Appendix to the paper "Random walks on free products of cyclic groups"

The peeling process on random planar maps coupled to an O(n) loop model (with an appendix by Linxiao Chen)

We extend the peeling exploration introduced in arxiv:1506.01590 to the setting of Boltzmann planar maps coupled to a rigid $O(n)$ loop model. Its law is related to a class of discrete Markov processes obtained by confining random walks to the positive integers with a new type of boundary condition. As an application we give a rigorous justification of the phase diagram of the model presented in arXiv:1106.0153. This entails two results pertaining to the so-called fixed-point equation: the first asserts that any solution determines a well-defined model, while the second result, contributed by …

On Random Walks and Diffusions Related to Parrondo's Games

In a series of papers, G. Harmer and D. Abbott study the behavior of random walks associated with games introduced in 1997 by J. M. R. Parrondo. These games illustrate an apparent paradox that random and deterministic mixtures of losing games may produce winning games. In this paper, classical cyclic random walks on the additive group of integers modulo $m$, a given integer, are used in a straightforward way to derive the strong law limits of a general class of games that contains the Parrondo games. We then consider the question of when random mixtures of fair games related to these walks may…

一维“石头-剪刀-布”循环元胞自动机的渐近行为

Asymptotic behaviour of the one-dimensional "rock-paper-scissors" cyclic cellular automaton

研究一维三态循环元胞自动机中状态主导有限区域的渐近概率，证明其等于被捕食者的初始概率，方法基于粒子系统和随机游走。

A note on fluctuations for internal diffusion limited aggregation

We consider a cluster growth model on Z^d, called internal diffusion limited aggregation (internal DLA). In this model, random walks start at the origin, one at a time, and stop moving when reaching a site not occupied by previous walks. It is known that the asymptotic shape of the cluster is spherical. Also, when dimension is 2 or more, and when the cluster has volume $n^d$, it is known that fluctuations of the radius are at most of order $n^{1/3}$. We improve this estimate to $n^{1/(d+1)}$, in dimension 3 or more. In so doing, we introduce a closely related cluster growth model, that we call…

Rotor Walks and Markov Chains

The rotor walk is a derandomized version of the random walk on a graph. On successive visits to any given vertex, the walker is routed to each of the neighboring vertices in some fixed cyclic order, rather than to a random sequence of neighbors. The concept generalizes naturally to Markov chains on a countable state space. Subject to general conditions, we prove that many natural quantities associated with the rotor walk (including normalized hitting frequencies, hitting times and occupation frequencies) concentrate around their expected values for the random walk. Furthermore, the concentrati…

From logarithmic to subdiffusive polynomial fluctuations for internal DLA and related growth models

We consider a cluster growth model on the d-dimensional lattice, called internal diffusion limited aggregation (internal DLA). In this model, random walks start at the origin, one at a time, and stop moving when reaching a site not occupied by previous walks. It is known that the asymptotic shape of the cluster is spherical. When dimension is 2 or more, we prove that fluctuations with respect to a sphere are at most a power of the logarithm of its radius in dimension d larger than or equal to 2. In so doing, we introduce a closely related cluster growth model, that we call the flashing process…

The Poisson boundary of lamplighter random walks on trees

Let T be the homogeneous tree with degree and G a finitely generated group whose Cayley graph is T. The associated lamplighter group is the wreath product of the cyclic group of order r with G. For a large class of random walks on this group, we prove almost sure convergence to a natural geometric boundary. If the probability law governing the random walk has finite first moment, then the probability space formed by this geometric boundary together with the limit distribution of the random walk is proved to be maximal, that is, the Poisson boundary. We also prove that the Dirichlet problem at …

On Learning Thresholds of Parities and Unions of Rectangles in Random Walk Models

In a recent breakthrough, [Bshouty et al., 2005] obtained the first passive-learning algorithm for DNFs under the uniform distribution. They showed that DNFs are learnable in the Random Walk and Noise Sensitivity models. We extend their results in several directions. We first show that thresholds of parities, a natural class encompassing DNFs, cannot be learned efficiently in the Noise Sensitivity model using only statistical queries. In contrast, we show that a cyclic version of the Random Walk model allows to learn efficiently polynomially weighted thresholds of parities. We also extend the …

Rotor walks on general trees

The rotor walk on a graph is a deterministic analogue of random walk. Each vertex is equipped with a rotor, which routes the walker to the neighbouring vertices in a fixed cyclic order on successive visits. We consider rotor walk on an infinite rooted tree, restarted from the root after each escape to infinity. We prove that the limiting proportion of escapes to infinity equals the escape probability for random walk, provided only finitely many rotors send the walker initially towards the root. For i.i.d. random initial rotor directions on a regular tree, the limiting proportion of escapes is …

Total variation cutoff for the flip-transpose top with random shuffle

We consider a random walk on the hyperoctahedral group $B_n$ generated by the signed permutations of the forms $(i,n)$ and $(-i,n)$ for $1\leq i\leq n$. We call this the flip-transpose top with random shuffle on $B_n$. We find the spectrum of the transition probability matrix for this shuffle. We prove that the mixing time for this shuffle is of order $n\log n$. We also show that this shuffle exhibits the cutoff phenomenon. In the appendix, we show that a similar random walk on the demihyperoctahedral group $D_n$ also has a cutoff at $\left(n-\frac{1}{2}\right)\log n$.

