Annales de l'Institut Henri Poincaré, Probabilités et Statistiques Articles (Project Euclid)
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The latest articles from Annales de l'Institut Henri Poincaré, Probabilités et Statistiques 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, 23 Mar 2011 09:35 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Between Paouris concentration inequality and variance conjecture
http://projecteuclid.org/euclid.aihp/1273584125
<strong>B. Fleury</strong><p><strong>Source: </strong>Ann. Inst. H. Poincaré Probab. Statist., Volume 46, Number 2, 299--312.</p><p><strong>Abstract:</strong><br/>
We prove an almost isometric reverse Hölder inequality for the Euclidean norm on an isotropic generalized Orlicz ball which interpolates Paouris concentration inequality and variance conjecture. We study in this direction the case of isotropic convex bodies with an unconditional basis and the case of general convex bodies.
</p>projecteuclid.org/euclid.aihp/1273584125_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTNon-fixation for biased Activated Random Walkshttps://projecteuclid.org/euclid.aihp/1524643235<strong>L. T. Rolla</strong>, <strong>L. Tournier</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 2, 938--951.</p><p><strong>Abstract:</strong><br/>
We prove that the model of Activated Random Walks on $\mathbb{Z}^{d}$ with biased jump distribution does not fixate for any positive density, if the sleep rate is small enough, as well as for any finite sleep rate, if the density is close enough to $1$. The proof uses a new criterion for non-fixation. We provide a pathwise construction of the process, of independent interest, used in the proof of this non-fixation criterion.
</p>projecteuclid.org/euclid.aihp/1524643235_20180425040043Wed, 25 Apr 2018 04:00 EDTRigidity of 3-colorings of the discrete torushttps://projecteuclid.org/euclid.aihp/1524643236<strong>Ohad Noy Feldheim</strong>, <strong>Ron Peled</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 2, 952--994.</p><p><strong>Abstract:</strong><br/>
We prove that a uniformly chosen proper $3$-coloring of the $d$-dimensional discrete torus has a very rigid structure when the dimension $d$ is sufficiently high. We show that with high probability the coloring takes just one color on almost all of either the even or the odd sub-torus. In particular, one color appears on nearly half of the torus sites. This model is the zero temperature case of the $3$-state anti-ferromagnetic Potts model from statistical physics.
Our work extends previously obtained results for the discrete torus with specific boundary conditions. The main challenge in this extension is to overcome certain topological obstructions which appear when no boundary conditions are imposed on the model. Locally, a proper $3$-coloring defines the discrete gradient of an integer-valued height function which changes by exactly one between adjacent sites. However, these locally-defined functions do not always yield a height function on the entire torus, as the gradients may accumulate to a non-zero quantity when winding around the torus. Our main result is that in high dimensions, a global height function is well defined with high probability, allowing to deduce the rigid structure of the coloring from previously known results. Moreover, the probability that the gradients accumulate to a vector $m$, corresponding to the winding in each of the $d$ directions, is at most exponentially small in the product of $\|m\|_{\infty}$ and the area of a cross-section of the torus.
In the course of the proof we develop discrete analogues of notions from algebraic topology. This theory is developed in some generality and may be of use in the study of other models.
</p>projecteuclid.org/euclid.aihp/1524643236_20180425040043Wed, 25 Apr 2018 04:00 EDT$\operatorname{ASEP}(q,j)$ converges to the KPZ equationhttps://projecteuclid.org/euclid.aihp/1524643237<strong>Ivan Corwin</strong>, <strong>Hao Shen</strong>, <strong>Li-Cheng Tsai</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 2, 995--1012.</p><p><strong>Abstract:</strong><br/>
We show that a generalized Asymmetric Exclusion Process called $\operatorname{ASEP}(q,j)$ introduced in ( Probab. Theory Related Fields 166 (2016) 887–933). converges to the Cole–Hopf solution to the KPZ equation under weak asymmetry scaling.
</p>projecteuclid.org/euclid.aihp/1524643237_20180425040043Wed, 25 Apr 2018 04:00 EDTStochastic orders and the frog modelhttps://projecteuclid.org/euclid.aihp/1524643238<strong>Tobias Johnson</strong>, <strong>Matthew Junge</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 2, 1013--1030.</p><p><strong>Abstract:</strong><br/>
The frog model starts with one active particle at the root of a graph and some number of dormant particles at all nonroot vertices. Active particles follow independent random paths, waking all inactive particles they encounter. We prove that certain frog model statistics are monotone in the initial configuration for two nonstandard stochastic dominance relations: the increasing concave and the probability generating function orders.
This extends many canonical theorems. We connect recurrence for random initial configurations to recurrence for deterministic configurations. Also, the limiting shape of activated sites on the integer lattice respects both of these orders. Other implications include monotonicity results on transience of the frog model where the number of frogs per vertex decays away from the origin, on survival of the frog model with death, and on the time to visit a given vertex in any frog model.
</p>projecteuclid.org/euclid.aihp/1524643238_20180425040043Wed, 25 Apr 2018 04:00 EDTHeight fluctuations of stationary TASEP on a ring in relaxation time scalehttps://projecteuclid.org/euclid.aihp/1524643239<strong>Zhipeng Liu</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 2, 1031--1057.</p><p><strong>Abstract:</strong><br/>
We consider the totally asymmetric simple exclusion process on a ring with stationary initial conditions. The crossover between KPZ dynamics and equilibrium dynamics occurs when time is proportional to the $3/2$ power of the ring size. We obtain the limit of the height function along the direction of the characteristic line in this time scale. The two-point covariance function in this scale is also discussed.
</p>projecteuclid.org/euclid.aihp/1524643239_20180425040043Wed, 25 Apr 2018 04:00 EDTVariational multiscale nonparametric regression: Smooth functionshttps://projecteuclid.org/euclid.aihp/1524643240<strong>Markus Grasmair</strong>, <strong>Housen Li</strong>, <strong>Axel Munk</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 2, 1058--1097.</p><p><strong>Abstract:</strong><br/>
For the problem of nonparametric regression of smooth functions, we reconsider and analyze a constrained variational approach, which we call the MultIscale Nemirovski–Dantzig (MIND) estimator. This can be viewed as a multiscale extension of the Dantzig selector ( Ann. Statist. 35 (2009) 2313–2351) based on early ideas of Nemirovski ( J. Comput. System Sci. 23 (1986) 1–11). MIND minimizes a homogeneous Sobolev norm under the constraint that the multiresolution norm of the residual is bounded by a universal threshold. The main contribution of this paper is the derivation of convergence rates of MIND with respect to $L^{q}$-loss, $1\le q\le\infty $, both almost surely and in expectation. To this end, we introduce the method of approximate source conditions. For a one-dimensional signal, these can be translated into approximation properties of B-splines. A remarkable consequence is that MIND attains almost minimax optimal rates simultaneously for a large range of Sobolev and Besov classes, which provides certain adaptation. Complimentary to the asymptotic analysis, we examine the finite sample performance of MIND by numerical simulations. A MATLAB package is available online.
</p>projecteuclid.org/euclid.aihp/1524643240_20180425040043Wed, 25 Apr 2018 04:00 EDTFrom optimal stopping boundaries to Rost’s reversed barriers and the Skorokhod embeddinghttps://projecteuclid.org/euclid.aihp/1524643241<strong>Tiziano De Angelis</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 2, 1098--1133.</p><p><strong>Abstract:</strong><br/>
We provide a new probabilistic proof of the connection between Rost’s solution of the Skorokhod embedding problem and a suitable family of optimal stopping problems for Brownian motion, with finite time-horizon. In particular we use stochastic calculus to show that the time reversal of the optimal stopping sets for such problems forms the so-called Rost’s reversed barrier.
</p>projecteuclid.org/euclid.aihp/1524643241_20180425040043Wed, 25 Apr 2018 04:00 EDTUniversality in a class of fragmentation-coalescence processeshttps://projecteuclid.org/euclid.aihp/1524643242<strong>A. E. Kyprianou</strong>, <strong>S. W. Pagett</strong>, <strong>T. Rogers</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 2, 1134--1151.</p><p><strong>Abstract:</strong><br/>
We introduce and analyse a class of fragmentation-coalescence processes defined on finite systems of particles organised into clusters. Coalescent events merge multiple clusters simultaneously to form a single larger cluster, while fragmentation breaks up a cluster into a collection of singletons. Under mild conditions on the coalescence rates, we show that the distribution of cluster sizes becomes non-random in the thermodynamic limit. Moreover, we discover that in the limit of small fragmentation rate these processes exhibit a universal cluster size distribution regardless of the details of the rates, following a power law with exponent $3/2$.
