The Annals of Probability Articles (Project Euclid)
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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 EDTBoundary 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 EDTQuantitative normal approximation of linear statistics of $\beta$-ensembleshttps://projecteuclid.org/euclid.aop/1571731432<strong>Gaultier Lambert</strong>, <strong>Michel Ledoux</strong>, <strong>Christian Webb</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 5, 2619--2685.</p><p><strong>Abstract:</strong><br/>
We present a new approach, inspired by Stein’s method, to prove a central limit theorem (CLT) for linear statistics of $\beta$-ensembles in the one-cut regime. Compared with the previous proofs, our result requires less regularity on the potential and provides a rate of convergence in the quadratic Kantorovich or Wasserstein-2 distance. The rate depends both on the regularity of the potential and the test functions, and we prove that it is optimal in the case of the Gaussian Unitary Ensemble (GUE) for certain polynomial test functions.
The method relies on a general normal approximation result of independent interest which is valid for a large class of Gibbs-type distributions. In the context of $\beta$-ensembles, this leads to a multi-dimensional CLT for a sequence of linear statistics which are approximate eigenfunctions of the infinitesimal generator of Dyson Brownian motion once the various error terms are controlled using the rigidity results of Bourgade, Erdős and Yau.
</p>projecteuclid.org/euclid.aop/1571731432_20191022040429Tue, 22 Oct 2019 04:04 EDTUniversality of local statistics for noncolliding random walkshttps://projecteuclid.org/euclid.aop/1571731433<strong>Vadim Gorin</strong>, <strong>Leonid Petrov</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 5, 2686--2753.</p><p><strong>Abstract:</strong><br/>
We consider the $N$-particle noncolliding Bernoulli random walk—a discrete time Markov process in $\mathbb{Z}^{N}$ obtained from a collection of $N$ independent simple random walks with steps $\in\{0,1\}$ by conditioning that they never collide. We study the asymptotic behavior of local statistics of this process started from an arbitrary initial configuration on short times $T\ll N$ as $N\to+\infty$. We show that if the particle density of the initial configuration is bounded away from $0$ and $1$ down to scales $\mathsf{D}\ll T$ in a neighborhood of size $\mathsf{Q}\gg T$ of some location $x$ (i.e., $x$ is in the “bulk”), and the initial configuration is balanced in a certain sense, then the space-time local statistics at $x$ are asymptotically governed by the extended discrete sine process (which can be identified with a translation invariant ergodic Gibbs measure on lozenge tilings of the plane). We also establish similar results for certain types of random initial data. Our proofs are based on a detailed analysis of the determinantal correlation kernel for the noncolliding Bernoulli random walk.
The noncolliding Bernoulli random walk is a discrete analogue of the $\boldsymbol{\beta}=2$ Dyson Brownian motion whose local statistics are universality governed by the continuous sine process. Our results parallel the ones in the continuous case. In addition, we naturally include situations with inhomogeneous local particle density on scale $T$, which nontrivially affects parameters of the limiting extended sine process, and in a particular case leads to a new behavior.
</p>projecteuclid.org/euclid.aop/1571731433_20191022040429Tue, 22 Oct 2019 04:04 EDTSampling perspectives on sparse exchangeable graphshttps://projecteuclid.org/euclid.aop/1571731436<strong>Christian Borgs</strong>, <strong>Jennifer T. Chayes</strong>, <strong>Henry Cohn</strong>, <strong>Victor Veitch</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 5, 2754--2800.</p><p><strong>Abstract:</strong><br/>
Recent work has introduced sparse exchangeable graphs and the associated graphex framework, as a generalization of dense exchangeable graphs and the associated graphon framework. The development of this subject involves the interplay between the statistical modeling of network data, the theory of large graph limits, exchangeability and network sampling. The purpose of the present paper is to clarify the relationships between these subjects by explaining each in terms of a certain natural sampling scheme associated with the graphex model. The first main technical contribution is the introduction of sampling convergence , a new notion of graph limit that generalizes left convergence so that it becomes meaningful for the sparse graph regime. The second main technical contribution is the demonstration that the (somewhat cryptic) notion of exchangeability underpinning the graphex framework is equivalent to a more natural probabilistic invariance expressed in terms of the sampling scheme.
