Annales de l'Institut Henri Poincaré, Probabilités et Statistiques Articles (Project 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 EDTDeep factorisation of the stable process II: Potentials and applicationshttps://projecteuclid.org/euclid.aihp/1519030831<strong>Andreas E. Kyprianou</strong>, <strong>Victor Rivero</strong>, <strong>Batı Şengül</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 1, 343--362.</p><p><strong>Abstract:</strong><br/> Here, we propose a different perspective of the deep factorisation in ( Electron. J. Probab. 21 (2016) Paper No. 23, 28) based on determining potentials. Indeed, we factorise the inverse of the MAP-exponent associated to a stable process via the Lamperti–Kiu transform. Here our factorisation is completely independent from the derivation in ( Electron. J. Probab. 21 (2016) Paper No. 23, 28), moreover there is no clear way to invert the factors in ( Electron. J. Probab. 21 (2016) Paper No. 23, 28) to derive our results. Our method gives direct access to the potential densities of the ascending and descending ladder MAP of the Lamperti-stable MAP in closed form. In the spirit of the interplay between the classical Wiener–Hopf factorisation and the fluctuation theory of the underlying Lévy process, our analysis will produce a collection of new results for stable processes. We give an identity for the law of the point of closest reach to the origin for a stable process with index $\alpha\in(0,1)$ as well as an identity for the the law of the point of furthest reach before absorption at the origin for a stable process with index $\alpha\in(1,2)$. Moreover, we show how the deep factorisation allows us to compute explicitly the limiting distribution of stable processes multiplicatively reflected in such a way that it remains in the strip $[-1,1]$. </p>projecteuclid.org/euclid.aihp/1519030831_20180219040043Mon, 19 Feb 2018 04:00 ESTQuenched invariance principle for random walk in time-dependent balanced random environmenthttps://projecteuclid.org/euclid.aihp/1519030832<strong>Jean-Dominique Deuschel</strong>, <strong>Xiaoqin Guo</strong>, <strong>Alejandro F. Ramírez</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 1, 363--384.</p><p><strong>Abstract:</strong><br/>
We prove a quenched central limit theorem for balanced random walks in time-dependent ergodic random environments which is not necessarily nearest-neighbor. We assume that the environment satisfies appropriate ergodicity and ellipticity conditions. The proof is based on the use of a maximum principle for parabolic difference operators.
</p>projecteuclid.org/euclid.aihp/1519030832_20180219040043Mon, 19 Feb 2018 04:00 ESTStein’s method for positively associated random variables with applications to the Ising and voter models, bond percolation, and contact processhttps://projecteuclid.org/euclid.aihp/1519030833<strong>Larry Goldstein</strong>, <strong>Nathakhun Wiroonsri</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 1, 385--421.</p><p><strong>Abstract:</strong><br/> We provide nonasymptotic $L^{1}$ bounds to the normal for four well-known models in statistical physics and particle systems in $\mathbb{Z}^{d}$; the ferromagnetic nearest-neighbor Ising model, the supercritical bond percolation model, the voter model and the contact process. In the Ising model, we obtain an $L^{1}$ distance bound between the total magnetization and the normal distribution at any temperature when the magnetic moment parameter is nonzero, and when the inverse temperature is below critical and the magnetic moment parameter is zero. In the percolation model we obtain such a bound for the total number of points in a finite region belonging to an infinite cluster in dimensions $d\ge2$, in the voter model for the occupation time of the origin in dimensions $d\ge7$, and for finite time integrals of nonconstant increasing cylindrical functions evaluated on the one dimensional supercritical contact process started in its