</p>projecteuclid.org/euclid.aihp/1524643242_20180425040043Wed, 25 Apr 2018 04:00 EDTIntertwinings of beta-Dyson Brownian motions of different dimensionshttps://projecteuclid.org/euclid.aihp/1524643243<strong>Kavita Ramanan</strong>, <strong>Mykhaylo Shkolnikov</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 2, 1152--1163.</p><p><strong>Abstract:</strong><br/>
We show that for all positive $\beta$ the semigroups of $\beta$-Dyson Brownian motions of different dimensions are intertwined. The proof relates $\beta$-Dyson Brownian motions directly to Jack symmetric polynomials and omits an approximation of the former by discrete space Markov chains, thereby disposing of the technical assumption $\beta\ge1$ in ( Probab. Theory Related Fields 163 (2015) 413–463). The corresponding results for $\beta$-Dyson Ornstein–Uhlenbeck processes are also presented.
</p>projecteuclid.org/euclid.aihp/1524643243_20180425040043Wed, 25 Apr 2018 04:00 EDTThe velocity of 1d Mott variable-range hopping with external fieldhttps://projecteuclid.org/euclid.aihp/1531296017<strong>Alessandra Faggionato</strong>, <strong>Nina Gantert</strong>, <strong>Michele Salvi</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 3, 1165--1203.</p><p><strong>Abstract:</strong><br/>
Mott variable-range hopping is a fundamental mechanism for low-temperature electron conduction in disordered solids in the regime of Anderson localization. In a mean field approximation, it reduces to a random walk (shortly, Mott random walk) on a random marked point process with possible long-range jumps.
We consider here the one-dimensional Mott random walk and we add an external field (or a bias to the right). We show that the bias makes the walk transient, and investigate its linear speed. Our main results are conditions for ballisticity (positive linear speed) and for sub-ballisticity (zero linear speed), and the existence in the ballistic regime of an invariant distribution for the environment viewed from the walker, which is mutually absolutely continuous with respect to the original law of the environment. If the point process is a renewal process, the aforementioned conditions result in a sharp criterion for ballisticity. Interestingly, the speed is not always continuous as a function of the bias.
</p>projecteuclid.org/euclid.aihp/1531296017_20180711040025Wed, 11 Jul 2018 04:00 EDTSpectral gap for the stochastic quantization equation on the 2-dimensional torushttps://projecteuclid.org/euclid.aihp/1531296018<strong>Pavlos Tsatsoulis</strong>, <strong>Hendrik Weber</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 3, 1204--1249.</p><p><strong>Abstract:</strong><br/>
We study the long time behavior of the stochastic quantization equation. Extending recent results by Mourrat and Weber (Global well-posedness of the dynamic $\phi^{4}$ in the plane (2015) Preprint) we first establish a strong non-linear dissipative bound that gives control of moments of solutions at all positive times independent of the initial datum. We then establish that solutions give rise to a Markov process whose transition semigroup satisfies the strong Feller property. Following arguments by Chouk and Friz (Support theorem for a singular SPDE: the case of gPAM (2016) Preprint) we also prove a support theorem for the laws of the solutions. Finally all of these results are combined to show that the transition semigroup satisfies the Doeblin criterion which implies exponential convergence to equilibrium.
Along the way we give a simple direct proof of the Markov property of solutions and an independent argument for the existence of an invariant measure using the Krylov–Bogoliubov existence theorem. Our method makes no use of the reversibility of the dynamics or the explicit knowledge of the invariant measure and it is therefore in principle applicable to situations where these are not available, e.g. the vector-valued case.
</p>projecteuclid.org/euclid.aihp/1531296018_20180711040025Wed, 11 Jul 2018 04:00 EDTAsymptotics of random domino tilings of rectangular Aztec diamondshttps://projecteuclid.org/euclid.aihp/1531296019<strong>Alexey Bufetov</strong>, <strong>Alisa Knizel</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 3, 1250--1290.</p><p><strong>Abstract:</strong><br/>
We consider asymptotics of a domino tiling model on a class of domains which we call rectangular Aztec diamonds. We prove the Law of Large Numbers for the corresponding height functions and provide explicit formulas for the limit. For a special class of examples, the explicit parametrization of the frozen boundary is given. It turns out to be an algebraic curve with very special properties. Moreover, we establish the convergence of the fluctuations of the height functions to the Gaussian Free Field in appropriate coordinates. Our main tool is a recently developed moment method for discrete particle systems.
</p>projecteuclid.org/euclid.aihp/1531296019_20180711040025Wed, 11 Jul 2018 04:00 EDTScaling limit and ageing for branching random walk in Pareto environmenthttps://projecteuclid.org/euclid.aihp/1531296020<strong>Marcel Ortgiese</strong>, <strong>Matthew I. Roberts</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 3, 1291--1313.</p><p><strong>Abstract:</strong><br/>
We consider a branching random walk on the lattice, where the branching rates are given by an i.i.d. Pareto random potential. We show that the system of particles, rescaled in an appropriate way, converges in distribution to a scaling limit that is interesting in its own right. We describe the limit object as a growing collection of “lilypads” built on a Poisson point process in $\mathbb{R}^{d}$. As an application of our main theorem, we show that the maximizer of the system displays the ageing property.
</p>projecteuclid.org/euclid.aihp/1531296020_20180711040025Wed, 11 Jul 2018 04:00 EDTThe strong Feller property for singular stochastic PDEshttps://projecteuclid.org/euclid.aihp/1531296021<strong>M. Hairer</strong>, <strong>J. Mattingly</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 3, 1314--1340.</p><p><strong>Abstract:</strong><br/>
We show that the Markov semigroups generated by a large class of singular stochastic PDEs satisfy the strong Feller property. These include for example the KPZ equation and the dynamical $\Phi^{4}_{3}$ model. As a corollary, we prove that the Brownian bridge measure is the unique invariant measure for the KPZ equation with periodic boundary conditions.
</p>projecteuclid.org/euclid.aihp/1531296021_20180711040025Wed, 11 Jul 2018 04:00 EDTBiased random walks on the interlacement sethttps://projecteuclid.org/euclid.aihp/1531296022<strong>Alexander Fribergh</strong>, <strong>Serguei Popov</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 3, 1341--1358.</p><p><strong>Abstract:</strong><br/>
We study a biased random walk on the interlacement set of $\mathbb{Z}^{d}$ for $d\geq3$. Although the walk is always transient, we can show, in the case $d=3$, that for any value of the bias the walk has a zero limiting speed and actually moves slower than any power.
</p>projecteuclid.org/euclid.aihp/1531296022_20180711040025Wed, 11 Jul 2018 04:00 EDTMarkov processes on the duals to infinite-dimensional classical Lie groupshttps://projecteuclid.org/euclid.aihp/1531296023<strong>Cesar Cuenca</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 3, 1359--1407.</p><p><strong>Abstract:</strong><br/>
We construct a four parameter $z,z',a,b$ family of Markov dynamics that preserve the $z$-measures on the boundary of the branching graph for classical Lie groups of type $B,C,D$. Our guiding principle is the “method of intertwiners” used previously in [ J. Funct. Anal. 263 (2012) 248–303] to construct Markov processes that preserve the $zw$-measures.
</p>projecteuclid.org/euclid.aihp/1531296023_20180711040025Wed, 11 Jul 2018 04:00 EDTWeak convergence of obliquely reflected diffusionshttps://projecteuclid.org/euclid.aihp/1531296024<strong>Andrey Sarantsev</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 3, 1408--1431.</p><p><strong>Abstract:</strong><br/>
Burdzy and Chen ( Electron. J. Probab. 3 (1998) 29–33) proved results on weak convergence of multidimensional normally reflected Brownian motions. We generalize their work by considering obliquely reflected diffusion processes. We require weak convergence of domains, which is stronger than convergence in Wijsman topology, but weaker than convergence in Hausdorff topology.
</p>projecteuclid.org/euclid.aihp/1531296024_20180711040025Wed, 11 Jul 2018 04:00 EDTFluctuations of bridges, reciprocal characteristics and concentration of measurehttps://projecteuclid.org/euclid.aihp/1531296025<strong>Giovanni Conforti</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 3, 1432--1463.</p><p><strong>Abstract:</strong><br/>
Conditions on the generator of a Markov process to control the fluctuations of its bridges are found. In particular, continuous time random walks on graphs and gradient diffusions are considered. Under these conditions, a concentration of measure inequality for the marginals of the bridge of a gradient diffusion and refined large deviation expansions for the tails of a random walk on a graph are derived. In contrast with the existing literature about bridges, all the estimates we obtain hold for non asymptotic time scales. New concentration of measure inequalities for pinned Poisson random vectors are also established. The quantities expressing our conditions are the so called reciprocal characteristics associated with the Markov generator.
