Refined estimates for some basic random walks on the symmetric and alternating groups

扩张图上随机游走的局部中心极限定理

A local central limit theorem for random walks on expander graphs

该研究针对扩张图上的随机游走，证明了带显式收敛速率的局部中心极限定理，同时推导得到了更优的总变差距离界，拓展了马尔可夫链中心极限定理相关的定量研究成果。

Asymptotic Normality of Superdiffusive Step-Reinforced Random Walks

In this article we establish for the superdiffusive regime $p \in (1/2,1)$ that the fluctuations of a general step-reinforced random walk around $a_n \hat{W}$, where $(a_n)_{n \in \mathbb{N}}$ is a non-negative sequence of order $n^p$ and $\hat{W}$ is a non-degenerate random variable, is Gaussian. This extends a known result by Kubota and Takei for the elephant random walk to the more general setting of step-reinforced random walks. Further, we provide an application of the asymptotic normality of $\hat{S}$ around $a_n \hat{W}$ to reinforced empirical processes as studied recently by Bertoin, …

coalescing heavy-tailed random walk 的 N 点相关函数的精确渐进行为

Sharp asymptotics for $N$-point correlation functions of coalescing heavy-tailed random walk

本文研究了 coalescing 重尾随机游走系统的 N 点相关函数的精确渐进行为，通过尾部估计和非碰撞概率分析，推导了重尾随机游走的精细估计。

Loop-Erased Random Walk Branch of Uniform Spanning Tree in Topological Polygons

We consider uniform spanning tree (UST) in topological polygons with $2N$ marked points on the boundary with alternating boundary conditions. In [LPW21], the authors derive the scaling limit of the Peano curve in the UST. They are variants of SLE$_8$. In this article, we derive the scaling limit of the loop-erased random walk branch (LERW) in the UST. They are variants of SLE$_2$. The conclusion is a generalization of [HLW20,Theorem 1.6] where the authors derive the scaling limit of the LERW branch of UST when $N=2$. When $N=2$, the limiting law is SLE$_2(-1,-1; -1, -1)$. However, the limiting…

Separation cutoff for Activated Random Walks

We consider Activated Random Walks on arbitrary finite networks, with particles being inserted at random and absorbed at the boundary. Despite the non-reversibility of the dynamics and the lack of knowledge on the stationary distribution, we explicitly determine the relaxation time of the process, and prove that separation cutoff is equivalent to the product condition. We also provide sharp estimates on the center and width of the cutoff window. Finally, we illustrate those results by establishing explicit separation cutoffs on various networks, including: (i) large finite subgraphs of any fix…

Indicator fractional stable motions

Using the framework of random walks in random scenery, Cohen and Samorodnitsky (2006) introduced a family of symmetric $α$-stable motions called local time fractional stable motions. When $α=2$, these processes are precisely fractional Brownian motions with $1/2<H<1$. Motivated by random walks in alternating scenery, we find a "complementary" family of symmetric $α$-stable motions which we call indicator fractional stable motions. These processes are complementary to local time fractional stable motions in that when $α=2$, one gets fractional Brownian motions with $0<H<1/2$.

二维格点随机游走的首次通过位置与钟形分布

First passage locations for two-dimensional lattice random walks and the bell-shape

本文研究二维格点随机游走的首次通过位置，证明其重标度分布为钟形，并引入两类有理函数以证明相关定理。

随机洗牌的转置前2的极限分布

Limit profile for the transpose top-2 with random shuffle

本文利用非交换傅里叶分析的比较方法，研究交错群上由特定3-循环生成的随机洗牌游走，确定其极限分布并给出交错群图的完整谱。

有限图上随机游走的首达时间分布

Hitting Time Distributions of Random Walks on Finite Graphs

本文研究有限图上随机游走首达时间的完整分布与方差，利用马尔可夫链、傅里叶和谱方法推导计算公式与递推关系，弥补现有研究仅关注期望值的不足。

Recurrence and Transience of Frogs with Drift on $\mathbb{Z}^d$

We study the frog model on $\mathbb{Z}^d$ with drift in dimension $d \geq 2$ and establish the existence of transient and recurrent regimes depending on the transition probabilities. We focus on a model in which the particles perform nearest neighbour random walks which are balanced in all but one direction. This gives a model with two parameters. We present conditions on the parameters for recurrence and transience, revealing interesting differences between dimension $d=2$ and dimension $d \geq 3$. Our proofs make use of (refined) couplings with branching random walks for the transience, and …

