Pierre Andreoletti
We consider Sinai's random walk in random environment. We prove that for an interval of time [1,n] Sinai's walk sojourns in a small neighborhood of the point of localization for the quasi totality of this amount of time. Moreover the local time at the point of localization normalized by $n$ converges in probability to a well defined random variable of the environment.
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We give an alternative proof of the localization of Sinai's random walk in random environment under weaker hypothesis than the ones used by Sinai. Moreover we give estimates that are stronger than the one of Sinai on the localization neighborhood and on the probability for the random walk to stay inside this neighborhood.
We consider Sinai's walk in i.i.d. random scenery and focus our attention on a conjecture of Révész \cite{r05} concerning the upper limits of Sinai's walk in random scenery when the scenery is bounded from above. A close study of the competition between the concentration property for Sinai's walk and negative values for the scenery enables us to prove that the conjecture is true if the scenery has "thin" negative tails and is false otherwise.
We consider Sinai's random walk in random environment. We prove that the logarithm of the local time is a good estimator of the random potential associated to the random environment. We give a constructive method allowing us to built the random environment from a single trajectory of the random walk.
Using the generators, we establish a connection between the Sinai's random walk and the so-called Brox process. We first find the Dirichlet form of the Brox diffusion, and then prove that it is the limit of the Dirichlet form of the Sinai's random walk. This also gives a natural way to connect between the Brox diffusion and the Brownian motion.
We consider Sinai's random walk in random environment. We prove that infinitely often (i.o.) the size of the concentration neighborhood of this random walk is almost surely bounded. As an application we get that i.o. the maximal distance between two favorite sites is almost surely bounded.
We identify the limit of the internal DLA cluster generated by Sinai's walk as the law of a functional of a Brownian motion which turns out to be a new interpretation of the Arcsine law.
We prove that the local times of a sequence of Sinai's random walks convergence to those of Brox's diffusion by proper scaling, which is accord with the result of Seignourel (2000). Our proof is based on the convergence of the branching processes in random environment by Kurtz (1979).
Sinai's random walk in random environment shows interesting patterns on the exponential time scale. We characterize the patterns that appear on infinitely many time scales after appropriate rescaling (a functional law of iterated logarithm). The curious rate function captures the difference between one-sided and two-sided behavior.
We are interested in Sinai's walk $(S\_n)\_{n\in\mathbb{N}}$. We prove that the annealed probability that $\sum\_{k=0}^n f(S\_k)$ is strictly positive for all $n\in[1,N]$ is equal to $1/(\log N)^{\frac{3-\sqrt{5}}{2}+o(1)}$, for a large class of functions $f$, and in particular for $f(x)=x$. The persistence exponent $\frac{3-\sqrt{5}}{2}$ first appears in a non-rigorous paper of Le Doussal, Monthus and Fischer, with motivations coming from physics. The proof relies on techniques of localization for Sinai's walk and uses results of Cheliotis about the sign changes of the bottom of valleys of a …
Annealed local limit theorem for Sinai's random walk in random environment
本文证明了Sinai随机环境中随机游走在退火律下的局部极限定理，并通过路径分解、更新理论及细致分析给出退火概率的精确渐近等价。
Sinai's walk is a recurrent one-dimensional nearest-neighbor random walk in random environment. It is known for a phenomenon of strong localization, namely, the walk spends almost all time at or near the bottom of deep valleys of the potential. Our main result shows a weakness of this localization phenomenon: with probability one, the zones where the walk stays for the most time can be far away from the sites where the walk spends the most time. In particular, this gives a negative answer to a problem of Erdős and Révész [Mathematical Structures--Computational Mathematics--Mathematical Modelli…