</p>projecteuclid.org/euclid.aihp/1531296025_20180711040025Wed, 11 Jul 2018 04:00 EDTConstruction of Malliavin differentiable strong solutions of SDEs under an integrability condition on the drift without the Yamada–Watanabe principlehttps://projecteuclid.org/euclid.aihp/1531296026<strong>David R. Baños</strong>, <strong>Sindre Duedahl</strong>, <strong>Thilo Meyer-Brandis</strong>, <strong>Frank Proske</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 3, 1464--1491.</p><p><strong>Abstract:</strong><br/>
In this paper we aim at employing a compactness criterion of Da Prato, Malliavin, Nualart (C. R. Math. Acad. Sci. Paris 315 (1992) 1287–1291) for square integrable Brownian functionals to construct strong solutions of SDE’s under an integrability condition on the drift coefficient. The obtained solutions turn out to be Malliavin differentiable and are used to derive a Bismut–Elworthy–Li formula for solutions of the Kolmogorov equation. We emphasise that our approach exhibits high flexibility to study a variety of other types of stochastic (partial) differential equations as e.g. stochastic differential equations driven by fractional Brownian motion.
</p>projecteuclid.org/euclid.aihp/1531296026_20180711040025Wed, 11 Jul 2018 04:00 EDTThick points of high-dimensional Gaussian free fieldshttps://projecteuclid.org/euclid.aihp/1531296027<strong>Linan Chen</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 3, 1492--1526.</p><p><strong>Abstract:</strong><br/>
This work aims to extend the existing results on thick points of logarithmic-correlated Gaussian Free Fields to Gaussian random fields that are more singular. To be specific, we adopt a sphere averaging regularization to study polynomial-correlated Gaussian Free Fields in higher-than-two dimensions. Under this setting, we introduce the definition of thick points which, heuristically speaking, are points where the value of the Gaussian Free Field is unusually large. We then establish a result on the Hausdorff dimension of the sets containing thick points.
</p>projecteuclid.org/euclid.aihp/1531296027_20180711040025Wed, 11 Jul 2018 04:00 EDTThe size of the last merger and time reversal in $\Lambda$-coalescentshttps://projecteuclid.org/euclid.aihp/1531296028<strong>Götz Kersting</strong>, <strong>Jason Schweinsberg</strong>, <strong>Anton Wakolbinger</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 3, 1527--1555.</p><p><strong>Abstract:</strong><br/>
We consider the number of blocks involved in the last merger of a $\Lambda$-coalescent started with $n$ blocks. We give conditions under which, as $n\to\infty$, the sequence of these random variables (a) is tight, (b) converges in distribution to a finite random variable or (c) converges to infinity in probability. Our conditions are optimal for $\Lambda$-coalescents that have a dust component. For general $\Lambda$, we relate the three cases to the existence, uniqueness and non-existence of invariant measures for the dynamics of the block-counting process, and in case (b) investigate the time-reversal of the block-counting process back from the time of the last merger.
</p>projecteuclid.org/euclid.aihp/1531296028_20180711040025Wed, 11 Jul 2018 04:00 EDTOptimal discretization of stochastic integrals driven by general Brownian semimartingalehttps://projecteuclid.org/euclid.aihp/1531296029<strong>Emmanuel Gobet</strong>, <strong>Uladzislau Stazhynski</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 3, 1556--1582.</p><p><strong>Abstract:</strong><br/>
We study the optimal discretization error of stochastic integrals, driven by a multidimensional continuous Brownian semimartingale. In this setting we establish a pathwise lower bound for the renormalized quadratic variation of the error and we provide a sequence of discretization stopping times, which is asymptotically optimal. The latter is defined as hitting times of random ellipsoids by the semimartingale at hand. In comparison with previous available results, we allow a quite large class of semimartingales (relaxing in particular the non degeneracy conditions usually requested) and we prove that the asymptotic lower bound is attainable.
</p>projecteuclid.org/euclid.aihp/1531296029_20180711040025Wed, 11 Jul 2018 04:00 EDTLow-rank diffusion matrix estimation for high-dimensional time-changed Lévy processeshttps://projecteuclid.org/euclid.aihp/1531296030<strong>Denis Belomestny</strong>, <strong>Mathias Trabs</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 3, 1583--1621.</p><p><strong>Abstract:</strong><br/>
The estimation of the diffusion matrix $\Sigma$ of a high-dimensional, possibly time-changed Lévy process is studied, based on discrete observations of the process with a fixed distance. A low-rank condition is imposed on $\Sigma$. Applying a spectral approach, we construct a weighted least-squares estimator with nuclear-norm-penalisation. We prove oracle inequalities and derive convergence rates for the diffusion matrix estimator. The convergence rates show a surprising dependency on the rank of $\Sigma$ and are optimal in the minimax sense for fixed dimensions. Theoretical results are illustrated by a simulation study.
</p>projecteuclid.org/euclid.aihp/1531296030_20180711040025Wed, 11 Jul 2018 04:00 EDTThe near-critical Gibbs measure of the branching random walkhttps://projecteuclid.org/euclid.aihp/1531296031<strong>Michel Pain</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 3, 1622--1666.</p><p><strong>Abstract:</strong><br/>
Consider the supercritical branching random walk on the real line in the boundary case and the associated Gibbs measure $\nu_{n,\beta}$ on the $n$th generation, which is also the polymer measure on a disordered tree with inverse temperature $\beta$. The convergence of the partition function $W_{n,\beta}$, after rescaling, towards a nontrivial limit has been proved by Aïdékon and Shi ( Ann. Probab. 42 (3) (2014) 959–993) in the critical case $\beta=1$ and by Madaule ( J. Theoret. Probab. 30 (1) (2017) 27–63) when $\beta>1$. We study here the near-critical case, where $\beta_{n}\to1$, and prove the convergence of $W_{n,\beta_{n}}$, after rescaling, towards a constant multiple of the limit of the derivative martingale. Moreover, trajectories of particles chosen according to the Gibbs measure $\nu_{n,\beta}$ have been studied by Madaule ( Stochastic Process. Appl. 126 (2) (2016) 470–502) in the critical case, with convergence towards the Brownian meander, and by Chen, Madaule and Mallein (On the trajectory of an individual chosen according to supercritical gibbs measure in the branching random walk (2015) Preprint) in the strong disorder regime, with convergence towards the normalized Brownian excursion. We prove here the convergence for trajectories of particles chosen according to the near-critical Gibbs measure and display continuous families of processes from the meander to the excursion or to the Brownian motion.
</p>projecteuclid.org/euclid.aihp/1531296031_20180711040025Wed, 11 Jul 2018 04:00 EDTCharacterization of a class of weak transport-entropy inequalities on the linehttps://projecteuclid.org/euclid.aihp/1531296032<strong>Nathael Gozlan</strong>, <strong>Cyril Roberto</strong>, <strong>Paul-Marie Samson</strong>, <strong>Yan Shu</strong>, <strong>Prasad Tetali</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 3, 1667--1693.</p><p><strong>Abstract:</strong><br/>
We study an weak transport cost related to the notion of convex order between probability measures. On the real line, we show that this weak transport cost is reached for a coupling that does not depend on the underlying cost function. As an application, we give a necessary and sufficient condition for weak transport-entropy inequalities (related to concentration of convex/concave functions) to hold on the line. In particular, we obtain a weak transport-entropy form of the convex Poincaré inequality in dimension one.
</p>projecteuclid.org/euclid.aihp/1531296032_20180711040025Wed, 11 Jul 2018 04:00 EDTLiouville quantum gravity on the unit diskhttps://projecteuclid.org/euclid.aihp/1531296033<strong>Yichao Huang</strong>, <strong>Rémi Rhodes</strong>, <strong>Vincent Vargas</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 3, 1694--1730.</p><p><strong>Abstract:</strong><br/>
Our purpose is to pursue the rigorous construction of Liouville Quantum Field Theory on Riemann surfaces initiated by F. David, A. Kupiainen and the last two authors in the context of the Riemann sphere and inspired by the 1981 seminal work by Polyakov. In this paper, we investigate the case of simply connected domains with boundary. We also make precise conjectures about the relationship of this theory to scaling limits of random planar maps with boundary conformally embedded onto the disk.
