Pierre Andreoletti
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.
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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 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.
Understanding the statistical properties of the aperiodic planar Lorentz gas stands as a grand challenge in the theory of dynamical systems. Here we study a greatly simplified but related model, proposed by Arvind Ayyer and popularized by Joel Lebowitz, in which a scatterer configuration on the torus is randomly updated between collisions. Taking advantage of recent progress in the theory of time-dependent billiards on the one hand and in probability theory on the other, we prove a vector-valued almost sure invariance principle for the model. Notably, the configuration sequence can be weakly d…
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.
The present paper has two goals. First to present a natural example of a new class of random fields which are the variable neighborhood random fields. The example we consider is a partially observed nearest neighbor binary Markov random field. The second goal is to establish sufficient conditions ensuring that the variable neighborhoods are almost surely finite. We discuss the relationship between the almost sure finiteness of the interaction neighborhoods and the presence/absence of phase transition of the underlying Markov random field. In the case where the underlying random field has no ph…
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 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.
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 prove a converse Lyapunov theorem for almost sure stabilizability and almost sure asymptotic stabilizability of controlled diffusions: given a stochastic system a.s. stochastic open loop stabilizable at the origin, we construct a lower semicontinuous positive definite function whose level sets form a local basis of viable neighborhoods of the equilibrium. This result provides, with the direct Lyapunov theorems proved in a companion paper, a complete Lyapunov-like characterization of the a.s. stabilizability.
In this paper we show that the closure of a random sample for a k-dimensional random vector is almost surely a deterministic set of all heavy points of the distribution. A heavy point is defined to be a point for which all its neighborhoods have positive probability.
Randomized coordinate gradient descent almost surely escapes strict saddle points
本文分析了非凸优化中随机坐标梯度下降的行为，通过将其表述为非线性随机动力系统并利用中心稳定流形定理，证明了迭代在标准假设下几乎必然逃离严格鞍点。
Singular diffusion limit of a tagged particle in zero range processes with Sinai-type random environment
该研究针对一维环面Sinai型随机环境下的零程过程，通过两步法推导标记粒子位置的奇异扩散极限，先引入正则化得到非平衡极限方程，再证明正则化参数趋于零时解依分布收敛于对应奇异扩散过程。
On Certificates for Almost Sure Reachability in Stochastic Systems
本文研究离散时间随机系统的几乎必然可达性证书，证明固定多项式或二次模板会丧失完备性，并刻画加性噪声线性系统中证书存在的矩阵条件。
In almost all shallow analytic neural network optimization landscapes, efficient minimizers have strongly convex neighborhoods
本文分析浅层解析神经网络回归问题的优化景观，发现有效域上几乎所有回归问题的局部极小值均具强凸邻域（优化景观为莫尔斯函数），而冗余域维度更小且局部极小值不孤立。
Spectra of Random Polynomial Matrices: the Petaloid Law
本文研究随机首一多项式矩阵行列式零点的极限分布，推广经典圆律、椭圆律和半圆律，提出花瓣律并给出极限密度与支撑的显式公式。
Model selection for Markov random fields on graphs under a mixing condition
提出一种在所有可能图上优化的全局模型选择准则，用于有限样本下估计随机向量条件依赖图，并在满足混合条件的非独立多元过程中证明其几乎必然一致性。
This paper presents a communication efficient distributed algorithm, $\mathcal{CIRFE}$ of the \emph{consensus}+\emph{innovations} type, to estimate a high-dimensional parameter in a multi-agent network, in which each agent is interested in reconstructing only a few components of the parameter. This problem arises for example when monitoring the high-dimensional distributed state of a large-scale infrastructure with a network of limited capability sensors and where each sensor is tasked with estimating some local components of the state. At each observation sampling epoch, each agent updates it…
Self-Repellent Random Walks on General Graphs -- Achieving Minimal Sampling Variance via Nonlinear Markov Chains
该研究提出一般图上的自排斥随机游走（SRRW），证明其经验分布收敛于目标平稳分布，且排斥性越强渐近采样方差越小，还通过数值模拟验证了时变α版本的综合优势。
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