unique invariant distribution. The tool developed for these purposes is a version of Stein’s method adapted to positively associated random variables. In one dimension, letting $\mathbf{{\xi}=(\xi_{1},\ldots,\xi_{m})}$ be a positively associated mean zero random vector with components that obey the bound $\vert \xi_{i}\vert \le B,i=1,\ldots,m$, and whose sum $W=\sum_{i=1}^{m}\xi_{i}$ has variance 1, it holds that \begin{eqnarray*}d_{1}(\mathcal{L}(W),\mathcal{L}(Z))\le5B+\sqrt{\frac{8}{\pi}}\sum_{i\neq j}\mathbb{E}[\xi_{i}\xi_{j}],\end{eqnarray*} where $Z$ has the standard normal distribution and $d_{1}(\cdot,\cdot)$ is the $L^{1}$ metric. Our methods apply in the multidimensional case with the $L^{1}$ metric replaced by a smooth function metric. </p>projecteuclid.org/euclid.aihp/1519030833_20180219040043Mon, 19 Feb 2018 04:00 ESTA new computation of the critical point for the planar random-cluster model with $q\ge1$https://projecteuclid.org/euclid.aihp/1519030834<strong>Hugo Duminil-Copin</strong>, <strong>Aran Raoufi</strong>, <strong>Vincent Tassion</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 1, 422--436.</p><p><strong>Abstract:</strong><br/>
We present a new computation of the critical value of the random-cluster model with cluster weight $q\ge1$ on $\mathbb{Z}^{2}$. This provides an alternative approach to the result in ( Probab. Theory Related Fields 153 (2012) 511–542). We believe that this approach has several advantages. First, most of the proof can easily be extended to other planar graphs with sufficient symmetries. Furthermore, it invokes RSW-type arguments which are not based on self-duality. And finally, it contains a new way of applying sharp threshold results which avoid the use of symmetric events and periodic boundary conditions. Some of the new methods presented in this paper have a larger scope than the planar random-cluster model, and may be useful to investigate sharp threshold phenomena for more general dependent percolation processes in arbitrary dimensions.
</p>projecteuclid.org/euclid.aihp/1519030834_20180219040043Mon, 19 Feb 2018 04:00 ESTJoint exceedances of random productshttps://projecteuclid.org/euclid.aihp/1519030835<strong>Anja Janßen</strong>, <strong>Holger Drees</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 1, 437--465.</p><p><strong>Abstract:</strong><br/>
We analyze the joint extremal behavior of $n$ random products of the form $\prod_{j=1}^{m}X_{j}^{a_{ij}}$, $1\leq i\leq n$, for non-negative, independent regularly varying random variables $X_{1},\ldots,X_{m}$ and general coefficients $a_{ij}\in\mathbb{R}$. Products of this form appear for example if one observes a linear time series with gamma type innovations at $n$ points in time. We combine arguments of linear optimization and a generalized concept of regular variation on cones to show that the asymptotic behavior of joint exceedance probabilities of these products is determined by the solution of a linear program related to the matrix $\mathbf{A}=(a_{ij})$.
</p>projecteuclid.org/euclid.aihp/1519030835_20180219040043Mon, 19 Feb 2018 04:00 ESTRange and critical generations of a random walk on Galton–Watson treeshttps://projecteuclid.org/euclid.aihp/1519030836<strong>Pierre Andreoletti</strong>, <strong>Xinxin Chen</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 1, 466--513.</p><p><strong>Abstract:</strong><br/>
In this paper we consider a random walk in random environment on a tree and focus on the boundary case for the underlying branching potential. We study the range $R_{n}$ of this walk up to time $n$ and obtain its correct asymptotic in probability which is of order $n/\log n$. This result is a consequence of the asymptotical behavior of the number of visited sites at generations of order $(\log n)^{2}$, which turn out to be the most visited generations. Our proof which involves a quenched analysis gives a description of the typical environments responsible for the behavior of $R_{n}$.
