We give a new and elementary computation of the spectral gap of the Kac walk on the N-sphere. The result is obtained as a by-product of a more general observation which allows to reduce the analysis of the spectral gap of an N-component system to that of the same system for N=3. The method applies to a number of random 'binary collision' processes with complete-graph structure, including non-homogeneous examples such as exclusion and colored exclusion processes with site disorder.
We introduce a Kac's type walk whose rate of binary collisions preserves the total momentum but not the kinetic energy. In the limit of large number of particles we describe the dynamics in terms of empirical measure and flow, proving the corresponding large deviation principle. The associated rate function has an explicit expression. As a byproduct of this analysis, we provide a gradient flow formulation of the Boltzmann-Kac equation.
Determining the mixing time of Kac's random walk on the sphere $\mathrm{S}^{n-1}$ is a long-standing open problem. We show that the total variation mixing time of Kac's walk on $\mathrm{S}^{n-1}$ is between $\frac{1}{2} \, n \log(n)$ and $200 \,n \log(n)$. Our bound is thus optimal up to a constant factor, improving on the best-known upper bound of $O(n^{5} \log(n)^{2})$ due to Jiang. Our main tool is a `non-Markovian' coupling recently introduced by the second author for obtaining the convergence rates of certain high dimensional Gibbs samplers in continuous state spaces.
In this article we prove existence of the asymptotic entropy for isotropic random walks on regular Fuchsian buildings. Moreover, we give formulae for the asymptotic entropy, and prove that it is equal to the rate of escape of the random walk with respect to the Green distance. When the building arises from a Fuchsian Kac-Moody group our results imply results for random walks induced by bi-invariant measures on these groups, however our results are proven in the general setting without the assumption of any group acting on the building. The main idea is to consider the retraction of the isotrop…
In this work, we generalize M. Kac's original many-particle binary stochastic model to derive a space homogeneous Boltzmann equation that includes a linear combination of higher-order collisional terms. First, we prove an abstract theorem about convergence from a finite hierarchy to an infinite hierarchy of coupled equations. We apply this convergence theorem on hierarchies for marginals corresponding to the generalized Kac model mentioned above. As a corollary, we prove propagation of chaos for the marginals associated to the generalized Kac model. In particular, the first marginal converges …
In this work, we analyze dimension reduction algorithms based on the Kac walk and discrete variants. (1) For $n$ points in $\mathbb{R}^{d}$, we design an optimal Johnson-Lindenstrauss (JL) transform based on the Kac walk which can be applied to any vector in time $O(d\log{d})$ for essentially the same restriction on $n$ as in the best-known transforms due to Ailon and Liberty [SODA, 2008], and Bamberger and Krahmer [arXiv, 2017]. Our algorithm is memory-optimal, and outperforms existing algorithms in regimes when $n$ is sufficiently large and the distortion parameter is sufficiently small. In …
Determining the total variation mixing time of Kac's random walk on the special orthogonal group $\mathrm{SO}(n)$ has been a long-standing open problem. In this paper, we construct a novel non-Markovian coupling for bounding this mixing time. The analysis of our coupling entails controlling the smallest singular value of a certain random matrix with highly dependent entries. The dependence of the entries in our matrix makes it not-amenable to existing techniques in random matrix theory. To circumvent this difficulty, we extend some recent bounds on the smallest singular values of matrices with…
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…
A Tale of Two Walks: Kipnis, Marchioro and Presutti Meet Kac in a Quantum World
The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；Nanjing University（南京大学）；National University of Singapore（新加坡国立大学）；California Institute of Technology（加州理工学院）；Portland State University（波特兰州立大学）；Hefei National Laboratory（合肥国家实验室）
An explicit estimate is derived for Kac's mean-field model of colliding hard spheres, which compares, in a Wasserstein distance, the empirical velocity distributions for two versions of the model based on different numbers of particles. For suitable initial data, with high probability, the two processes agree to within a tolerance of order $N^{-1/d}$, where $N$ is the smaller particle number and $d$ is the dimension, provided that $d\ge3$. From this estimate we can deduce that the spatially homogeneous Boltzmann equation is well posed in a class of measure-valued processes and provides a good …
We analyze the large deviations for a discrete energy Kac-like walk. In particular, we exhibit a path, with probability exponentially small in the number of particles, that looses energy.