</p>projecteuclid.org/euclid.aihp/1531296033_20180711040025Wed, 11 Jul 2018 04:00 EDTInterpolation process between standard diffusion and fractional diffusionhttps://projecteuclid.org/euclid.aihp/1531296034<strong>Cédric Bernardin</strong>, <strong>Patrícia Gonçalves</strong>, <strong>Milton Jara</strong>, <strong>Marielle Simon</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 3, 1731--1757.</p><p><strong>Abstract:</strong><br/>
We consider a Hamiltonian lattice field model with two conserved quantities, energy and volume, perturbed by stochastic noise preserving the two previous quantities. It is known that this model displays anomalous diffusion of energy of fractional type due to the conservation of the volume ( Nonlinearity 25 (4) (2012) 1099–1133; Arch. Ration. Mech. Anal. 220 (2) (2016) 505–542). We superpose to this system a second stochastic noise conserving energy but not volume. If the intensity of this noise is of order one, normal diffusion of energy is restored while it is without effect if intensity is sufficiently small. In this paper we investigate the nature of the energy fluctuations for a critical value of the intensity. We show that the latter are described by an Ornstein–Uhlenbeck process driven by a Lévy process which interpolates between Brownian motion and the maximally asymmetric $3/2$-stable Lévy process. This result extends and solves a problem left open in ( J. Stat. Phys. 159 (6) (2015) 1327–1368).
</p>projecteuclid.org/euclid.aihp/1531296034_20180711040025Wed, 11 Jul 2018 04:00 EDTGaussian fluctuations for the classical XY modelhttps://projecteuclid.org/euclid.aihp/1539849782<strong>Charles M. Newman</strong>, <strong>Wei Wu</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 1759--1777.</p><p><strong>Abstract:</strong><br/>
We study the classical XY model in bounded domains of $\mathbb{Z}^{d}$ with Dirichlet boundary conditions. We prove that when the temperature goes to zero faster than a certain rate as the lattice spacing goes to zero, the fluctuation field converges to a Gaussian white noise. This and related results also apply to a large class of gradient field models.
</p>projecteuclid.org/euclid.aihp/1539849782_20181018040325Thu, 18 Oct 2018 04:03 EDTContinuum percolation in high dimensionshttps://projecteuclid.org/euclid.aihp/1539849783<strong>Jean-Baptiste Gouéré</strong>, <strong>Régine Marchand</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 1778--1804.</p><p><strong>Abstract:</strong><br/>
Consider a Boolean model $\Sigma$ in $\mathbb{R}^{d}$. The centers are given by a homogeneous Poisson point process with intensity $\lambda$ and the radii of distinct balls are i.i.d. with common distribution $\nu$. The critical covered volume is the proportion of space covered by $\Sigma$ when the intensity $\lambda$ is critical for percolation. We study the asymptotic behaviour, as $d$ tends to infinity, of the critical covered volume. It appears that, in contrast to what happens in the constant radii case studied by Penrose, geometrical dependencies do not always vanish in high dimension.
</p>projecteuclid.org/euclid.aihp/1539849783_20181018040325Thu, 18 Oct 2018 04:03 EDTConvergence to equilibrium in the free Fokker–Planck equation with a double-well potentialhttps://projecteuclid.org/euclid.aihp/1539849784<strong>Catherine Donati-Martin</strong>, <strong>Benjamin Groux</strong>, <strong>Mylène Maïda</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 1805--1818.</p><p><strong>Abstract:</strong><br/>
We consider the one-dimensional free Fokker–Planck equation
\[\frac{\partial \mu_{t}}{\partial t}=\frac{\partial }{\partial x}[\mu_{t}\cdot (\frac{1}{2}V'-H\mu_{t})],\] where $H$ denotes the Hilbert transform and $V$ is a particular double-well quartic potential, namely $V(x)=\frac{1}{4}x^{4}+\frac{c}{2}x^{2}$, with $c\ge -2$. We prove that the solution $(\mu_{t})_{t\ge 0}$ of this PDE converges in Wasserstein distance of any order $p\ge 1$ to the equilibrium measure $\mu_{V}$ as $t$ goes to infinity. This provides a first result of convergence for this equation in a non-convex setting. The proof involves free probability and complex analysis techniques.
</p>projecteuclid.org/euclid.aihp/1539849784_20181018040325Thu, 18 Oct 2018 04:03 EDTPercolation and isoperimetry on roughly transitive graphshttps://projecteuclid.org/euclid.aihp/1539849785<strong>Elisabetta Candellero</strong>, <strong>Augusto Teixeira</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 1819--1847.</p><p><strong>Abstract:</strong><br/>
In this paper we study percolation on a roughly transitive graph $G$ with polynomial growth and isoperimetric dimension larger than one. For these graphs we are able to prove that $p_{c}<1$, or in other words, that there exists a percolation phase. The main results of the article work for both dependent and independent percolation processes, since they are based on a quite robust renormalization technique. When $G$ is transitive, the fact that $p_{c}<1$ was already known before. But even in that case our proof yields some new results and it is entirely probabilistic, not involving the use of Gromov’s theorem on groups of polynomial growth. We finish the paper giving some examples of dependent percolation for which our results apply.
</p>projecteuclid.org/euclid.aihp/1539849785_20181018040325Thu, 18 Oct 2018 04:03 EDTMulti-arm incipient infinite clusters in 2D: Scaling limits and winding numbershttps://projecteuclid.org/euclid.aihp/1539849786<strong>Chang-Long Yao</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 1848--1876.</p><p><strong>Abstract:</strong><br/>
We study the alternating $k$-arm incipient infinite cluster (IIC) of site percolation on the triangular lattice $\mathbb{T}$. Using Camia and Newman’s result that the scaling limit of critical site percolation on $\mathbb{T}$ is CLE$_{6}$, we prove the existence of the scaling limit of the $k$-arm IIC for $k=1,2,4$. Conditioned on the event that there are open and closed arms connecting the origin to $\partial\mathbb{D}_{R}$, we show that the winding number variance of the arms is $(3/2+o(1))\log R$ as $R\rightarrow\infty$, which confirms a prediction of Wieland and Wilson [ Phys. Rev. E 68 (2003) 056101]. Our proof uses two-sided radial SLE$_{6}$ and coupling argument. Using this result we get an explicit form for the CLT of the winding numbers, and get analogous result for the 2-arm IIC, thus improving our earlier result.
</p>projecteuclid.org/euclid.aihp/1539849786_20181018040325Thu, 18 Oct 2018 04:03 EDTBrownian motion and random walk above quenched random wallhttps://projecteuclid.org/euclid.aihp/1539849787<strong>Bastien Mallein</strong>, <strong>Piotr Miłoś</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 1877--1916.</p><p><strong>Abstract:</strong><br/>
We study the persistence exponent for the first passage time of a random walk below the trajectory of another random walk. More precisely, let $\{B_{n}\}$ and $\{W_{n}\}$ be two centered, weakly dependent random walks. We establish that $\mathbb{P}(\forall_{n\leq N}B_{n}\geq W_{n}\vert W)=N^{-\gamma+o(1)}$ for a non-random $\gamma\geq1/2$. In the classical setting, $W_{n}\equiv0$, it is well-known that $\gamma=1/2$. We prove that for any non-trivial $W$ one has $\gamma>1/2$ and the exponent $\gamma$ depends only on $\operatorname{Var}(B_{1})/\operatorname{Var}(W_{1})$. Our result holds also in the continuous setting, when $B$ and $W$ are independent and possibly perturbed Brownian motions or Ornstein–Uhlenbeck processes. In the latter case the probability decays at exponential rate.
</p>projecteuclid.org/euclid.aihp/1539849787_20181018040325Thu, 18 Oct 2018 04:03 EDTMesoscopic central limit theorem for general $\beta$-ensembleshttps://projecteuclid.org/euclid.aihp/1539849788<strong>Florent Bekerman</strong>, <strong>Asad Lodhia</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 1917--1938.</p><p><strong>Abstract:</strong><br/>
We prove that the linear statistics of eigenvalues of $\beta$-log gases satisfying the one-cut and off-critical assumption with a potential $V\in C^{7}(\mathbb{R})$ satisfy a central limit theorem at all mesoscopic scales $\alpha\in(0;1)$. We prove this for compactly supported test functions $f\in C^{6}(\mathbb{R})$ using loop equations at all orders along with rigidity estimates.
</p>projecteuclid.org/euclid.aihp/1539849788_20181018040325Thu, 18 Oct 2018 04:03 EDTScaling limits of stochastic processes associated with resistance formshttps://projecteuclid.org/euclid.aihp/1539849789<strong>D. A. Croydon</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 1939--1968.</p><p><strong>Abstract:</strong><br/>
We establish that if a sequence of spaces equipped with resistance metrics and measures converge with respect to the Gromov–Hausdorff-vague topology, and a certain non-explosion condition is satisfied, then the associated stochastic processes also converge. This result generalises previous work on trees, fractals, and various models of random graphs. We further conjecture that it will be applicable to the random walk on the incipient infinite cluster of critical bond percolation on the high-dimensional integer lattice.