</p>projecteuclid.org/euclid.aihp/1519030836_20180219040043Mon, 19 Feb 2018 04:00 ESTProduct blocking measures and a particle system proof of the Jacobi triple producthttps://projecteuclid.org/euclid.aihp/1519030837<strong>Márton Balázs</strong>, <strong>Ross Bowen</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 1, 514--528.</p><p><strong>Abstract:</strong><br/>
We review product form blocking measures in the general framework of nearest neighbor asymmetric one dimensional misanthrope processes. This class includes exclusion, zero range, bricklayers, and many other models. We characterize the cases when such measures exist in infinite volume, and when finite boundaries need to be added. By looking at inter-particle distances, we extend the construction to some 0-1 valued particle systems e.g., $q$-ASEP and the Katz-Lebowitz-Spohn process, even outside the misanthrope class. Along the way we provide a full ergodic decomposition of the product blocking measure into components that are characterized by a non-trivial conserved quantity. Substituting in simple exclusion and zero range has an interesting consequence: a purely probabilistic proof of the Jacobi triple product, a famous identity that mostly occurs in number theory and the combinatorics of partitions. Surprisingly, here it follows very naturally from the exclusion – zero range correspondence.
</p>projecteuclid.org/euclid.aihp/1519030837_20180219040043Mon, 19 Feb 2018 04:00 ESTLimit theorems for affine Markov walks conditioned to stay positivehttps://projecteuclid.org/euclid.aihp/1519030838<strong>Ion Grama</strong>, <strong>Ronan Lauvergnat</strong>, <strong>Émile Le Page</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 1, 529--568.</p><p><strong>Abstract:</strong><br/>
Consider the real Markov walk $S_{n}=X_{1}+\cdots+X_{n}$ with increments $(X_{n})_{n\geq 1}$ defined by a stochastic recursion starting at $X_{0}=x$. For a starting point $y>0$, denote by $\tau_{y}$ the exit time of the process $(y+S_{n})_{n\geq 1}$ from the positive part of the real line. We investigate the asymptotic behaviour of the probability of the event $\tau_{y}\geq n$ and of the conditional law of $y+S_{n}$ given $\tau_{y}\geq n$ as $n\to+\infty$.
</p>projecteuclid.org/euclid.aihp/1519030838_20180219040043Mon, 19 Feb 2018 04:00 ESTSpeed of convergence in first passage percolation and geodesicity of the average distancehttps://projecteuclid.org/euclid.aihp/1524643222<strong>Romain Tessera</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 2, 569--586.</p><p><strong>Abstract:</strong><br/>
We give an elementary proof that Talagrand’s sub-Gaussian concentration inequality implies a limit shape theorem for first passage percolation on any Cayley graph of $\mathbb{Z}^{d}$, with a speed of convergence $\lesssim (\frac{\log n}{n})^{1/2}$. Our approach, which does not use the subadditive theorem, is based on proving that the average distance $\mathbb{E}d_{\omega}$ on $\mathbb{Z}^{d}$ is close to being geodesic. Our key observation, of independent interest, is that the problem of estimating the rate of convergence for the average distance is equivalent (in a precise sense) to estimating its “level of geodesicity”.
</p>projecteuclid.org/euclid.aihp/1524643222_20180425040043Wed, 25 Apr 2018 04:00 EDTEstimating the extremal index through local dependencehttps://projecteuclid.org/euclid.aihp/1524643223<strong>Helena Ferreira</strong>, <strong>Marta Ferreira</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 2, 587--605.</p><p><strong>Abstract:</strong><br/>
The extremal index is an important parameter in the characterization of extreme values of a stationary sequence. Our new estimation approach for this parameter is based on the extremal behavior under the local dependence condition D$^{(k)}$($u_{n}$). We compare a process satisfying one of this hierarchy of increasingly weaker local mixing conditions with a process of cycles satisfying the D$^{(2)}$($u_{n}$) condition. We also analyze local dependence within moving maxima processes and derive a necessary and sufficient condition for D$^{(k)}$($u_{n}$). In order to evaluate the performance of the proposed estimators, we apply an empirical diagnostic for local dependence conditions, we conduct a simulation study and compare with existing methods. An application to a financial time series is also presented.