</p>projecteuclid.org/euclid.aop/1571731436_20191022040429Tue, 22 Oct 2019 04:04 EDTSelf-avoiding walk on nonunimodular transitive graphshttps://projecteuclid.org/euclid.aop/1571731437<strong>Tom Hutchcroft</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 5, 2801--2829.</p><p><strong>Abstract:</strong><br/>
We study self-avoiding walk on graphs whose automorphism group has a transitive nonunimodular subgroup. We prove that self-avoiding walk is ballistic, that the bubble diagram converges at criticality, and that the critical two-point function decays exponentially in the distance from the origin. This implies that the critical exponent governing the susceptibility takes its mean-field value, and hence that the number of self-avoiding walks of length $n$ is comparable to the $n$th power of the connective constant. We also prove that the same results hold for a large class of repulsive walk models with a self-intersection based interaction, including the weakly self-avoiding walk. All of these results apply in particular to the product $T_{k}\times\mathbb{Z}^{d}$ of a $k$-regular tree ($k\geq3$) with $\mathbb{Z}^{d}$, for which these results were previously only known for large $k$.
</p>projecteuclid.org/euclid.aop/1571731437_20191022040429Tue, 22 Oct 2019 04:04 EDTHeat kernel estimates for symmetric jump processes with mixed polynomial growthshttps://projecteuclid.org/euclid.aop/1571731438<strong>Joohak Bae</strong>, <strong>Jaehoon Kang</strong>, <strong>Panki Kim</strong>, <strong>Jaehun Lee</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 5, 2830--2868.</p><p><strong>Abstract:</strong><br/>
In this paper, we study the transition densities of pure-jump symmetric Markov processes in $\mathbb{R}^{d}$, whose jumping kernels are comparable to radially symmetric functions with mixed polynomial growths. Under some mild assumptions on their scale functions, we establish sharp two-sided estimates of the transition densities (heat kernel estimates) for such processes. This is the first study on global heat kernel estimates of jump processes (including non-Lévy processes) whose weak scaling index is not necessarily strictly less than 2. As an application, we proved that the finite second moment condition on such symmetric Markov process is equivalent to the Khintchine-type law of iterated logarithm at infinity.
</p>projecteuclid.org/euclid.aop/1571731438_20191022040429Tue, 22 Oct 2019 04:04 EDTGeometric structures of late points of a two-dimensional simple random walkhttps://projecteuclid.org/euclid.aop/1571731439<strong>Izumi Okada</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 5, 2869--2893.</p><p><strong>Abstract:</strong><br/>
As Dembo (In Lectures on Probability Theory and Statistics (2005) 1–101 Springer, and International Congress of Mathematicians, Vol. III (2006) 535–558, Eur. Math. Soc.) suggested, we consider the problem of late points for a simple random walk in two dimensions. It has been shown that the exponents for the number of pairs of late points coincide with those of favorite points and high points in the Gaussian free field, whose exact values are known. We determine the exponents for the number of $j$-tuples of late points on average.
</p>projecteuclid.org/euclid.aop/1571731439_20191022040429Tue, 22 Oct 2019 04:04 EDTDistribution flows associated with positivity preserving coercive formshttps://projecteuclid.org/euclid.aop/1571731440<strong>Xian Chen</strong>, <strong>Zhi-Ming Ma</strong>, <strong>Xue Peng</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 5, 2894--2929.</p><p><strong>Abstract:</strong><br/>
For a given quasi-regular positivity preserving coercive form, we construct a family of ($\sigma$-finite) distribution flows associated with the semigroup of the form. The canonical cadlag process equipped with the distribution flows behaves like a strong Markov process. Moreover, employing distribution flows we can construct optional measures and establish Revuz correspondence between additive functionals and smooth measures. The results obtained in this paper will enable us to perform a kind of stochastic analysis related to positivity preserving coercive forms.
</p>projecteuclid.org/euclid.aop/1571731440_20191022040429Tue, 22 Oct 2019 04:04 EDTWeak Poincaré inequalities for convergence rate of degenerate diffusion processeshttps://projecteuclid.org/euclid.aop/1571731441<strong>Martin Grothaus</strong>, <strong>Feng-Yu Wang</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 5, 2930--2952.</p><p><strong>Abstract:</strong><br/>
For a contraction $C_{0}$-semigroup on a separable Hilbert space, the decay rate is estimated by using the weak Poincaré inequalities for the symmetric and antisymmetric part of the generator. As applications, nonexponential convergence rate is characterized for a class of degenerate diffusion processes, so that the study of hypocoercivity is extended. Concrete examples are presented.