</p>projecteuclid.org/euclid.aihp/1539849789_20181018040325Thu, 18 Oct 2018 04:03 EDTGlobal well-posedness of complex Ginzburg–Landau equation with a space–time white noisehttps://projecteuclid.org/euclid.aihp/1539849790<strong>Masato Hoshino</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 1969--2001.</p><p><strong>Abstract:</strong><br/>
We show the global-in-time well-posedness of the complex Ginzburg–Landau (CGL) equation with a space–time white noise on the 3-dimensional torus. Our method is based on Mourrat and Weber (Global well-posedness of the dynamic $\Phi_{3}^{4}$ model on the torus), where Mourrat and Weber showed the global well-posedness for the dynamical $\Phi_{3}^{4}$ model. We prove a priori $L^{2p}$ estimate for the paracontrolled solution as in the deterministic case [ Phys. D 71 (1994) 285–318].
</p>projecteuclid.org/euclid.aihp/1539849790_20181018040325Thu, 18 Oct 2018 04:03 EDTLocal large deviations principle for occupation measures of the stochastic damped nonlinear wave equationhttps://projecteuclid.org/euclid.aihp/1539849791<strong>D. Martirosyan</strong>, <strong>V. Nersesyan</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 2002--2041.</p><p><strong>Abstract:</strong><br/>
We consider the damped nonlinear wave (NLW) equation driven by a noise which is white in time and colored in space. Assuming that the noise is non-degenerate in all Fourier modes, we establish a large deviations principle (LDP) for the occupation measures of the trajectories. The lower bound in the LDP is of a local type, which is related to the weakly dissipative nature of the equation and is a novelty in the context of randomly forced PDE’s. The proof is based on an extension of methods developed in ( Comm. Pure Appl. Math. 68 (12) (2015) 2108–2143) and (Large deviations and mixing for dissipative PDE’s with unbounded random kicks (2014) Preprint) in the case of kick forced dissipative PDE’s with parabolic regularization property such as, for example, the Navier–Stokes system and the complex Ginzburg–Landau equations. We also show that a high concentration towards the stationary measure is impossible, by proving that the rate function that governs the LDP cannot have the trivial form (i.e., vanish on the stationary measure and be infinite elsewhere).
</p>projecteuclid.org/euclid.aihp/1539849791_20181018040325Thu, 18 Oct 2018 04:03 EDTMultifractality of jump diffusion processeshttps://projecteuclid.org/euclid.aihp/1539849792<strong>Xiaochuan Yang</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 2042--2074.</p><p><strong>Abstract:</strong><br/>
We study the local regularity and multifractal nature of the sample paths of jump diffusion processes, which are solutions to a class of stochastic differential equations with jumps. This article extends the recent work of Barral et al. who constructed a pure jump monotone Markov process with random multifractal spectrum. The class of processes studied here is much larger and exhibits novel features on the extreme values of the spectrum. This class includes Bass’ stable-like processes and non-degenerate stable-driven SDEs.
</p>projecteuclid.org/euclid.aihp/1539849792_20181018040325Thu, 18 Oct 2018 04:03 EDTA characterization of a class of convex log-Sobolev inequalities on the real linehttps://projecteuclid.org/euclid.aihp/1539849793<strong>Yan Shu</strong>, <strong>Michał Strzelecki</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 2075--2091.</p><p><strong>Abstract:</strong><br/>
We give a sufficient and necessary condition for a probability measure $\mu$ on the real line to satisfy the logarithmic Sobolev inequality for convex functions. The condition is expressed in terms of the unique left-continuous and non-decreasing map transporting the symmetric exponential measure onto $\mu$. The main tool in the proof is the theory of weak transport costs.
As a consequence, we obtain dimension-free concentration bounds for the lower and upper tails of convex functions of independent random variables which satisfy the convex log-Sobolev inequality.
</p>projecteuclid.org/euclid.aihp/1539849793_20181018040325Thu, 18 Oct 2018 04:03 EDTIsoperimetry in supercritical bond percolation in dimensions three and higherhttps://projecteuclid.org/euclid.aihp/1539849794<strong>Julian Gold</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 2092--2158.</p><p><strong>Abstract:</strong><br/>
We study the isoperimetric subgraphs of the infinite cluster $\mathbf{C}_{\infty}$ for supercritical bond percolation on $\mathbb{Z}^{d}$ with $d\geq3$. Specifically, we consider subgraphs of $\mathbf{C}_{\infty}\cap[-n,n]^{d}$ having minimal open edge boundary to volume ratio. We prove a shape theorem for these subgraphs: when suitably rescaled, they converge almost surely to a translate of a deterministic shape. This deterministic shape is itself an isoperimetric set for a norm we construct. As a corollary, we obtain sharp asymptotics on a natural modification of the Cheeger constant for $\mathbf{C}_{\infty}\cap[-n,n]^{d}$, settling a conjecture of Benjamini for the version of the Cheeger constant defined here.
</p>projecteuclid.org/euclid.aihp/1539849794_20181018040325Thu, 18 Oct 2018 04:03 EDTClassical and quantum part of the environment for quantum Langevin equationshttps://projecteuclid.org/euclid.aihp/1539849795<strong>Stéphane Attal</strong>, <strong>Ivan Bardet</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 2159--2176.</p><p><strong>Abstract:</strong><br/>
Among quantum Langevin equations describing the unitary time evolution of a quantum system in contact with a quantum bath, we completely characterize those equations which are actually driven by classical noises. The characterization is purely algebraic, in terms of the coefficients of the equation. In a second part, we consider general quantum Langevin equations and we prove that they can always be split into a maximal part driven by classical noises and a purely quantum one.
</p>projecteuclid.org/euclid.aihp/1539849795_20181018040325Thu, 18 Oct 2018 04:03 EDTHow can a clairvoyant particle escape the exclusion process?https://projecteuclid.org/euclid.aihp/1539849796<strong>Rangel Baldasso</strong>, <strong>Augusto Teixeira</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 2177--2202.</p><p><strong>Abstract:</strong><br/>
We study a detection problem in the following setting: On the one-dimensional integer lattice, at time zero, place detectors on each site independently with probability $\rho \in{[0,1)}$ and let they evolve as a simple symmetric exclusion process. At time zero, place a target at the origin. The target moves only at integer times, and can move to any site that is within distance $R$ from its current position. Assume also that the target can predict the future movement of all detectors. We prove that, for $R$ large enough (depending on the value of $\rho $) it is possible for the target to avoid detection forever with positive probability. The proof of this result uses two ingredients of independent interest. First we establish a renormalisation scheme that can be used to prove percolation for dependent oriented models under a certain decoupling condition. This result is general and does not rely on the specifities of the model. As an application, we prove our main theorem for different dynamics, such as independent random walks and independent renewal chains. We also prove existence of oriented percolation for random interlacements and for its vacant set for large dimensions. The second step of the proof is a space–time decoupling for the exclusion process.
</p>projecteuclid.org/euclid.aihp/1539849796_20181018040325Thu, 18 Oct 2018 04:03 EDTThe geometry of a critical percolation cluster on the UIPThttps://projecteuclid.org/euclid.aihp/1539849797<strong>Matthias Gorny</strong>, <strong>Édouard Maurel-Segala</strong>, <strong>Arvind Singh</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 2203--2238.</p><p><strong>Abstract:</strong><br/>
We consider a critical Bernoulli site percolation on the uniform infinite planar triangulation. We study the tail distributions of the peeling time, perimeter, and volume of the hull of a critical cluster. The exponents obtained here differ by a factor $2$ from those computed previously by Angel and Curien [ Ann. Inst. Henri Poincaré Probab. Stat. 51 (2015) 405–431] in the case of critical site percolation on the uniform infinite half-plane triangulation.
</p>projecteuclid.org/euclid.aihp/1539849797_20181018040325Thu, 18 Oct 2018 04:03 EDTOn the large deviations of traces of random matriceshttps://projecteuclid.org/euclid.aihp/1539849798<strong>Fanny Augeri</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 2239--2285.</p><p><strong>Abstract:</strong><br/>
We present large deviations principles for the moments of the empirical spectral measure of Wigner matrices and empirical measure of $\beta $-ensembles in three cases: the case of $\beta $-ensembles associated with a convex potential with polynomial growth, the case of Gaussian Wigner matrices, and the case of Wigner matrices without Gaussian tails, that is Wigner matrices whose entries have tail distributions decreasing as $e^{-ct^{\alpha }}$, for some constant $c>0$ and with $\alpha \in (0,2)$.