Anchored expansion, percolation and speed

Benjamini, Lyons and Schramm [Random Walks and Discrete Potential Theory (1999) 56-84] considered properties of an infinite graph G, and the simple random walk on it, that are preserved by random perturbations. In this paper we solve several problems raised by those authors. The anchored expansion constant is a variant of the Cheeger constant; its positivity implies positive lower speed for the simple random walk, as shown by Virag [Geom. Funct. Anal. 10 (2000) 1588-1605]. We prove that if G has a positive anchored expansion constant, then so does every infinite cluster of independent percolat…

虚阿贝尔群上噪声敏感性

Noise sensitivity on virtually abelian groups

研究虚阿贝尔群上非周期性随机游走的噪声敏感性，证明其在总变分下噪声敏感的充要条件是群无非零同态到无限循环群。

Speed of random walks, isoperimetry and compression of finitely generated groups

We give a solution to the inverse problem (given a function, find a corresponding group) for large classes of speed, entropy, isoperimetric profile, return probability and $L_p$-compression functions of finitely generated groups of exponential volume growth. For smaller classes, we give solutions among solvable groups. As corollaries, we prove a recent conjecture of Amir on joint evaluation of speed and entropy exponents and we obtain a new proof of the existence of uncountably many pairwise non-quasi-isometric solvable groups, originally due to Cornulier and Tessera. We also obtain a formula …

Boundary of the Range of a random walk and the Fölner property

The range process $R_n$ of a random walk is the collection of sites visited by the random walk up to time $n$. In this work we deal with the question of whether the range process of a random walk or the range process of a cocycle over an ergodic transformation is almost surely a Fölner sequence and show the following results: % (a) The size of the inner boundary $|\partial R_n|$ of the range of recurrent aperiodic random walks on $\mathbb{Z}^2$ with finite variance and aperiodic random walks in $\mathbb{Z}$ in the standard domain of attraction of the Cauchy distribution, divided by $\frac{n}{\…

仿射群的随机自由半群

Random free semigroups of affine groups

研究可解非幂零情形下半群的随机自由性，聚焦相关仿射模型，表明自由性长期行为由阿贝尔影子决定，证明结合仿射随机游走局部收缩定理与乒乓论证。

圆周上的随机游走与丢番图逼近

Random walks on the circle and Diophantine approximation

本文研究圆周群上的随机游走，将有限状态空间Markov链的中心极限定理推广至无理跨度情形，并揭示了Kolmogorov度量下收敛速率从多项式到指数衰减的新现象。

Coxeter交换图上的随机游走

Random walks on Coxeter interchange graphs

本文研究Zaslavsky符号图上的Coxeter竞赛图对应的交换图随机游走，通过路径耦合方法结合度量重加权证明其快速混合，并探讨其与Coxeter置换多面体的几何关联。

在$\mathfrak{sl}_2(\mathbf{F}_p)$上的随机李括号

Random Lie bracket on $\mathfrak{sl}_2(\mathbf{F}_p)$

研究在$\mathfrak{sl}_2(\mathbf{F}_p)$上通过随机李括号的随机游走，揭示其极限分布和收敛速度，并通过引入随机扭实现对整个李代数的覆盖。

On the affine random walk on the torus

Let $μ$ be a borelian probability measure on $\mathbf{G}:=\mathrm{SL}_d(\mathbb{Z}) \ltimes \mathbb{T}^d$. Define, for $x\in \mathbb{T}^d$, a random walk starting at $x$ denoting for $n\in \mathbb{N}$, \[ \left\{\begin{array}{rcl} X_0 &=&x\\ X_{n+1} &=& a_{n+1} X_n + b_{n+1} \end{array}\right. \] where $((a_n,b_n))\in \mathbf{G}^\mathbb{N}$ is an iid sequence of law $μ$. Then, we denote by $\mathbb{P}_x$ the measure on $(\mathbb{T}^d)^\mathbb{N}$ that is the image of $μ^{\otimes \mathbb{N}}$ by the map $\left((g_n) \mapsto (x,g_1 x, g_2 g_1 x, \dots , g_n \dots g_1 x, \dots)\right)$ and for an…

On the spectrum of lamplighter groups and percolation clusters

Let $G$ be a finitely generated group and $X$ its Cayley graph with respect to a finite, symmetric generating set $S$. Furthermore, let $H$ be a finite group and $H \wr G$ the lamplighter group (wreath product) over $G$ with group of "lamps" $H$. We show that the spectral measure (Plancherel measure) of any symmetric "switch--walk--switch" random walk on $H \wr G$ coincides with the expected spectral measure (integrated density of states) of the random walk with absorbing boundary on the cluster of the group identity for Bernoulli site percolation on $X$ with parameter $p = 1/|H|$. The return …

Phase transition for the interchange and quantum Heisenberg models on the Hamming graph

We study a family of random permutation models on the Hamming graph $H(2,n)$ (i.e., the $2$-fold Cartesian product of complete graphs), containing the interchange process and the cycle-weighted interchange process with parameter $θ> 0$. This family contains the random walk representation of the quantum Heisenberg ferromagnet. We show that in these models the cycle structure of permutations undergoes a \textit{phase transition} -- when the number of transpositions defining the permutation is $\leq c n^2$, for small enough $c > 0$, all cycles are microscopic, while for more than $\geq C n^2$ tra…

带词长的有限循环群的尖锐对数Sobolev不等式

Sharp log-Sobolev inequalities and quartic stability on finite cyclic groups

本文证明了对于均匀概率测度下的循环群Z_n，带词长ψ_n(k)=min(k,n-k)的拉普拉斯算子满足尖锐对数Sobolev不等式，常数2π与n无关（n≥4）。

五维交换过程中的无限循环

Infinite cycles in the interchange process in five dimensions

本研究针对五维及以上整数格点上的交换过程，通过分析循环时间随机游走的扩散性与局部逃逸性质，证明了大时间下存在无限循环，证实了Tóth于1993年提出的相关猜想。