We study in this short note aging properties of Sinai's (nearest neighbour) random walk in random environment. With $\PP^o$ denoting the annealed law of the RWRE $X_n$, our main result is a full proof of the following statement due to P. Le Doussal, C. Monthus and D. S. Fisher: $$\lim_{η\to0} \lim_{n\to\infty} \PP^o (\frac{|X_{n^h} - X_n|}{(\log n)^2} < η) = \frac{1}{h^2} [ {5/3} - {2/3} e^{-(h-1)} ]. $$
For Sinaï's walk (X_k) we show that the empirical measure of the environment seen from the particle (\bar\w_k) converges in law to some random measure S. This limit measure is explicitly given in terms of the infinite valley, which construction goes back to Golosov. As a consequence an "in law" ergodic theorem holds for additive functionals of (\bar\w_k) . When the limit in this "in law" ergodic theorem is deterministic, it holds in probability. This allows some extensions to the recurrent case of the ballistic "environment's method" dating back to Kozlov and Molchanov. In particular, we show …
We prove that the upward ladder height subordinator $H$ associated to a real valued Lévy process $ξ$ has Laplace exponent $ϕ$ that varies regularly at $\infty$ (resp. at 0) if and only if the underlying Lévy process $ξ$ satisfies Sinai's condition at 0 (resp. at $\infty$). Sinai's condition for real valued Lévy processes is the continuous time analogue of Sinai's condition for random walks. We provide several criteria in terms of the characteristics of $ξ$ to determine whether or not it satisfies Sinai's condition. Some of these criteria are deduced from tail estimates of the Lévy measure of $…
A coupling between random walks in random environments and Brox's diffusion
本文通过构造 Sinai 随机游走与 Brox 扩散之间的特定耦合，并利用粗糙路径框架下鞅问题的收敛结果，量化了 Sinai 随机游走重标度后向 Brox 扩散的分布收敛。
Sinai's walk can be thought of as a random walk on $\mathbb {Z}$ with random potential $V$, with $V$ weakly converging under diffusive rescaling to a two-sided Brownian motion. We consider here the generator $\mathbb {L}_N$ of Sinai's walk on $[-N,N]\cap \mathbb {Z}$ with Dirichlet conditions on $-N,N$. By means of potential theory, for each $h>0$, we show the relation between the spectral properties of $\mathbb {L}_N$ for eigenvalues of order $o(\exp(-h\sqrt{N}))$ and the distribution of the $h$-extrema of the rescaled potential $V_N(x)\equiv V(Nx)/\sqrt{N}$ defined on $[-1,1]$. Information a…
Sinaĭ excursions: An analogue of Sparre Andersen's formula for the area process of a random walk
本文研究随机游走桥面积过程为游程的精确渐近概率，建立 Sparre Andersen 公式的类比，并借助 von Sterneck 子集计数公式改进已有界，回应润湿模型中的相关问题。
We apply the techniques developed in Comets and Popov (2003) to present a new proof to Sinai's theorem (Sinai, 1982) on one-dimensional random walk in random environment (RWRE), working in a scale-free way to avoid rescaling arguments and splitting the proof in two independent parts: a quenched one, related to the measure $P_ω$ conditioned on a fixed, typical realization $ω$ of the environment, and an annealed one, related to the product measure $\mathbb{P}$ of the environment $ω$. The quenched part still holds even if we use another measure (possibly dependent) for the environment.
We clarify the asymptotic of the limsup of the size of the neighborhood of concentration of Sinai's walk improving the result in \cite{Pierre3}. Also we get the almost sure limit of the number of points visited more than a small but fixed proportion of a given amount of time.
Dynamics for exclusion processes with random environments of particle jump rates
本文研究随机环境下排斥粒子系统的云分解，揭示Sinai机制下深势阱导致粒子堵塞，形成速度指数递减的稳定云，并给出分布极限。
In this article, a localisation result is proved for the biased random walk on the range of a simple random walk in high dimensions (d \geq 5). This demonstrates that, unlike in the supercritical percolation setting, a slowdown effect occurs as soon a non-trivial bias is introduced. The proof applies a decomposition of the underlying simple random walk path at its cut-times to relate the associated biased random walk to a one-dimensional random walk in a random environment in Sinai's regime.