</p>projecteuclid.org/euclid.aihp/1524643223_20180425040043Wed, 25 Apr 2018 04:00 EDTPerturbation by non-local operatorshttps://projecteuclid.org/euclid.aihp/1524643224<strong>Zhen-Qing Chen</strong>, <strong>Jie-Ming Wang</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 2, 606--639.</p><p><strong>Abstract:</strong><br/>
Suppose that $d\ge1$ and $0<\beta<\alpha<2$. We establish the existence and uniqueness of the fundamental solution $q^{b}(t,x,y)$ to a class of (typically non-symmetric) non-local operators $\mathcal{L}^{b}=\Delta^{\alpha/2}+\mathcal{S}^{b}$, where
\[\mathcal{S}^{b}f(x):=\mathcal{A}(d,-\beta)\int_{\mathbb{R}^{d}}(f(x+z)-f(x)-\nabla f(x)\cdot z\mathbb{1}_{\{|z|\leq1\}})\frac{b(x,z)}{|z|^{d+\beta}}\,dz\] and $b(x,z)$ is a bounded measurable function on $\mathbb{R}^{d}\times\mathbb{R}^{d}$ with $b(x,z)=b(x,-z)$ for $x,z\in\mathbb{R}^{d}$. Here $\mathcal{A}(d,-\beta)$ is a normalizing constant so that $\mathcal{S}^{b}=\Delta^{\beta/2}$ when $b(x,z)\equiv1$. We show that if $b(x,z)\geq-\frac{\mathcal{A}(d,-\alpha)}{\mathcal{A}(d,-\beta)}|z|^{\beta-\alpha}$, then $q^{b}(t,x,y)$ is a strictly positive continuous function and it uniquely determines a conservative Feller process $X^{b}$, which has strong Feller property. The Feller process $X^{b}$ is the unique solution to the martingale problem of $(\mathcal{L}^{b},\mathcal{S}(\mathbb{R}^{d}))$, where $\mathcal{S}(\mathbb{R}^{d})$ denotes the space of tempered functions on $\mathbb{R}^{d}$. Furthermore, sharp two-sided estimates on $q^{b}(t,x,y)$ are derived. In stark contrast with the gradient perturbations, these estimates exhibit different behaviors for different types of $b(x,z)$. The model considered in this paper contains the following as a special case. Let $Y$ and $Z$ be (rotationally) symmetric $\alpha$-stable process and symmetric $\beta$-stable processes on $\mathbb{R}^{d}$, respectively, that are independent to each other. Solution to stochastic differential equations $dX_{t}=dY_{t}+c(X_{t-})\,dZ_{t}$ has infinitesimal generator $\mathcal{L}^{b}$ with $b(x,z)=|c(x)|^{\beta}$.
</p>projecteuclid.org/euclid.aihp/1524643224_20180425040043Wed, 25 Apr 2018 04:00 EDTKPZ and Airy limits of Hall–Littlewood random plane partitionshttps://projecteuclid.org/euclid.aihp/1524643225<strong>Evgeni Dimitrov</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 2, 640--693.</p><p><strong>Abstract:</strong><br/>
In this paper we consider a probability distribution $\mathbb{P}^{q,t}_{\mathrm{HL}}$ on plane partitions, which arises as a one-parameter generalization of the standard $q^{\mathrm{volume}}$ measure. This generalization is closely related to the classical multivariate Hall–Littlewood polynomials, and it was first introduced by Vuletić in ( Trans. Am. Math. Soc. 361 (2009) 2789–2804).
We prove that as the plane partitions become large ($q$ goes to $1$, while the Hall–Littlewood parameter $t$ is fixed), the scaled bottom slice of the random plane partition converges to a deterministic limit shape, and that one-point fluctuations around the limit shape are asymptotically given by the GUE Tracy–Widom distribution. On the other hand, if $t$ simultaneously converges to its own critical value of $1$, the fluctuations instead converge to the one-dimensional Kardar–Parisi–Zhang (KPZ) equation with the so-called narrow wedge initial data.