</p>projecteuclid.org/euclid.aihp/1539849798_20181018040325Thu, 18 Oct 2018 04:03 EDTTransporting random measures on the line and embedding excursions into Brownian motionhttps://projecteuclid.org/euclid.aihp/1539849799<strong>Günter Last</strong>, <strong>Wenpin Tang</strong>, <strong>Hermann Thorisson</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 2286--2303.</p><p><strong>Abstract:</strong><br/>
We consider two jointly stationary and ergodic random measures $\xi$ and $\eta$ on the real line $\mathbb{R}$ with equal intensities. An allocation is an equivariant random mapping from $\mathbb{R}$ to $\mathbb{R}$. We give sufficient and partially necessary conditions for the existence of allocations transporting $\xi$ to $\eta$. An important ingredient of our approach is a transport kernel balancing $\xi$ and $\eta$, provided these random measures are mutually singular. In the second part of the paper, we apply this result to the path decomposition of a two-sided Brownian motion into three independent pieces: a time reversed Brownian motion on $(-\infty,0]$, an excursion distributed according to a conditional Itô measure and a Brownian motion starting after this excursion. An analogous result holds for Bismut’s excursion measure.
</p>projecteuclid.org/euclid.aihp/1539849799_20181018040325Thu, 18 Oct 2018 04:03 EDTA temporal central limit theorem for real-valued cocycles over rotationshttps://projecteuclid.org/euclid.aihp/1539849800<strong>Michael Bromberg</strong>, <strong>Corinna Ulcigrai</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 2304--2334.</p><p><strong>Abstract:</strong><br/>
We consider deterministic random walks on the real line driven by irrational rotations, or equivalently, skew product extensions of a rotation by $\alpha$ where the skewing cocycle is a piecewise constant mean zero function with a jump by one at a point $\beta$. When $\alpha$ is badly approximable and $\beta$ is badly approximable with respect to $\alpha$, we prove a Temporal Central Limit theorem (in the terminology recently introduced by D. Dolgopyat and O. Sarig), namely we show that for any fixed initial point, the occupancy random variables , suitably rescaled, converge to a Gaussian random variable. This result generalizes and extends a theorem by J. Beck for the special case when $\alpha$ is quadratic irrational, $\beta$ is rational and the initial point is the origin, recently reproved and then generalized to cover any initial point using geometric renormalization arguments by Avila–Dolgopyat–Duryev–Sarig ( Israel J. Math. 207 (2015) 653–717) and Dolgopyat–Sarig ( J. Stat. Phys. 166 (2017) 680–713). We also use renormalization, but in order to treat irrational values of $\beta$, instead of geometric arguments, we use the renormalization associated to the continued fraction algorithm and dynamical Ostrowski expansions. This yields a suitable symbolic coding framework which allows us to reduce the main result to a CLT for non homogeneous Markov chains.
</p>projecteuclid.org/euclid.aihp/1539849800_20181018040325Thu, 18 Oct 2018 04:03 EDTKinetically constrained lattice gases: Tagged particle diffusionhttps://projecteuclid.org/euclid.aihp/1539849801<strong>O. Blondel</strong>, <strong>C. Toninelli</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 2335--2348.</p><p><strong>Abstract:</strong><br/>
Kinetically constrained lattice gases (KCLG) are interacting particle systems on the integer lattice $\mathbb{Z}^{d}$ with hard core exclusion and Kawasaki type dynamics. Their peculiarity is that jumps are allowed only if the configuration satisfies a constraint which asks for enough empty sites in a certain local neighborhood. KCLG have been introduced and extensively studied in physics literature as models of glassy dynamics. We focus on the most studied class of KCLG, the Kob Andersen (KA) models. We analyze the behavior of a tracer (i.e. a tagged particle) at equilibrium. We prove that for all dimensions $d\geq2$ and for any equilibrium particle density, under diffusive rescaling the motion of the tracer converges to a $d$-dimensional Brownian motion with non-degenerate diffusion matrix. Therefore we disprove the occurrence of a diffusive/non diffusive transition which had been conjectured in physics literature. Our technique is flexible enough and can be extended to analyse the tracer behavior for other choices of constraints.
</p>projecteuclid.org/euclid.aihp/1539849801_20181018040325Thu, 18 Oct 2018 04:03 EDTLocation of the path supremum for self-similar processes with stationary incrementshttps://projecteuclid.org/euclid.aihp/1539849802<strong>Yi Shen</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 2349--2360.</p><p><strong>Abstract:</strong><br/>
In this paper we consider the distribution of the location of the path supremum in a fixed interval for self-similar processes with stationary increments. A point process is constructed and its relation to the distribution of the location of the path supremum is studied. Using this framework, we show that the distribution has a spectral-type representation, in the sense that it is always a mixture of a special group of absolutely continuous distributions, plus point masses on the two boundaries. An upper bound for the value of the density function is established. We further discuss self-similar Lévy processes as an example. Most of the results in this paper can be generalized to a group of random locations, including the location of the largest jump, etc.
</p>projecteuclid.org/euclid.aihp/1539849802_20181018040325Thu, 18 Oct 2018 04:03 EDTScaling limits for the critical Fortuin–Kasteleyn model on a random planar map I: Cone timeshttps://projecteuclid.org/euclid.aihp/1547802394<strong>Ewain Gwynne</strong>, <strong>Cheng Mao</strong>, <strong>Xin Sun</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 1, 1--60.</p><p><strong>Abstract:</strong><br/>
Sheffield (2011) introduced an inventory accumulation model which encodes a random planar map decorated by a collection of loops sampled from the critical Fortuin–Kasteleyn (FK) model. He showed that a certain two-dimensional random walk associated with the infinite-volume version of the model converges in the scaling limit to a correlated planar Brownian motion. We improve on this scaling limit result by showing that the times corresponding to FK loops (or “flexible orders”) in the inventory accumulation model converge in the scaling limit to the $\pi/2$-cone times of the correlated Brownian motion. This statement implies a scaling limit result for the joint law of the areas and boundary lengths of the bounded complementary connected components of the FK loops on the infinite-volume planar map. In light of the encoding of Duplantier, Miller, and Sheffield (2014), the limiting object coincides with the joint law of the areas and boundary lengths of the bounded complementary connected components of a collection of CLE loops on an independent Liouville quantum gravity surface.
</p>projecteuclid.org/euclid.aihp/1547802394_20190118040648Fri, 18 Jan 2019 04:06 ESTOn the fourth moment condition for Rademacher chaoshttps://projecteuclid.org/euclid.aihp/1547802395<strong>Christian Döbler</strong>, <strong>Kai Krokowski</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 1, 61--97.</p><p><strong>Abstract:</strong><br/>
Adapting the spectral viewpoint suggested in ( Ann. Probab. 40 (6) (2012) 2439–2459) in the context of symmetric Markov diffusion generators and recently exploited in the non-diffusive setup of a Poisson random measure ( Ann. Probab. (2017)), we investigate the fourth moment condition for discrete multiple integrals with respect to general, i.e. non-symmetric and non-homogeneous, Rademacher sequences and show that, in this situation, the fourth moment alone does not govern the asymptotic normality. Indeed, here one also has to take into consideration the maximal influence of the corresponding kernel functions. In particular, we show that there is no exact fourth moment theorem for discrete multiple integrals of order $m\geq2$ with respect to a symmetric Rademacher sequence. This behavior, which is in contrast to the Gaussian ( Ann. Probab. 33 (1) (2005) 177–193) and Poisson ( Ann. Probab. (2017)) situation, closely resembles that of degenerate, non-symmetric $U$-statistics from the classical paper ( J. Multivariate Anal. 34 (2) (1990) 275–289).
</p>projecteuclid.org/euclid.aihp/1547802395_20190118040648Fri, 18 Jan 2019 04:06 ESTProducts of random matrices from polynomial ensembleshttps://projecteuclid.org/euclid.aihp/1547802396<strong>Mario Kieburg</strong>, <strong>Holger Kösters</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 1, 98--126.</p><p><strong>Abstract:</strong><br/>
Very recently we have shown that the spherical transform is a convenient tool for studying the relation between the joint density of the singular values and that of the eigenvalues for bi-unitarily invariant random matrices. In the present work we discuss the implications of these results for products of random matrices. In particular, we derive a transformation formula for the joint densities of a product of two independent bi-unitarily invariant random matrices, the first from a polynomial ensemble and the second from a polynomial ensemble of derivative type. This allows us to re-derive and generalize a number of recent results in random matrix theory, including a transformation formula for the kernels of the corresponding determinantal point processes. Starting from these results, we construct a continuous family of random matrix ensembles interpolating between the products of different numbers of Ginibre matrices and inverse Ginibre matrices. Furthermore, we make contact to the asymptotic distribution of the Lyapunov exponents of the products of a large number of bi-unitarily invariant random matrices of fixed dimension.