Within the framework of the previous paper [8]. we develop a generalized stochastic calculus for processes associated to higher order diffusion operators. Applications to the study of a Cauchy problem, a Feynman-Kac formula and a representation formula for higher derivatives of analytic functions are also given.
本文研究了不可压缩粒子系统在碰撞过程中的混沌传播和平均场极限。我们将考虑的主要例子是产生玻尔兹曼方程的卡茨和麦基恩\cite{Kac1956,McKean1967}的多粒子跳跃过程。我们解决了卡茨提出的猜想\cite{Kac1956}，激发他的计划，在严格的连接之间的长期行为的碰撞多粒子系统和它的平均场极限之一，有界以及无界碰撞率。受Grünbaum \cite{Grunbaum}的启发，我们开发了一种抽象方法，将混沌传播问题简化为证明生成算子的纯函数估计（{ \em一致性估计}），以及非线性极限方程流的可微性估计（{ \em稳定性估计}）。这使我们能够在平均场极限方程的水平上而不是在粒子系统的水平上（如卡茨所提出的）利用耗散性。利用这种方法，我们证明了：（1）对一族态的混沌性的定量估计，即在时间上一致的估计。(2)如\cite{CCLLV}中定义的{ \it熵无序性}的传播。(3)估计松弛至平衡的时间，即\emph{independent of the number of particles in the system}。我们的结果涵盖了碰撞率无界的两个主要玻尔兹曼物理碰撞过程：硬球和\emph{true}麦克斯韦分子相互作用。这些模型的\emph{stability estimates}的证明需要大量的分析工作和新的估计。
In this paper we survey the theory of random walks on buildings and associated groups of Lie type and Kac-Moody groups. We begin with an introduction to the theory of Coxeter systems and buildings, taking a largely combinatorial perspective. We then survey the theory of random walks on buildings, and show how this theory leads to limit theorems for random walks on the associated groups.
We give bounds in total variation distance for random walks associated to pure central states on free orthogonal quantum groups. As a consequence, we prove that the analogue of the uniform plane Kac walk on this quantum group has a cut-off at $N\ln(N)/2(1-\cos(θ))$. This is the first result of this type for genuine compact quantum groups. We also obtain similar results for mixtures of rotations and quantum permutations.
We study dynamic random conductance models on $\mathbb{Z}^2$ in which the environment evolves as a reversible Markov process that is stationary under space-time shifts. We prove under a second moment assumption that two conditionally independent random walks in the same environment collide infinitely often almost surely. These results apply in particular to random walks on dynamical percolation.
For an integer $k\ge 2$, let $S^{(1)}, S^{(2)}, \dots, S^{(k)}$ be $k$ independent simple symmetric random walks on $\mathbb{Z}$. A pair $(n,z)$ is called a collision event if there are at least two distinct random walks, namely, $S^{(i)},S^{(j)}$ satisfying $S^{(i)}_n= S^{(j)}_n=z$. We show that under the same scaling as in Donsker's theorem, the sequence of random measures representing these collision events converges to a non-trivial random measure on $[0,1]\times \mathbb{R}$. Moreover, the limit random measure can be characterized using Wiener chaos. The proof is inspired by methods from s…
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 …
Let $\mathcal{U}$ be the uniform spanning tree on $\mathbb{Z}^3$, whose probability law is denoted by $\mathbf{P}$. For $\mathbf{P}$-a.s. realization of $\mathcal{U}$, the recurrence of the the simple random walk on $\mathcal{U}$ is proved in [5] and it is also demonstrated in [8] that two independent simple random walks on $\mathcal{U}$ collide infinitely often. In this article, we will give a quantitative estimate on the number of collisions of two independent simple random walks on $\mathcal{U}$, which provides another proof of the infinite collision property of $\mathcal{U}$.
Convergence of directed forests, spanning on random subsets of lattices or on point processes, towards the Brownian web has made the subject of an abundant literature, a large part of which relies on a criterion proposed by Fontes, Isopi, Newman and Ravishankar (2004). One of their convergence condition, called (B2), states that the probability of the event that there exists three distinct paths for a time interval of length $t(>0)$, all starting within a segment of length $\varepsilon$, is of small order of $\varepsilon$. This condition is often verified by applying an FKG type correlation in…
On the phase transition for the number of collisions on comb graphs
National University of Singapore（新加坡国立大学）；The Hong Kong University（香港大学）；The Hong Kong University of Science and Technology（香港科技大学）；Technische Universität München（慕尼黑工业大学）