</p>projecteuclid.org/euclid.aop/1571731441_20191022040429Tue, 22 Oct 2019 04:04 EDT1-stable fluctuations in branching Brownian motion at critical temperature I: The derivative martingalehttps://projecteuclid.org/euclid.aop/1571731442<strong>Pascal Maillard</strong>, <strong>Michel Pain</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 5, 2953--3002.</p><p><strong>Abstract:</strong><br/>
Let $(Z_{t})_{t\geq 0}$ denote the derivative martingale of branching Brownian motion, that is, the derivative with respect to the inverse temperature of the normalized partition function at critical temperature. A well-known result by Lalley and Sellke ( Ann. Probab. 15 (1987) 1052–1061) says that this martingale converges almost surely to a limit $Z_{\infty }$, positive on the event of survival. In this paper our concern is the fluctuations of the derivative martingale around its limit. A corollary of our results is the following convergence, confirming and strengthening a conjecture by Mueller and Munier ( Phys. Rev. E 90 (2014) 042143): \begin{equation*}\sqrt{t}\bigg(Z_{\infty }-Z_{t}+\frac{\log t}{\sqrt{2\pi t}}Z_{\infty }\bigg)\xrightarrow[t\to \infty ]{}S_{Z_{\infty }}\quad\text{in law},\end{equation*} where $S$ is a spectrally positive 1-stable Lévy process independent of $Z_{\infty }$.
In a first part of the paper, a relatively short proof of (a slightly stronger form of) this convergence is given based on the functional equation satisfied by the characteristic function of $Z_{\infty }$ together with tail asymptotics of this random variable. We then set up more elaborate arguments which yield a more thorough understanding of the trajectories of the particles contributing to the fluctuations. In this way we can upgrade our convergence result to functional convergence. This approach also sets the ground for a follow-up paper, where we study the fluctuations of more general functionals including the renormalized critical additive martingale.
All proofs in this paper are given under the moment assumption $\mathbb{E}[L(\log L)^{3}]<\infty $, where the random variable $L$ follows the offspring distribution of the branching Brownian motion. We believe this hypothesis to be optimal.
</p>projecteuclid.org/euclid.aop/1571731442_20191022040429Tue, 22 Oct 2019 04:04 EDTOn the transient (T) condition for random walk in mixing environmenthttps://projecteuclid.org/euclid.aop/1571731443<strong>Enrique Guerra Aguilar</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 5, 3003--3054.</p><p><strong>Abstract:</strong><br/>
We prove a ballistic strong law of large numbers and an invariance principle for random walks in strong mixing environments, under condition $(T)$ of Sznitman (cf. Ann. Probab. 29 (2001) 724–765). This weakens for the first time Kalikow’s ballisticity assumption on mixing environments and proves the existence of arbitrary finite order moments for the approximate regeneration time of F. Comets and O. Zeitouni ( Israel J. Math. 148 (2005) 87–113). The main technical tool in the proof is the introduction of renormalization schemes, which had only been considered for i.i.d. environments.
</p>projecteuclid.org/euclid.aop/1571731443_20191022040429Tue, 22 Oct 2019 04:04 EDTFinitary isomorphisms of Poisson point processeshttps://projecteuclid.org/euclid.aop/1571731444<strong>Terry Soo</strong>, <strong>Amanda Wilkens</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 5, 3055--3081.</p><p><strong>Abstract:</strong><br/>
As part of a general theory for the isomorphism problem for actions of amenable groups, Ornstein and Weiss ( J. Anal. Math. 48 (1987) 1–141) proved that any two Poisson point processes are isomorphic as measure-preserving actions. We give an elementary construction of an isomorphism between Poisson point processes that is finitary.
</p>projecteuclid.org/euclid.aop/1571731444_20191022040429Tue, 22 Oct 2019 04:04 EDTThe tail expansion of Gaussian multiplicative chaos and the Liouville reflection coefficienthttps://projecteuclid.org/euclid.aop/1571731445<strong>Rémi Rhodes</strong>, <strong>Vincent Vargas</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 5, 3082--3107.</p><p><strong>Abstract:</strong><br/>
In this short note we derive a precise tail expansion for Gaussian multiplicative chaos (GMC) associated to the 2d Gaussian Free Fiedl (GFF) on the unit disk with zero average on the unit circle (and variants). More specifically, we show that to first order the tail is a constant times an inverse power with an explicit value for the tail exponent as well as an explicit value for the constant in front of the inverse power; we also provide a second order bound for the tail expansion. The main interest of our work consists of two points. First, our derivation is based on a simple method, which we believe is universal in the sense that it can be generalized to all dimensions and to all log-correlated fields. Second, in the 2d case we consider, the value of the constant in front of the inverse power is (up to explicit terms) nothing but the Liouville reflection coefficient taken at a special value. The explicit computation of the constant was performed in the recent rigorous derivation with A. Kupiainen of the DOZZ formula (Kupiainen, Rhodes and Vargas (2017a, 2017b)); to our knowledge, it is the first time one derives rigorously an explicit value for such a constant in the tail expansion of a GMC measure. We have deliberately kept this paper short to emphasize the method so that it becomes an easily accessible toolbox for computing tails in GMC theory.