</p>projecteuclid.org/euclid.aihp/1547802396_20190118040648Fri, 18 Jan 2019 04:06 ESTBarrier estimates for a critical Galton–Watson process and the cover time of the binary treehttps://projecteuclid.org/euclid.aihp/1547802397<strong>David Belius</strong>, <strong>Jay Rosen</strong>, <strong>Ofer Zeitouni</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 1, 127--154.</p><p><strong>Abstract:</strong><br/>
For the critical Galton–Watson process with geometric offspring distributions we provide sharp barrier estimates for barriers which are (small) perturbations of linear barriers. These are useful in analyzing the cover time of finite graphs in the critical regime by random walk, and the Brownian cover times of compact two-dimensional manifolds. As an application of the barrier estimates, we prove that if $C_{L}$ denotes the cover time of the binary tree of depth $L$ by simple walk, then $\sqrt{C_{L}/2^{L+1}}-\sqrt{2\log2}L+\log L/\sqrt{2\log2}$ is tight. The latter improves results of Aldous ( J. Math. Anal. Appl. 157 (1991) 271–283), Bramson and Zeitouni ( Ann. Probab. 37 (2009) 615–653) and Ding and Zeitouni ( Stochastic Process. Appl. 122 (2012) 2117–2133). In a subsequent article we use these barrier estimates to prove tightness of the Brownian cover time for compact two-dimensional manifolds.
</p>projecteuclid.org/euclid.aihp/1547802397_20190118040648Fri, 18 Jan 2019 04:06 ESTLocal limits of large Galton–Watson trees rerooted at a random vertexhttps://projecteuclid.org/euclid.aihp/1547802398<strong>Benedikt Stufler</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 1, 155--183.</p><p><strong>Abstract:</strong><br/>
We discuss various forms of convergence of the vicinity of a uniformly at random selected vertex in random simply generated trees, as the size tends to infinity. For the standard case of a critical Galton–Watson tree conditioned to be large the limit is the invariant random sin-tree constructed by Aldous (1991). In the condensation regime, we describe in complete generality the asymptotic local behaviour from a random vertex up to its first ancestor with large degree. Beyond this distinguished ancestor, different behaviour may occur, depending on the branching weights. In a subregime of complete condensation, we obtain convergence toward a novel limit tree, that describes the asymptotic shape of the vicinity of the full path from a random vertex to the root vertex. This includes the case where the offspring distribution follows a power law up to a factor that varies slowly at infinity.
</p>projecteuclid.org/euclid.aihp/1547802398_20190118040648Fri, 18 Jan 2019 04:06 ESTBranching diffusion representation of semilinear PDEs and Monte Carlo approximationhttps://projecteuclid.org/euclid.aihp/1547802399<strong>Pierre Henry-Labordère</strong>, <strong>Nadia Oudjane</strong>, <strong>Xiaolu Tan</strong>, <strong>Nizar Touzi</strong>, <strong>Xavier Warin</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 1, 184--210.</p><p><strong>Abstract:</strong><br/>
We provide a representation result of parabolic semi-linear PDEs, with polynomial nonlinearity, by branching diffusion processes. We extend the classical representation for KPP equations, introduced by Skorokhod [ Theory Probab. Appl. 9 (1964) 445–449], Watanabe [ J. Math. Kyoto Univ. 4 (1965) 385–398] and McKean [ Comm. Pure Appl. Math. 28 (1975) 323–331], by allowing for polynomial nonlinearity in the pair $(u,Du)$, where $u$ is the solution of the PDE with space gradient $Du$. Similar to the previous literature, our result requires a non-explosion condition which restrict to “small maturity” or “small nonlinearity” of the PDE. Our main ingredient is the Malliavin automatic differentiation technique as in [ Ann. Appl. Probab. 27 (2017) 3305–3341], based on the Malliavin integration by parts, which allows to account for the nonlinearities in the gradient. As a consequence, the particles of our branching diffusion are marked by the nature of the nonlinearity. This new representation has very important numerical implications as it is suitable for Monte Carlo simulation. Indeed, this provides the first numerical method for high dimensional nonlinear PDEs with error estimate induced by the dimension-free central limit theorem. The complexity is also easily seen to be of the order of the squared dimension. The final section of this paper illustrates the efficiency of the algorithm by some high dimensional numerical experiments.
</p>projecteuclid.org/euclid.aihp/1547802399_20190118040648Fri, 18 Jan 2019 04:06 ESTLarge deviations for the two-dimensional stochastic Navier–Stokes equation with vanishing noise correlationhttps://projecteuclid.org/euclid.aihp/1547802400<strong>Sandra Cerrai</strong>, <strong>Arnaud Debussche</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 1, 211--236.</p><p><strong>Abstract:</strong><br/>
We are dealing with the validity of a large deviation principle for the two-dimensional Navier–Stokes equation, with periodic boundary conditions, perturbed by a Gaussian random forcing. We are here interested in the regime where both the strength of the noise and its correlation are vanishing, on a length scale $\epsilon$ and $\delta(\epsilon)$, respectively, with $0<\epsilon,\delta(\epsilon)\ll1$. Depending on the relationship between $\epsilon$ and $\delta(\epsilon)$ we will prove the validity of the large deviation principle in different functional spaces.
</p>projecteuclid.org/euclid.aihp/1547802400_20190118040648Fri, 18 Jan 2019 04:06 ESTBrownian disks and the Brownian snakehttps://projecteuclid.org/euclid.aihp/1547802401<strong>Jean-François Le Gall</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 1, 237--313.</p><p><strong>Abstract:</strong><br/>
We provide a new construction of the Brownian disks, which have been defined by Bettinelli and Miermont as scaling limits of quadrangulations with a boundary when the boundary size tends to infinity. Our method is very similar to the construction of the Brownian map, but it makes use of the positive excursion measure of the Brownian snake which has been introduced recently. This excursion measure involves a continuous random tree whose vertices are assigned nonnegative labels, which correspond to distances from the boundary in our approach to the Brownian disk. We provide several applications of our construction. In particular, we prove that the uniform measure on the boundary can be obtained as the limit of the suitably normalized volume measure on a small tubular neighborhood of the boundary. We also prove that connected components of the complement of the Brownian net are Brownian disks, as it was suggested in the recent work of Miller and Sheffield. Finally, we show that connected components of the complement of balls centered at the distinguished point of the Brownian map are independent Brownian disks, conditionally on their volumes and perimeters.
</p>projecteuclid.org/euclid.aihp/1547802401_20190118040648Fri, 18 Jan 2019 04:06 ESTConditioning a Brownian loop-soup cluster on a portion of its boundaryhttps://projecteuclid.org/euclid.aihp/1547802402<strong>Wei Qian</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 1, 314--340.</p><p><strong>Abstract:</strong><br/>
We show that if one conditions a cluster in a Brownian loop-soup $L$ (of any intensity) in a two-dimensional domain by a portion $\partial $ of its outer boundary, then in the remaining domain, the union of all the loops of $L$ that touch $\partial $ satisfies the conformal restriction property while the other loops in $L$ form an independent loop-soup. This result holds when one discovers $\partial $ in a natural Markovian way, such as in the exploration procedures that have been defined in order to actually construct the Conformal Loop Ensembles as outer boundaries of loop-soup clusters. This result implies among other things that a phase transition occurs at $c=14/15$ for the connectedness of the loops that touch $\partial $.
Our results can be viewed as an extension of some of the results in our paper ( J. Eur. Math. Soc. (2019) to appear) in the following two directions: There, a loop-soup cluster was conditioned on its entire outer boundary while we discover here only part of this boundary. And, while it was explained in ( J. Eur. Math. Soc. (2019) to appear) that the strong decomposition using a Poisson point process of excursions that we derived there should be specific to the case of the critical loop-soup, we show here that in the subcritical cases, a weaker property involving the conformal restriction property nevertheless holds.
</p>projecteuclid.org/euclid.aihp/1547802402_20190118040648Fri, 18 Jan 2019 04:06 ESTIntertwinings and Stein’s magic factors for birth–death processeshttps://projecteuclid.org/euclid.aihp/1547802403<strong>Bertrand Cloez</strong>, <strong>Claire Delplancke</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 1, 341--377.</p><p><strong>Abstract:</strong><br/>
This article investigates second order intertwinings between semigroups of birth–death processes and discrete gradients on $\mathbb{N}$. It goes one step beyond a recent work of Chafaï and Joulin which establishes and applies to the analysis of birth–death semigroups a first order intertwining. Similarly to the first order relation, the second order intertwining involves birth–death and Feynman–Kac semigroups and weighted gradients on $\mathbb{N}$, and can be seen as a second derivative relation. As our main application, we provide new quantitative bounds on the Stein factors of discrete distributions. To illustrate the relevance of this approach, we also derive approximation results for the mixture of Poisson and geometric laws.