排斥过程上随机游走弹道性的尖锐阈值

Sharp threshold for the ballisticity of the random walk on the exclusion process

本文研究排斥过程平流下随机游走的弹道性，证明回溯概率在临界密度处发生尖锐转变，速度严格单调且零速度区域坍缩为单点，从而解决一个已有猜想。

On the fragmentation of a torus by random walk

We consider a simple random walk on a discrete torus (Z/NZ)^d with dimension d at least 3 and large side length N. For a fixed constant u > 0, we study the percolative properties of the vacant set, consisting of the set of vertices not visited by the random walk in its first [uN^d] steps. We prove the existence of two distinct phases of the vacant set in the following sense: if u > 0 is chosen large enough, all components of the vacant set contain no more than a power of log(N) vertices with high probability as N tends to infinity. On the other hand, for small u > 0, there exists a macroscopic…

Fluctuations of bridges, reciprocal characteristics, and concentration of measure

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…

KMT coupling for random walk bridges

In this paper we prove an analogue of the Komlós-Major-Tusnády (KMT) embedding theorem for random walk bridges. The random bridges we consider are constructed through random walks with i.i.d jumps that are conditioned on the locations of their endpoints. We prove that such bridges can be strongly coupled to Brownian bridges of appropriate variance when the jumps are either continuous or integer valued under some mild technical assumptions on the jump distributions. Our arguments follow a similar dyadic scheme to KMT's original proof, but they require more refined estimates and stronger assumpt…

Supplementary Material for Random Cayley Graphs Project

This document contains supplementary material for the main articles in our Random Cayley Graphs project. We prove refined results about simple random walks on the integers and on the cycle. We are primarily interested in the entropy of these random walks at certain times and how this entropy changes when the time changes slightly. Additionally, we prove some large deviation and exit time estimates. We prove some results on the size of discrete lattice balls and how this size changes when the radius changes slightly. We do this in a general $L_q$ norm, with $q \in [1,\infty]$. We also prove som…

Localization and limit laws of a three-state alternate quantum walk on a two-dimensional lattice

A two-dimensional discrete-time quantum walk (DTQW) can be realized by alternating a two-state DTQW in one spatial dimension followed by an evolution in the other dimension. This was shown to reproduce a probability distribution for a certain configuration of a four-state DTQW on a two-dimensional lattice. In this work we present a three-state alternate DTQW with a parameterized coin-flip operator and show that it can produce localization that is also observed for a certain other configuration of the four-state DTQW and non-reproducible using the two-state alternate DTQW. We will present two l…

Gaussian, stable, tempered stable and mixed limit laws for random walks in cooling random environments

Random Walks in Cooling Random Environments (RWCRE) is a model of random walks in dynamic random environments where the entire environment is resampled along a fixed sequence of times, called the "cooling sequence," and is kept fixed in between those times. This model interpolates between that of a homogenous random walk, where the environment is reset at every step, and Random Walks in (static) Random Environments (RWRE), where the environment is never resampled. In this work we focus on the limiting distributions of one-dimensional RWCRE in the regime where the fluctuations of the correspond…

General linear-fractional branching processes with discrete time

We study a linear-fractional Bienaymé-Galton-Watson process with a general type space. The corresponding tree contour process is described by an alternating random walk with the downward jumps having a geometric distribution. This leads to the linear-fractional distribution formula for an arbitrary observation time, which allows us to establish transparent limit theorems for the subcritical, critical and supercritical cases. Our results extend recent findings for the linear-fractional branching processes with countably many types.