</p>projecteuclid.org/euclid.aop/1571731445_20191022040429Tue, 22 Oct 2019 04:04 EDTStrong differential subordinates for noncommutative submartingaleshttps://projecteuclid.org/euclid.aop/1571731446<strong>Yong Jiao</strong>, <strong>Adam Osȩkowski</strong>, <strong>Lian Wu</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 5, 3108--3142.</p><p><strong>Abstract:</strong><br/>
We introduce a notion of strong differential subordination of noncommutative semimartingales, extending Burkholder’s definition from the classical case (Ann. Probab. 22 (1994) 995–1025). Then we establish the maximal weak-type $(1,1)$ inequality under the additional assumption that the dominating process is a submartingale. The proof rests on a significant extension of the maximal weak-type estimate of Cuculescu and a Gundy-type decomposition of an arbitrary noncommutative submartingale. We also show the corresponding strong-type $(p,p)$ estimate for $1<p<\infty $ under the assumption that the dominating process is a nonnegative submartingale. This is accomplished by combining several techniques, including interpolation-flavor method, Doob–Meyer decomposition and noncommutative analogue of good-$\lambda$ inequalities.
</p>projecteuclid.org/euclid.aop/1571731446_20191022040429Tue, 22 Oct 2019 04:04 EDTCone points of Brownian motion in arbitrary dimensionhttps://projecteuclid.org/euclid.aop/1571731447<strong>Yotam Alexander</strong>, <strong>Ronen Eldan</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 5, 3143--3169.</p><p><strong>Abstract:</strong><br/>
We show that the convex hull of the path of Brownian motion in $n$-dimensions, up to time $1$, is a smooth set. As a consequence we conclude that a Brownian motion in any dimension almost surely has no cone points for any cone whose dual cone is nontrivial.
</p>projecteuclid.org/euclid.aop/1571731447_20191022040429Tue, 22 Oct 2019 04:04 EDTCutoff for the mean-field zero-range processhttps://projecteuclid.org/euclid.aop/1571731448<strong>Mathieu Merle</strong>, <strong>Justin Salez</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 5, 3170--3201.</p><p><strong>Abstract:</strong><br/>
We study the mixing time of the unit-rate zero-range process on the complete graph, in the regime where the number $n$ of sites tends to infinity while the density of particles per site stabilizes to some limit $\rho >0$. We prove that the worst-case total-variation distance to equilibrium drops abruptly from $1$ to $0$ at time $n(\rho +\frac{1}{2}\rho^{2})$. More generally, we determine the mixing time from an arbitrary initial configuration. The answer turns out to depend on the largest initial heights in a remarkably explicit way. 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 combines metastability, separation of timescales, fluid limits, propagation of chaos, entropy and a spectral estimate by Morris ( Ann. Probab. 34 (2006) 1645–1664).
</p>projecteuclid.org/euclid.aop/1571731448_20191022040429Tue, 22 Oct 2019 04:04 EDTAsymptotic zero distribution of random orthogonal polynomialshttps://projecteuclid.org/euclid.aop/1571731449<strong>Thomas Bloom</strong>, <strong>Duncan Dauvergne</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 5, 3202--3230.</p><p><strong>Abstract:</strong><br/>
We consider random polynomials of the form $H_{n}(z)=\sum_{j=0}^{n}\xi_{j}q_{j}(z)$ where the $\{\xi_{j}\}$ are i.i.d. nondegenerate complex random variables, and the $\{q_{j}(z)\}$ are orthonormal polynomials with respect to a compactly supported measure $\tau $ satisfying the Bernstein–Markov property on a regular compact set $K\subset\mathbb{C}$. We show that if $\mathbb{P}(|\xi_{0}|>e^{|z|})=o(|z|^{-1})$, then the normalized counting measure of the zeros of $H_{n}$ converges weakly in probability to the equilibrium measure of $K$. This is the best possible result, in the sense that the roots of $G_{n}(z)=\sum_{j=0}^{n}\xi_{j}z^{j}$ fail to converge in probability to the appropriate equilibrium measure when the above condition on the $\xi_{j}$ is not satisfied.
