sheffield constructed a metric on $\sqrt{8/3}$-Liouville quantum gravity (LQG) under which $\sqrt{8/3}$-LQG surfaces (e.g., the LQG sphere, wedge, cone and disk) are isometric to their Brownian surface counterparts (e.g., the Brownian map, half-plane, plane and disk).
We identify the metric gluings of certain collections of independent $\sqrt{8/3}$-LQG surfaces with boundaries identified together according to LQG length along their boundaries. Our results imply in particular that the metric gluing of two independent instances of the Brownian half-plane along their positive boundaries is isometric to a certain LQG wedge decorated by an independent chordal $\mathrm{SLE}_{8/3}$ curve. If one identifies the two sides of the boundary of a single Brownian half-plane, one obtains a certain LQG cone decorated by an independent whole-plane $\mathrm{SLE}_{8/3}$. If one identifies the entire boundaries of two Brownian half-planes, one obtains a different LQG cone and the interface between them is a two-sided variant of whole-plane $\mathrm{SLE}_{8/3}$.
Combined with another work of the authors, the present work identifies the scaling limit of self-avoiding walk on random quadrangulations with $\mathrm{SLE}_{8/3}$ on $\sqrt{8/3}$-LQG.
</p>projecteuclid.org/euclid.aop/1562205710_20190703220211Wed, 03 Jul 2019 22:02 EDTThe circular law for sparse non-Hermitian matriceshttps://projecteuclid.org/euclid.aop/1562205711<strong>Anirban Basak</strong>, <strong>Mark Rudelson</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 4, 2359--2416.</p><p><strong>Abstract:</strong><br/>
For a class of sparse random matrices of the form $A_{n}=(\xi_{i,j}\delta_{i,j})_{i,j=1}^{n}$, where $\{\xi_{i,j}\}$ are i.i.d. centered sub-Gaussian random variables of unit variance, and $\{\delta_{i,j}\}$ are i.i.d. Bernoulli random variables taking value $1$ with probability $p_{n}$, we prove that the empirical spectral distribution of $A_{n}/\sqrt{np_{n}}$ converges weakly to the circular law, in probability, for all $p_{n}$ such that $p_{n}=\omega({\log^{2}n}/{n})$. Additionally if $p_{n}$ satisfies the inequality $np_{n}>\exp(c\sqrt{\log n})$ for some constant $c$, then the above convergence is shown to hold almost surely. The key to this is a new bound on the smallest singular value of complex shifts of real valued sparse random matrices. The circular law limit also extends to the adjacency matrix of a directed Erdős–Rényi graph with edge connectivity probability $p_{n}$.
</p>projecteuclid.org/euclid.aop/1562205711_20190703220211Wed, 03 Jul 2019 22:02 EDTStrong convergence of eigenangles and eigenvectors for the circular unitary ensemblehttps://projecteuclid.org/euclid.aop/1562205712<strong>Kenneth Maples</strong>, <strong>Joseph Najnudel</strong>, <strong>Ashkan Nikeghbali</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 4, 2417--2458.</p><p><strong>Abstract:</strong><br/>
It is known that a unitary matrix can be decomposed into a product of complex reflections, one for each dimension, and that these reflections are independent and uniformly distributed on the space where they live if the initial matrix is Haar-distributed. If we take an infinite sequence of such reflections, and consider their successive products, then we get an infinite sequence of unitary matrices of increasing dimension, all of them following the circular unitary ensemble.
In this coupling, we show that the eigenvalues of the matrices converge almost surely to the eigenvalues of the flow, which are distributed according to a sine-kernel point process, and we get some estimates of the rate of convergence. Moreover, we also prove that the eigenvectors of the matrices converge almost surely to vectors which are distributed as Gaussian random fields on a countable set.
</p>projecteuclid.org/euclid.aop/1562205712_20190703220211Wed, 03 Jul 2019 22:02 EDTOn macroscopic holes in some supercritical strongly dependent percolation modelshttps://projecteuclid.org/euclid.aop/1562205713<strong>Alain-Sol Sznitman</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 4, 2459--2493.</p><p><strong>Abstract:</strong><br/>
We consider $\mathbb{Z}^{d}$, $d\ge 3$. We investigate the vacant set $\mathcal{V}^{u}$ of random interlacements in the strongly percolative regime, the vacant set $\mathcal{V}$ of the simple random walk and the excursion set $E^{\ge \alpha }$ of the Gaussian free field in the strongly percolative regime. We consider the large deviation probability that the adequately thickened component of the boundary of a large box centered at the origin in the respective vacant sets or excursion set leaves in the box a macroscopic volume in its complement. We derive asymptotic upper and lower exponential bounds for theses large deviation probabilities. We also derive geometric information on the shape of the left-out volume. It is plausible, but open at the moment, that certain critical levels coincide, both in the case of random interlacements and of the Gaussian free field. If this holds true, the asymptotic upper and lower bounds that we obtain are matching in principal order for all three models, and the macroscopic holes are nearly spherical. We heavily rely on the recent work by Maximilian Nitzschner (2018) and the author for the coarse graining procedure, which we employ in the derivation of the upper bounds.