We prove an annealed weak limit of the trajectory of the random walks in cooling random environment (RWCRE) under both slow (polynomial) and fast (exponential) cooling. We identify the weak limit when the underlying static environment is recurrent (Sinai's model). Avena and den Hollander have previously proved law of large numbers and Gaussian fluctuation of RWCRE. We find that the weak limit of the trajectory exists as a time-rescaled Brownian motion in the slow cooling case but the limit degenerates to a constant function in the fast cooling case.
We study the random walk in random environment on {0,1,2,...}, where the environment is subject to a vanishing (random) perturbation. The two particular cases we consider are: (i) random walk in random environment perturbed from Sinai's regime; (ii) simple random walk with random perturbation. We give almost sure results on how far the random walker will be from the origin after a long time t, for almost every environment. We give both upper and lower almost sure bounds. These bounds are of order $(\log t)^β$, for $β\in (1,\infty)$, depending on the perturbation. In addition, in the ergodic ca…
We consider d independent walkers on Z, m of them performing simple symmetric random walk and r = d -- m of them performing recurrent RWRE (Sinai walk), in I independent random environments. We show that the product is recurrent, almost surely, if and only if m $\le$ 1 or m = d = 2. In the transient case with r $\ge$ 1, we prove that the walkers meet infinitely often, almost surely, if and only if m = 2 and r $\ge$ I = 1. In particular, while I does not have an influence for the recurrence or transience, it does play a role for the probability to have infinitely many meetings. To obtain these …
We consider, in the continuous time version, $γ$ independent random walks on $\mathbb{Z_+}$ in random environment in the Sinai's regime. Let $T_\gam$ be the first meeting time of one pair of the $γ$ random walks starting at different positions. We first show that the tail of the quenched distribution of $T_γ$, after a suitable rescaling, converges in probability, to some functional of the Brownian motion. Then we compute the law of this functional. Eventually, we obtain results about the moments of this meeting time. Being $\Eo$ the quenched expectation, we show that, for almost all environmen…
We investigate the hydrodynamical behavior of a system of random walks with zero-range interactions moving in a common `Sinai-type' random environment on a one dimensional torus. The hydrodynamic equation found is a quasilinear SPDE with a `rough' random drift term coming from a scaling of the random environment and a homogenization of the particle interaction. Part of the motivation for this work is to understand how the space-time limit of the particle mass relates to that of the known single particle Brox diffusion limit. In this respect, given the hydrodynamic limit shown, we describe form…
Localization induced by Sinai disorder in mass-exchange models
该研究引入带淬火无序的保守连续质量交换模型，通过生灭链求解其平稳质量分布，揭示β√N参数对分布类型的区分，表征涨落与局域相关定律，还分析了极值质量的标度特性。
We introduce random walks in a sparse random environment on $\mathbb Z$ and investigate basic asymptotic properties of this model, such as recurrence-transience, asymptotic speed, and limit theorems in both the transient and recurrent regimes. The new model combines features of several existing models of random motion in random media and admits a transparent physical interpretation. More specifically, a random walk in a sparse random environment can be characterized as a "locally strong" perturbation of a simple random walk by a random potential induced by "rare impurities," which are randomly…
We consider the one-sided exit problem for (fractionally) integrated random walks and Lévy processes. We prove that the rate of decrease of the non-exit probability -- the so-called survival exponent -- is universal in this class of processes. In particular, the survival exponent can be inferred from the (fractionally) integrated Brownian motion. This, in particular, extends Sinai's result on the survival exponent for the integrated simple random walk to general random walks with some finite exponential moment. Further, we prove existence and monotonicity of the survival exponent of fractional…
Random walks and Lorentz processes
本文利用概率论方法研究洛伦兹过程，证明平面齐次随机游走的局部扰动不改变极限律，并得到有限视界平面洛伦兹过程局部扰动的概率结论，同时为Lenci的常返性结果提供新证明。
Let $S_n$ be a centered random walk with a finite variance, and define the new sequence $A_n:=\sum_{i=1}^n S_i$, which we call an integrated random walk. We are interested in the asymptotics of $$p_N:=P(\min_{1 \le k \le N} A_k \ge 0)$$ as $N \to \infty$. Sinai (1992) proved that $p_N \asymp N^{-1/4}$ if $S_n$ is a simple random walk. We show that $p_N \asymp N^{-1/4}$ for some other types of random walks that include double-sided exponential and double-sided geometric walks, both not necessarily symmetric. We also prove that $p_N \le c N^{-1/4}$ for lattice walks and for upper exponential wal…
We consider the simple random walk in i.i.d. nonnegative potentials on the multidimensional cubic lattice. Our goal is to investigate the cost paid by the simple random walk for traveling from the origin to a remote location in a landscape of potentials. In particular, we obtain concentration inequalities for the travel cost in unbounded nonnegative potentials.