关于具有多项式增长的群上随机游走最大位移的集中不等式

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。

Random Walks on Dynamic Graphs: Mixing Times, HittingTimes, and Return Probabilities

We establish and generalise several bounds for various random walk quantities including the mixing time and the maximum hitting time. Unlike previous analyses, our derivations are based on rather intuitive notions of local expansion properties which allows us to capture the progress the random walk makes through $t$-step probabilities. We apply our framework to dynamically changing graphs, where the set of vertices is fixed while the set of edges changes in each round. For random walks on dynamic connected graphs for which the stationary distribution does not change over time, we show that the…

Spectral analysis of Sinai's walk for small eigenvalues

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…

Asymptotic behavior of the length of the longest increasing subsequences of random walks

We numerically estimate the leading asymptotic behavior of the length $L_{n}$ of the longest increasing subsequence of random walks with step increments following Student's $t$-distribution with parameter in the range $1/2 \leq ν\leq 5$. We find that the expected value $\mathbb{E}(L_{n}) \sim n^θ\ln{n}$ with $θ$ decreasing from $θ(ν=1/2) \approx 0.70$ to $θ(ν\geq 5/2) \approx 0.50$. For random walks with distribution of step increments of finite variance ($ν> 2$), this confirms previous observation of $\mathbb{E}(L_{n}) \sim \sqrt{n}\ln{n}$ to leading order. We note that this asymptotic behavi…

A functional central limit theorem for branching random walks, almost sure weak convergence, and applications to random trees

Let $W_{\infty}(β)$ be the limit of the Biggins martingale $W_n(β)$ associated to a supercritical branching random walk with mean number of offspring $m$. We prove a functional central limit theorem stating that as $n\to\infty$ the process $$ D_n(u):= m^{\frac 12 n} \left(W_{\infty}\left(\frac{u}{\sqrt n}\right) - W_{n}\left(\frac{u}{\sqrt n}\right) \right) $$ converges weakly, on a suitable space of analytic functions, to a Gaussian random analytic function with random variance. Using this result we prove central limit theorems for the total path length of random trees. In the setting of bina…

Solutions to kinetic-type evolution equations: beyond the boundary case

We study the asymptotic behavior as $t \to \infty$ of a time-dependent family $(μ_t)_{t \geq 0}$ of probability measures on $\mathbb{R}$ solving the kinetic-type evolution equation $\partial_t μ_t + μ_t = Q(μ_t)$ where $Q$ is a smoothing transformation on $\mathbb{R}$. This problem has been investigated earlier, e.g. by Bassetti and Ladelli [Ann. Appl. Probab. 22(5): 1928-1961, 2012] and Bogus, Buraczewski and Marynych [Stochastic Process. Appl. 130(2):677-693, 2020]. Combining the refined analysis of the latter paper, which provides a probabilistic description of the solution $μ_t$ as the law…

Plane bipolar orientations and quadrant walks

Bipolar orientations of planar maps have recently attracted some interest in combinatorics, probability theory and theoretical physics. Plane bipolar orientations with $n$ edges are known to be counted by the $n$th Baxter number $b(n)$, which can be defined by a linear recurrence relation with polynomial coefficients. Equivalently, the associated generating function $\sum_n b(n)t^n$ is D-finite. In this paper, we address a much refined enumeration problem, where we record for every $r$ the number of faces of degree $r$. When these degrees are bounded, we show that the associated generating fun…

Formation of large-scale random structure by competitive erosion

We study the following one-dimensional model of annihilating particles. Beginning with all sites of $\mathbb{Z}$ uncolored, a blue particle performs simple random walk from $0$ until it reaches a nonzero red or uncolored site, and turns that site blue; then, a red particle performs simple random walk from $0$ until it reaches a nonzero blue or uncolored site, and turns that site red. We prove that after $n$ blue and $n$ red particles alternately perform such walks, the total number of colored sites is of order $n^{1/4}$. The resulting random color configuration, after rescaling by $n^{1/4}$ an…

Kac的旋转矩阵漫步在n² log n步内混合

Kac's walk on rotation matrices mixes in $n^2 \log n$ steps

研究Kac漫步的总变分混合时间，提出基于精细两阶段耦合的新框架，证明在O(n² log n)步内混合，适用于高维马尔可夫链。

无尽的平方自由数位走行

Dead ends in square-free digit walks

该研究证明了在平方自由数位走行中，死胡同的渐近密度远低于预测值，且通过形式化方法给出了闭合表达式。

随机\(k\)-循环的击中时间混合

Hitting time mixing for random $k$-cycles

研究对称群\(\mathfrak{S}_n\)上由\(k\)-循环共轭类生成的随机游走的击中时间混合，结合固定时间近似与标记方案及奇偶兼容耦合，证明特定\(k\)范围结果并对所有\(2\leq k\leq n - 1\)提出猜想。