We also consider random polynomials of the form $\sum_{k=0}^{n}\xi_{k}f_{n,k}z^{k}$, where the coefficients $f_{n,k}$ are complex constants satisfying certain conditions, and the random variables $\{\xi_{k}\}$ satisfy $\mathbb{E}\log (1+|\xi_{0}|)<\infty $. In this case, we establish almost sure convergence of the normalized counting measure of the zeros to an appropriate limiting measure. Again, this is the best possible result in the same sense as above.
</p>projecteuclid.org/euclid.aop/1571731449_20191022040429Tue, 22 Oct 2019 04:04 EDTIntertwining, excursion theory and Krein theory of strings for non-self-adjoint Markov semigroupshttps://projecteuclid.org/euclid.aop/1571731450<strong>Pierre Patie</strong>, <strong>Mladen Savov</strong>, <strong>Yixuan Zhao</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 5, 3231--3277.</p><p><strong>Abstract:</strong><br/>
In this paper, we start by showing that the intertwining relationship between two minimal Markov semigroups acting on Hilbert spaces implies that any recurrent extensions, in the sense of Itô, of these semigroups satisfy the same intertwining identity. Under mild additional assumptions on the intertwining operator, we prove that the converse also holds. This connection, which relies on the representation of excursion quantities as developed by Fitzsimmons and Getoor ( Illinois J. Math. 50 (2006) 413–437), enables us to give an interesting probabilistic interpretation of intertwining relationships between Markov semigroups via excursion theory: two such recurrent extensions that intertwine share, under an appropriate normalization, the same local time at the boundary point. Moreover, in the case when one of the (non-self-adjoint) semigroup intertwines with the one of a quasi-diffusion, we obtain an extension of Krein’s theory of strings by showing that its densely defined spectral measure is absolutely continuous with respect to the measure appearing in the Stieltjes representation of the Laplace exponent of the inverse local time. Finally, we illustrate our results with the class of positive self-similar Markov semigroups and also the reflected generalized Laguerre semigroups. For the latter, we obtain their spectral decomposition and provide, under some conditions, an explicit hypocoercivity $L^{2}$-rate of convergence to equilibrium which is expressed as the spectral gap perturbed by the spectral projection norms.
</p>projecteuclid.org/euclid.aop/1571731450_20191022040429Tue, 22 Oct 2019 04:04 EDTLocal law and complete eigenvector delocalization for supercritical Erdős–Rényi graphshttps://projecteuclid.org/euclid.aop/1571731451<strong>Yukun He</strong>, <strong>Antti Knowles</strong>, <strong>Matteo Marcozzi</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 5, 3278--3302.</p><p><strong>Abstract:</strong><br/>
We prove a local law for the adjacency matrix of the Erdős–Rényi graph $G(N,p)$ in the supercritical regime $pN\geq C\log N$ where $G(N,p)$ has with high probability no isolated vertices. In the same regime, we also prove the complete delocalization of the eigenvectors. Both results are false in the complementary subcritical regime. Our result improves the corresponding results from ( Ann. Probab. 41 (2013) 2279–2375) by extending them all the way down to the critical scale $pN=O(\log N)$.
A key ingredient of our proof is a new family of multilinear large deviation estimates for sparse random vectors, which carefully balance mixed $\ell^{2}$ and $\ell^{\infty }$ norms of the coefficients with combinatorial factors, allowing us to prove strong enough concentration down to the critical scale $pN=O(\log N)$. These estimates are of independent interest and we expect them to be more generally useful in the analysis of very sparse random matrices.
</p>projecteuclid.org/euclid.aop/1571731451_20191022040429Tue, 22 Oct 2019 04:04 EDTCutoff for random to random card shufflehttps://projecteuclid.org/euclid.aop/1571731452<strong>Megan Bernstein</strong>, <strong>Evita Nestoridi</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 5, 3303--3320.</p><p><strong>Abstract:</strong><br/>
In this paper, we use the eigenvalues of the random to random card shuffle to prove a sharp upper bound for the total variation mixing time. Combined with the lower bound due to Subag, we prove that this walk exhibits cutoff at $\frac{3}{4}n\log n-\frac{1}{4}n\log\log{n}$ with window of order $n$, answering a conjecture of Diaconis.
</p>projecteuclid.org/euclid.aop/1571731452_20191022040429Tue, 22 Oct 2019 04:04 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 EDT