We study downward deviations of the boundary of the range of a transient walk on the Euclidean lattice. We describe the optimal strategy adopted by the walk in order to shrink the boundary of its range. The technics we develop apply equally well to the range, and provide pathwise statements for the {\it Swiss cheese} picture of Bolthausen, van den Berg and den Hollander \cite{BBH}.
Concentration Inequalities for Branching Random Walks
本文提出了一种扩展的集中不等式，适用于更广泛的随机过程，特别是针对具有漂移项的分支随机游走建立了新的理论框架。
We consider an infinite system of particles in one dimension, each particle performs independant Sinai's random walk in random environment. Considering an instant $t$, large enough, we prove a result in probability showing that the particles are trapped in the neighborhood of well defined points of the lattice depending on the random environment the time $t$ and the starting point of the particles.
Generalized gamma distributions arise as limits in many settings involving random graphs, walks, trees, and branching processes. Peköz, Röllin, and Ross (2016, arXiv:1309.4183 [math.PR]) exploited characterizing distributional fixed point equations to obtain uniform error bounds for generalized gamma approximations using Stein's method. Here we show how monotone couplings arising with these fixed point equations can be used to obtain sharper tail bounds that, in many cases, outperform competing moment-based bounds and the uniform bounds obtainable with Stein's method. Applications are given to…
We show that one-dimensional random walks in a quasi-periodic environment with Liouville frequency generically have an erratic statistical behavior. In the recurrent case we show that neither quenched nor annealed limit theorems hold and both drift and variance exhibit wild oscillations, being logarithmic at some times and almost linear at other times. In the transient case we show that the annealed Central Limit Theorem fails generically. These results are in stark contrast with the Diophantine case where the Central Limit Theorem with linear drift and variance was established by Sinai.
We consider a continuous time random walk $X$ in random environment on $\Z^+$ such that its potential can be approximated by the function $V: \R^+\to \R$ given by $V(x)=\sig W(x) -\frac{b}{1-\alf}x^{1-\alf}$ where $\sig W$ a Brownian motion with diffusion coefficient $\sig>0$ and parameters $b$, $\alf$ are such that $b>0$ and $0<\alf<1/2$. We show that $¶$-a.s.\ (where $¶$ is the averaged law) $\lim_{t\to \infty} \frac{X_t}{(C^*(\ln\ln t)^{-1}\ln t)^{\frac{1}{\alf}}}=1$ with $C^*=\frac{2\alf b}{\sig^2(1-2\alf)}$. In fact, we prove that by showing that there is a trap located around $(C^*(\ln\l…
In previous work by Avena and den Hollander, a model of a one-dimensional random walk in a dynamic random environment was proposed where the random environment is resampled from a given law along a growing sequence of deterministic times. In the regime where the increments of the resampling times diverge, which is referred to as the cooling regime, a weak law of large numbers and certain fluctuation properties were derived under the annealed measure. In the present paper we show that a strong law of large numbers and a quenched large deviation principle hold as well. In the cooling regime, the…
We study a general class of random walks driven by a uniquely ergodic Markovian environment. Under a coupling condition on the environment we obtain strong ergodicity properties and concentration inequalities for the environment as seen from the position of the walker, i.e the environment process. We also obtain ergodicity of the uniquely ergodic measure of the environment process as well as continuity as a function of the jump rates of the walker. As a consequence we obtain several limit theorems, such as law of large numbers, Einstein relation, central limit theorem and concentration propert…
We provide a characterization for anti-concentration of inhomogeneous random walks in non-abelian groups. In application we extend the classical bounds by Erdos-Littlewood-Offord and Sarkozy-Szemeredi to non-abelian settings.