</p>projecteuclid.org/euclid.aihp/1547802403_20190118040648Fri, 18 Jan 2019 04:06 ESTMixing and decorrelation in infinite measure: The case of the periodic Sinai billiardhttps://projecteuclid.org/euclid.aihp/1547802404<strong>Françoise Pène</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 1, 378--411.</p><p><strong>Abstract:</strong><br/>
We investigate the question of the rate of mixing for observables of a $\mathbb{Z}^{d}$-extension of a probability preserving dynamical system with good spectral properties. We state general mixing results, including expansions of every order. The main motivation of this article is the study of mixing rates for smooth observables of the $\mathbb{Z}^{2}$-periodic Sinai billiard, with different kinds of results depending on whether the horizon is finite or infinite. We establish a first order mixing result when the horizon is infinite. In the finite horizon case, we establish an asymptotic expansion of every order, enabling the study of the mixing rate even for observables with null integrals. This result is related to an Edgeworth expansion in the local limit theorem.
</p>projecteuclid.org/euclid.aihp/1547802404_20190118040648Fri, 18 Jan 2019 04:06 ESTThe local limit of random sorting networkshttps://projecteuclid.org/euclid.aihp/1547802405<strong>Omer Angel</strong>, <strong>Duncan Dauvergne</strong>, <strong>Alexander E. Holroyd</strong>, <strong>Bálint Virág</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 1, 412--440.</p><p><strong>Abstract:</strong><br/>
A sorting network is a geodesic path from $12\cdots n$ to $n\cdots21$ in the Cayley graph of $S_{n}$ generated by adjacent transpositions. For a uniformly random sorting network, we establish the existence of a local limit of the process of space–time locations of transpositions in a neighbourhood of $an$ for $a\in [0,1]$ as $n\to \infty $. Here time is scaled by a factor of $1/n$ and space is not scaled.
The limit is a swap process $U$ on $\mathbb{Z}$. We show that $U$ is stationary and mixing with respect to the spatial shift and has time-stationary increments. Moreover, the only dependence on $a$ is through time scaling by a factor of $\sqrt{a(1-a)}$.
To establish the existence of $U$, we find a local limit for staircase-shaped Young tableaux. These Young tableaux are related to sorting networks through a bijection of Edelman and Greene.
</p>projecteuclid.org/euclid.aihp/1547802405_20190118040648Fri, 18 Jan 2019 04:06 ESTFinite rank perturbations in products of coupled random matrices: From one correlated to two Wishart ensembleshttps://projecteuclid.org/euclid.aihp/1547802406<strong>Gernot Akemann</strong>, <strong>Tomasz Checinski</strong>, <strong>Dang-Zheng Liu</strong>, <strong>Eugene Strahov</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 1, 441--479.</p><p><strong>Abstract:</strong><br/>
We compare finite rank perturbations of the following three ensembles of complex rectangular random matrices: First, a generalised Wishart ensemble with one random and two fixed correlation matrices introduced by Borodin and Péché, second, the product of two independent random matrices where one has correlated entries, and third, the case when the two random matrices become also coupled through a fixed matrix. The singular value statistics of all three ensembles is shown to be determinantal and we derive double contour integral representations for their respective kernels. Three different kernels are found in the limit of infinite matrix dimension at the origin of the spectrum. They depend on finite rank perturbations of the correlation and coupling matrices and are shown to be integrable. The first kernel (I) is found for two independent matrices from the second, and two weakly coupled matrices from the third ensemble. It generalises the Meijer $G$-kernel for two independent and uncorrelated matrices. The third kernel (III) is obtained for the generalised Wishart ensemble and for two strongly coupled matrices. It further generalises the perturbed Bessel kernel of Desrosiers and Forrester. Finally, kernel (II), found for the ensemble of two coupled matrices, provides an interpolation between the kernels (I) and (III), generalising previous findings of part of the authors.
</p>projecteuclid.org/euclid.aihp/1547802406_20190118040648Fri, 18 Jan 2019 04:06 ESTFunctional limit theorem for the self-intersection local time of the fractional Brownian motionhttps://projecteuclid.org/euclid.aihp/1547802407<strong>Arturo Jaramillo</strong>, <strong>David Nualart</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 1, 480--527.</p><p><strong>Abstract:</strong><br/>
Let $\{B_{t}\}_{t\geq0}$ be a $d$-dimensional fractional Brownian motion with Hurst parameter $0<H<1$, where $d\geq2$. Consider the approximation of the self-intersection local time of $B$, defined as
\[I_{T}^{\varepsilon}=\int_{0}^{T}\int_{0}^{t}p_{\varepsilon}(B_{t}-B_{s})\,ds\,dt,\] where $p_{\varepsilon}(x)$ is the heat kernel. We prove that the process $\{I_{T}^{\varepsilon}-\mathbb{E}[I_{T}^{\varepsilon}]\}_{T\geq0}$, rescaled by a suitable normalization, converges in law to a constant multiple of a standard Brownian motion for $\frac{3}{2d}<H\leq\frac{3}{4}$ and to a multiple of a sum of independent Hermite processes for $\frac{3}{4}<H<1$, in the space $C[0,\infty)$, endowed with the topology of uniform convergence on compacts.
</p>projecteuclid.org/euclid.aihp/1547802407_20190118040648Fri, 18 Jan 2019 04:06 ESTUniversality of Ghirlanda–Guerra identities and spin distributions in mixed $p$-spin modelshttps://projecteuclid.org/euclid.aihp/1547802408<strong>Yu-Ting Chen</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 1, 528--550.</p><p><strong>Abstract:</strong><br/>
We prove universality of the Ghirlanda–Guerra identities and spin distributions in the mixed $p$-spin models. The assumption for the universality of the identities requires exactly that the coupling constants have zero means and finite variances, and the result applies to the Sherrington–Kirkpatrick model. As an application, we obtain weakly convergent universality of spin distributions in the generic $p$-spin models under the condition of two matching moments. In particular, certain identities for 3-overlaps and 4-overlaps under the Gaussian disorder follow. Under the stronger mode of total variation convergence, we find that universality of spin distributions in the mixed $p$-spin models holds if mild dilution of connectivity by the Viana–Bray diluted spin glass Hamiltonians is present and the first three moments of coupling constants in the mixed $p$-spin Hamiltonians match. These universality results are in stark contrast to the characterization of spin distributions in the undiluted mixed $p$-spin models, which is known up to now that four matching moments are required in general.
</p>projecteuclid.org/euclid.aihp/1547802408_20190118040648Fri, 18 Jan 2019 04:06 ESTConvergence of the free Boltzmann quadrangulation with simple boundary to the Brownian diskhttps://projecteuclid.org/euclid.aihp/1547802409<strong>Ewain Gwynne</strong>, <strong>Jason Miller</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 1, 551--589.</p><p><strong>Abstract:</strong><br/>
We prove that the free Boltzmann quadrangulation with simple boundary and fixed perimeter, equipped with its graph metric, natural area measure, and the path which traces its boundary converges in the scaling limit to the free Boltzmann Brownian disk. The topology of convergence is the so-called Gromov–Hausdorff–Prokhorov-uniform (GHPU) topology, the natural analog of the Gromov–Hausdorff topology for curve-decorated metric measure spaces. From this we deduce that a random quadrangulation of the sphere decorated by a $2l$-step self-avoiding loop converges in law in the GHPU topology to the random curve-decorated metric measure space obtained by gluing together two independent Brownian disks along their boundaries.
</p>projecteuclid.org/euclid.aihp/1547802409_20190118040648Fri, 18 Jan 2019 04:06 ESTErgodicity of a system of interacting random walks with asymmetric interactionhttps://projecteuclid.org/euclid.aihp/1547802410<strong>Luisa Andreis</strong>, <strong>Amine Asselah</strong>, <strong>Paolo Dai Pra</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 1, 590--606.</p><p><strong>Abstract:</strong><br/>
We study $N$ interacting random walks on the positive integers. Each particle has drift $\delta$ towards infinity, a reflection at the origin, and a drift towards particles with lower positions. This inhomogeneous mean field system is shown to be ergodic only when the interaction is strong enough. We focus on this latter regime, and point out the effect of piles of particles, a phenomenon absent in models of interacting diffusion in continuous space.
</p>projecteuclid.org/euclid.aihp/1547802410_20190118040648Fri, 18 Jan 2019 04:06 EST