</p>projecteuclid.org/euclid.aop/1562205713_20190703220211Wed, 03 Jul 2019 22:02 EDTInvariant measure for random walks on ergodic environments on a striphttps://projecteuclid.org/euclid.aop/1562205714<strong>Dmitry Dolgopyat</strong>, <strong>Ilya Goldsheid</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 4, 2494--2528.</p><p><strong>Abstract:</strong><br/>
Environment viewed from the particle is a powerful method of analyzing random walks (RW) in random environment (RE). It is well known that in this setting the environment process is a Markov chain on the set of environments. We study the fundamental question of existence of the density of the invariant measure of this Markov chain with respect to the measure on the set of environments for RW on a strip. We first describe all positive subexponentially growing solutions of the corresponding invariant density equation in the deterministic setting and then derive necessary and sufficient conditions for the existence of the density when the environment is ergodic in both the transient and the recurrent regimes. We also provide applications of our analysis to the question of positive and null recurrence, the study of the Green functions and to random walks on orbits of a dynamical system.
</p>projecteuclid.org/euclid.aop/1562205714_20190703220211Wed, 03 Jul 2019 22:02 EDTExtremal theory for long range dependent infinitely divisible processeshttps://projecteuclid.org/euclid.aop/1562205715<strong>Gennady Samorodnitsky</strong>, <strong>Yizao Wang</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 4, 2529--2562.</p><p><strong>Abstract:</strong><br/>
We prove limit theorems of an entirely new type for certain long memory regularly varying stationary infinitely divisible random processes. These theorems involve multiple phase transitions governed by how long the memory is. Apart from one regime, our results exhibit limits that are not among the classical extreme value distributions. Restricted to the one-dimensional case, the distributions we obtain interpolate, in the appropriate parameter range, the $\alpha$-Fréchet distribution and the skewed $\alpha$-stable distribution. In general, the limit is a new family of stationary and self-similar random sup-measures with parameters $\alpha\in(0,\infty)$ and $\beta\in(0,1)$, with representations based on intersections of independent $\beta$-stable regenerative sets. The tail of the limit random sup-measure on each interval with finite positive length is regularly varying with index $-\alpha$. The intriguing structure of these random sup-measures is due to intersections of independent $\beta$-stable regenerative sets and the fact that the number of such sets intersecting simultaneously increases to infinity as $\beta$ increases to one. The results in this paper extend substantially previous investigations where only $\alpha\in(0,2)$ and $\beta\in(0,1/2)$ have been considered.
</p>projecteuclid.org/euclid.aop/1562205715_20190703220211Wed, 03 Jul 2019 22:02 EDTDensity of the set of probability measures with the martingale representation propertyhttps://projecteuclid.org/euclid.aop/1562205716<strong>Dmitry Kramkov</strong>, <strong>Sergio Pulido</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 4, 2563--2581.</p><p><strong>Abstract:</strong><br/>
Let $\psi$ be a multidimensional random variable. We show that the set of probability measures $\mathbb{Q}$ such that the $\mathbb{Q}$-martingale $S^{\mathbb{Q}}_{t}=\mathbb{E}^{\mathbb{Q}}[\psi \lvert\mathcal{F}_{t}]$ has the Martingale Representation Property (MRP) is either empty or dense in $\mathcal{L}_{\infty}$-norm. The proof is based on a related result involving analytic fields of terminal conditions $(\psi(x))_{x\in U}$ and probability measures $(\mathbb{Q}(x))_{x\in U}$ over an open set $U$. Namely, we show that the set of points $x\in U$ such that $S_{t}(x)=\mathbb{E}^{\mathbb{Q}(x)}[\psi(x)\lvert\mathcal{F}_{t}]$ does not have the MRP, either coincides with $U$ or has Lebesgue measure zero. Our study is motivated by the problem of endogenous completeness in financial economics.
</p>projecteuclid.org/euclid.aop/1562205716_20190703220211Wed, 03 Jul 2019 22:02 EDTOn the dimension of Bernoulli convolutionshttps://projecteuclid.org/euclid.aop/1562205717<strong>Emmanuel Breuillard</strong>, <strong>Péter P. Varjú</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 4, 2582--2617.</p><p><strong>Abstract:</strong><br/>
The Bernoulli convolution with parameter $\lambda\in(0,1)$ is the probability measure $\mu_{\lambda}$ that is the law of the random variable $\sum_{n\ge0}\pm\lambda^{n}$, where the signs are independent unbiased coin tosses.
We prove that each parameter $\lambda\in(1/2,1)$ with $\dim\mu_{\lambda}<1$ can be approximated by algebraic parameters $\eta\in(1/2,1)$ within an error of order $\exp(-\deg(\eta)^{A})$ such that $\dim\mu_{\eta}<1$, for any number $A$. As a corollary, we conclude that $\dim\mu_{\lambda}=1$ for each of $\lambda=\ln2,e^{-1/2},\pi/4$. These are the first explicit examples of such transcendental parameters. Moreover, we show that Lehmer’s conjecture implies the existence of a constant $a<1$ such that $\dim\mu_{\lambda}=1$ for all $\lambda\in(a,1)$.
</p>projecteuclid.org/euclid.aop/1562205717_20190703220211Wed, 03 Jul 2019 22:02 EDT