The algebraic part of our arguments is closely related to the formalism of Macdonald processes ( Probab. Theory Relat. Fields 158 (1) (2014) 225–400). The analytic part consists of detailed asymptotic analysis of the arising Fredholm determinants.
</p>projecteuclid.org/euclid.aihp/1524643225_20180425040043Wed, 25 Apr 2018 04:00 EDTDavie’s type uniqueness for a class of SDEs with jumpshttps://projecteuclid.org/euclid.aihp/1524643226<strong>Enrico Priola</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 2, 694--725.</p><p><strong>Abstract:</strong><br/>
A result of A.M. Davie (Int. Math. Res. Not. 24 (2007) rnm124) states that a multidimensional stochastic equation $dX_{t}=b(t,X_{t})\,dt+dW_{t}$, $X_{0}=x$, driven by a Wiener process $W=(W_{t})$ with a coefficient $b$ which is only bounded and measurable has a unique solution for almost all choices of the driving Wiener path. We consider a similar problem when $W$ is replaced by a Lévy process $L=(L_{t})$ and $b$ is $\beta$-Hölder continuous in the space variable, $\beta\in(0,1)$. We assume that $L_{1}$ has a finite moment of order $\theta$, for some ${\theta}>0$. Using a new càdlàg regularity result for strong solutions, we prove that strong existence and uniqueness for the SDE together with $L^{p}$-Lipschitz continuity of the strong solution with respect to $x$ imply a Davie’s type uniqueness result for almost all choices of the Lévy path. We apply this result to a class of SDEs driven by non-degenerate $\alpha$-stable Lévy processes, $\alpha\in(0,2)$ and $\beta>1-\alpha/2$.
</p>projecteuclid.org/euclid.aihp/1524643226_20180425040043Wed, 25 Apr 2018 04:00 EDTStochastic integral equations for Walsh semimartingaleshttps://projecteuclid.org/euclid.aihp/1524643227<strong>Tomoyuki Ichiba</strong>, <strong>Ioannis Karatzas</strong>, <strong>Vilmos Prokaj</strong>, <strong>Minghan Yan</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 2, 726--756.</p><p><strong>Abstract:</strong><br/>
We construct planar semimartingales that include the Walsh Brownian motion as a special case, and derive Harrison–Shepp-type equations and a change-of-variable formula in the spirit of Freidlin–Sheu for these so-called “Walsh semimartingales”. We examine the solvability of the resulting system of stochastic integral equations. In appropriate Markovian settings we study two types of connections to martingale problems, questions of uniqueness in distribution for such processes, and a few examples.
</p>projecteuclid.org/euclid.aihp/1524643227_20180425040043Wed, 25 Apr 2018 04:00 EDTStochastic Ising model with flipping sets of spins and fast decreasing temperaturehttps://projecteuclid.org/euclid.aihp/1524643228<strong>Roy Cerqueti</strong>, <strong>Emilio De Santis</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 2, 757--789.</p><p><strong>Abstract:</strong><br/>
This paper deals with the stochastic Ising model with a temperature shrinking to zero as time goes to infinity. A generalization of the Glauber dynamics is considered, on the basis of the existence of simultaneous flips of some spins. Such dynamics act on a wide class of graphs which are periodic and embedded in $\mathbb{R}^{d}$. The interactions between couples of spins are assumed to be quenched i.i.d. random variables following a Bernoulli distribution with support $\{-1,+1\}$. The specific problem here analyzed concerns the assessment of how often (finitely or infinitely many times, almost surely) a given spin flips. Adopting the classification proposed in ( Comm. Math. Phys. 214 (2002) 373–387), we present conditions in order to have models of type $\mathcal{F}$ (any spin flips finitely many times), $\mathcal{I}$ (any spin flips infinitely many times) and $\mathcal{M}$ (a mixed case). Several examples are provided in all dimensions and for different cases of graphs. The most part of the obtained results holds true for the case of zero-temperature and some of them for the cubic lattice $\mathbb{L}_{d}=(\mathbb{Z}^{d},\mathbb{E}_{d})$ as well.