Random walk in random environment (RWRE) is a fundamental model of statistical mechanics, describing the movement of a particle in a highly disordered and inhomogeneous medium as a random walk with random jump probabilities. It has been introduced in a series of papers by Chernov and Temkin as a model for DNA chain replication and crystal growth, and also as a model for turbulent behavior in fluids through a Lorentz gas description by Sinai. It is a simple but powerful model for a variety of complex large-scale disordered phenomena arising from fields such as physics, biology and engineering. …
Recurrence, transience and anti-concentration of Rademacher random walks
本文研究了一维Rademacher随机游走的暂留与复发性，证明了步长序列有界时游走弱递归，步长增长缓慢时可能暂留，但步长快速增长仍可保持弱递归，且当步长增长速度超过n^{1/2}时游走暂留。
We revisit the problem of estimates of moments of random n-dimensional matrices of Wigner ensemble by using the approach elaborated by Ya. Sinai and A. Soshnikov and further developed by A. Ruzmaikina. Our main subject is given by the structure of closed even walks and their graphs that arise in these studies. We show that the total degree of a vertex of such a graph depends not only on the self-intersections degree of but also on the total number of all non-closed instants of self-intersections of the walk. This result is used to fill the gaps of earlier considerations.
We generalise the coarse Ricci curvature method of Ollivier by considering the coarse Ricci curvature of multiple steps in the Markov chain. This implies new spectral bounds and concentration inequalities. We also extend this approach to the bounds for MCMC empirical averages obtained by Joulin and Ollivier. We prove a recursive lower bound on the coarse Ricci curvature of multiple steps in the Markov chain, making our method broadly applicable. Applications include the split-merge random walk on partitions, Glauber dynamics with random scan and systemic scan for statistical physical spin mode…
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 vec…
Concentration inequalities for maximal displacement of random walks on groups of polynomial growth
本文研究了具有多项式增长的群上随机游走最大位移的集中不等式，证明了在特定群类上随机游走的扩散性质。
After a general introduction to the field, we describe some recent results concerning disorder effects on both `random walk models', where the random walk is a dynamical process generated by local transition rules, and on `polymer models', where each random walk trajectory representing the configuration of a polymer chain is associated to a global Boltzmann weight. For random walk models, we explain, on the specific examples of the Sinai model and of the trap model, how disorder induces anomalous diffusion, aging behaviours and Golosov localization, and how these properties can be understood v…
We study the mean and variance of the number of self-intersections of the equilateral isotropic random walk in the plane, as well as the corresponding quantities for isotropic equilateral random polygons (random walks conditioned to return to their starting point after a given number of steps). The expected number of self-intersections is $(2/π^2)n\log n + O(n)$ for both walks and polygons with $n$ steps. The variance is $O(n^2 \log n)$ for both walks and polygons, which shows that the number of self-intersections exhibits concentration around the mean.
A Random-Walk Concentration Principle for Occupancy Processes on Finite Graphs
该研究提出有限图离散时间占位过程的两个随机游走集中定理，可应用于动态随机图等边过程，仅需利普希茨连续性，适用于稠密与稀疏图，能估计子图密度偏差。
Unique ergodicity of simple symmetric random walks on the circle
本文针对圆周上由无理旋转驱动的简单对称随机游走，给出任意无理角度下唯一平稳分布的一个新证明，简化并推广了 Sinai、Conze 与 Guivarc'h 的经典结果。
Chernoff bounds for branching random walks
本文提出了一种更一般的分支随机游走框架，并由此推导出BRW的Chernoff型界，同时讨论了随机游走与BRW之间的关系。
The goal of this article is two-fold: in a first part, we prove Azuma-Hoeffding type concentration inequalities around the drift for the displacement of non-elementary random walks on hyperbolic spaces. For a proper hyperbolic space $M$, we obtain explicit bounds that depend only on $M$, the size of support of the measure as in the classical case of sums of independent random variables, and on the norm of the driving probability measure in the left regular representation of the group of isometries. We obtain uniform bounds in the case of hyperbolic groups and effective bounds for simple linear…
We consider a directed random walk on the backbone of the supercritical oriented percolation cluster in dimensions $d+1$ with $d \ge 3$ being the spatial dimension. For this random walk we prove an annealed local central limit theorem and a quenched local limit theorem. The latter shows that the quenched transition probabilities of the random walk converge to the annealed transition probabilities reweighted by a function of the medium centred at the target site. This function is the density of the unique measure which is invariant for the point of view of the particle, is absolutely continuous…
In this article, universal concentration estimates are established for the local times of random walks on weighted graphs in terms of the resistance metric. As a particular application of these, a modulus of continuity for local times is provided in the case when the graphs in question satisfy a certain volume growth condition with respect to the resistance metric. Moreover, it is explained how these results can be applied to self-similar fractals, for which they are shown to be useful for deriving scaling limits for local times and asymptotic bounds for the cover time distribution.
Mixing of a generic simple symmetric random walk on the circle
本文针对圆上一般简单对称随机游走，给出了唯一遍历性的替代证明，并得到混合性的一般结果，部分解决了近期开放问题。
Uniform spectral gaps, non-abelian Littlewood-Offord and anti-concentration for random walks
本文研究了随机游走在半单代数群上的反集中问题，通过证明一致谱间隙和非阿贝尔利特伍德-奥弗德不等式，揭示了随机游走的扩张性质。
We give new criteria for ballistic behavior of random walks in random environment which are perturbations of the simple symmetric random walk on $\mathbb Z^d$ in dimensions $d\ge 4$. Our results extend those of Sznitman [Ann. Probab. 31, no. 1, 285-322 (2003)] and the recent ones of Ramírez and Saglietti [Preprint, arXiv:1808.01523], and allow us to exhibit new examples in dimensions $d\ge 4$ of ballistic random walks which do not satisfy Kalikow's condition. Our criteria implies ballisticity whenever the average of the local drift of the walk is not too small compared with an appropriate mome…
In this paper we study random walks on dynamical random environments in $1 + 1$ dimensions. Assuming that the environment is invariant under space-time shifts and fulfills a mild mixing hypothesis, we establish a law of large numbers and a concentration inequality around the asymptotic speed. The mixing hypothesis imposes a polynomial decay rate of covariances on the environment with sufficiently high exponent but does not impose uniform mixing. Examples of environments for which our methods apply include the contact process and Markovian environments with a positive spectral gap, such as the …
We explore a method introduced by Chatterjee and Ledoux in a paper on eigenvalues of principle submatrices. The method provides a tool to prove concentration of measure in cases where there is a Markov chain meeting certain conditions, and where the spectral gap of the chain is known. We provide several additional applications of this method. These applications include results on operator compressions using the Kac walk on $SO(n)$ and a Kac walk coupled to a thermostat, and a concentration of measure result for the length of the longest increasing subsequence of a random walk distributed under…
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