</p>projecteuclid.org/euclid.aihp/1524643228_20180425040043Wed, 25 Apr 2018 04:00 EDTMonotonicity and condensation in homogeneous stochastic particle systemshttps://projecteuclid.org/euclid.aihp/1524643229<strong>Thomas Rafferty</strong>, <strong>Paul Chleboun</strong>, <strong>Stefan Grosskinsky</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 2, 790--818.</p><p><strong>Abstract:</strong><br/>
We study stochastic particle systems that conserve the particle density and exhibit a condensation transition due to particle interactions. We restrict our analysis to spatially homogeneous systems on fixed finite lattices with stationary product measures, which includes previously studied zero-range or misanthrope processes. All known examples of such condensing processes are non-monotone, i.e. the dynamics do not preserve a partial ordering of the state space and the canonical measures (with a fixed number of particles) are not monotonically ordered. For our main result we prove that condensing homogeneous particle systems with finite critical density are necessarily non-monotone. On fixed finite lattices condensation can occur even when the critical density is infinite, in this case we give an example of a condensing process that numerical evidence suggests is monotone, and give a partial proof of its monotonicity.
</p>projecteuclid.org/euclid.aihp/1524643229_20180425040043Wed, 25 Apr 2018 04:00 EDTDegrees of freedom for piecewise Lipschitz estimatorshttps://projecteuclid.org/euclid.aihp/1524643230<strong>Frederik Riis Mikkelsen</strong>, <strong>Niels Richard Hansen</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 2, 819--841.</p><p><strong>Abstract:</strong><br/>
A representation of the degrees of freedom akin to Stein’s lemma is given for a class of estimators of a mean value parameter in $\mathbb{R}^{n}$. Contrary to previous results our representation holds for a range of discontinues estimators. It shows that even though the discontinuities form a Lebesgue null set, they cannot be ignored when computing degrees of freedom. Estimators with discontinuities arise naturally in regression if data driven variable selection is used. Two such examples, namely best subset selection and lasso-OLS, are considered in detail in this paper. For lasso-OLS the general representation leads to an estimate of the degrees of freedom based on the lasso solution path, which in turn can be used for estimating the risk of lasso-OLS. A similar estimate is proposed for best subset selection. The usefulness of the risk estimates for selecting the number of variables is demonstrated via simulations with a particular focus on lasso-OLS.
</p>projecteuclid.org/euclid.aihp/1524643230_20180425040043Wed, 25 Apr 2018 04:00 EDTMultidimensional two-component Gaussian mixtures detectionhttps://projecteuclid.org/euclid.aihp/1524643231<strong>Béatrice Laurent</strong>, <strong>Clément Marteau</strong>, <strong>Cathy Maugis-Rabusseau</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 2, 842--865.</p><p><strong>Abstract:</strong><br/>
Let $(X_{1},\ldots,X_{n})$ be a $d$-dimensional i.i.d. sample from a distribution with density $f$. The problem of detection of a two-component mixture is considered. Our aim is to decide whether $f$ is the density of a standard Gaussian random $d$-vector ($f=\phi_{d}$) against $f$ is a two-component mixture: $f=(1-\varepsilon)\phi_{d}+\varepsilon\phi_{d}(\cdot -\mu)$ where $(\varepsilon,\mu)$ are unknown parameters. Optimal separation conditions on $\varepsilon$, $\mu$, $n$ and the dimension $d$ are established, allowing to separate both hypotheses with prescribed errors. Several testing procedures are proposed and two alternative subsets are considered.
</p>projecteuclid.org/euclid.aihp/1524643231_20180425040043Wed, 25 Apr 2018 04:00 EDTSharpness of the phase transition and lower bounds for the critical intensity in continuum percolation on $\mathbb{R}^{d}$https://projecteuclid.org/euclid.aihp/1524643232<strong>Sebastian Ziesche</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 2, 866--878.</p><p><strong>Abstract:</strong><br/>
We consider the Boolean model $Z$ on $\mathbb{R}^{d}$ with random compact grains of bounded diameter, i.e. $Z:=\bigcup_{i\in\mathbb{N}}(Z_{i}+X_{i})$ where $\{X_{1},X_{2},\dots\}$ is a Poisson point process of intensity $t$ and $(Z_{1},Z_{2},\dots)$ is an i.i.d. sequence of compact grains (not necessarily balls) with diameters a.s. bounded by some constant. We will show that exponential decay holds in the sub-critical regime, that means the volume and radius of the cluster of the typical grain in $Z$ have an exponential tail. To achieve this we adapt the arguments of (A new proof of the sharpness of the phase transition for Bernoulli percolation on $\mathbb{Z}^{d}$ (2015) Preprint) and apply a new construction of the cluster of the typical grain together with arguments related to branching processes.
In the second part of the paper, we obtain new lower bounds for the Boolean model with deterministic grains. Some of these bounds are rigorous, while others are obtained via simulation. The simulated bounds come with confidence intervals and are much more precise than the rigorous ones. They improve known results ( J. Chem. Phys. 137 (2012) 074106) in dimension six and above.
</p>projecteuclid.org/euclid.aihp/1524643232_20180425040043Wed, 25 Apr 2018 04:00 EDTDistributional limits of positive, ergodic stationary processes and infinite ergodic transformationshttps://projecteuclid.org/euclid.aihp/1524643233<strong>Jon Aaronson</strong>, <strong>Benjamin Weiss</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 2, 879--906.</p><p><strong>Abstract:</strong><br/>
In this note we identify the distributional limits of non-negative, ergodic stationary processes, showing that all are possible. Consequences for infinite ergodic theory are also explored and new examples of distributionally stable – and $\alpha$-rationally ergodic – transformations are presented.
</p>projecteuclid.org/euclid.aihp/1524643233_20180425040043Wed, 25 Apr 2018 04:00 EDTEta-diagonal distributions and infinite divisibility for R-diagonalshttps://projecteuclid.org/euclid.aihp/1524643234<strong>Hari Bercovici</strong>, <strong>Alexandru Nica</strong>, <strong>Michael Noyes</strong>, <strong>Kamil Szpojankowski</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 2, 907--937.</p><p><strong>Abstract:</strong><br/>
The class of $R$-diagonal $*$-distributions is fairly well understood in free probability. In this class, we consider the concept of infinite divisibility with respect to the operation $\boxplus$ of free additive convolution. We exploit the relation between free probability and the parallel (and simpler) world of Boolean probability. It is natural to introduce the concept of an $\eta$-diagonal distribution that is the Boolean counterpart of an $R$-diagonal distribution. We establish a number of properties of $\eta$-diagonal distributions, then we examine the canonical bijection relating $\eta$-diagonal distributions to infinitely divisible $R$-diagonal ones. The overall result is a parametrization of an arbitrary $\boxplus$-infinitely divisible $R$-diagonal distribution that can arise in a $C^{*}$-probability space by a pair of compactly supported Borel probability measures on $[0,\infty)$. Among the applications of this parametrization, we prove that the set of $\boxplus$-infinitely divisible $R$-diagonal distributions is closed under the operation $\boxtimes$ of free multiplicative convolution.
</p>projecteuclid.org/euclid.aihp/1524643234_20180425040043Wed, 25 Apr 2018 04:00 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 EDT