We show that almost any one-dimensional projection of a suitably scaled random walk on a hypercube, inscribed in a hypersphere, converges weakly to an Ornstein-Uhlenbeck process as the dimension of the sphere tends to infinity. We also observe that the same result holds when the random walk is replaced with spherical Brownian motion. This latter result can be viewed as a "functional" generalisation of Poincaré's observation for projections of uniform measure on high dimensional spheres; the former result is an analogous generalisation of the Bernoulli-Laplace central limit theorem. Given the relation of these two classic results to the central limit theorem for convex bodies, the modest results provided here would appear to motivate a functional generalisation.
We give Harnack inequalities for the hitting distributions of a large family of symmetric random walks on $\Z^2$, and their projections onto the lattice torus $\Z^2_K$. This extends a framework for the simple random walk in Dembo, et al. (2006), and generalizes the results in Bass, Rosen (2007) to the toral projection.
We examine the sets of late points of a symmetric random walk on $Z^2$ projected onto the torus $Z^2_K$, culminating in a limit theorem for the cover time of the toral random walk. This extends the work done for the simple random walk in Dembo, et al. (2006) to a large class of random walks projected onto the lattice torus. The approach uses comparisons between planar and toral hitting times and distributions on annuli, and uses only random walk methods.
We examine escape and entrance times, Green's functions, local times, and hitting distributions of discs and annuli of a symmetric random walk on $\Z^2$ projected onto the periodic lattice $\Z^2_K$. This extends a framework for the simple planar random walk in Dembo, et al. (2006) to the large class of planar random walks in Bass, Rosen (2007). The approach uses comparisons between $\Z^2$ and $\Z^2_K$ hitting times and distributions on annuli, and uses only random walk methods.
Suppose we are given an infinite, finitely generated group $G$ and a transient random walk on the wreath product $(\mathbb{Z}/ 2\mathbb{Z})\wr G$, such that its projection on $G$ is transient and has finite first moment. This random walk can be interpreted as a lamplighter random walk on $G$. Our aim is to show that the random walk on the wreath product escapes to infinity with respect to a suitable (pseudo-)metric faster than its projection onto $G$. We also address the case where the pseudo-metric is the length of a shortest ``travelling salesman tour''. In this context, and excluding some d…
We show that the largest subsurface projection distance between a marking and its image under the nth step of a random walk grows logarithmically in n, with probability approaching 1 as n tends to infinity. Our setup is general and also applies to (relatively) hyperbolic groups and to $\mathrm{Out}(F_n)$. We then use this result to prove Rivin's conjecture that for a random walk $(w_n)$ on the mapping class group, the shortest geodesic in the hyperbolic mapping torus $M_{w_n}$ has length on the order of $1/ \log^2(n)$.
We prove the existence of scaling limits for the projection on the backbone of the random walks on the Incipient Infinite Cluster and the Invasion Percolation Cluster on a regular tree. We treat these projected random walks as randomly trapped random walks (as defined in [BCČR15]) and thus describe these scaling limits as spatially subordinated Brownian motions
The growth exponent $α$ for loop-erased or Laplacian random walk on the integer lattice is defined by saying that the expected time to reach the sphere of radius $n$ is of order $n^α$. We prove that in two dimensions, the growth exponent is strictly greater than one. The proof uses a known estimate on the third moment of the escape probability and an improvement on the discrete Beurling projection theorem.
We study the set of directions asymptotically explored by a spatially homogeneous random walk in $d$-dimensional Euclidean space. We survey some pertinent results of Kesten and Erickson, make some further observations, and present some examples. We also explore links to the asymptotics of one-dimensional projections, and to the growth of the convex hull of the random walk.
We analyze the differences between the horizontal and the vertical component of the simple random walk on the 2-dimensional comb. In particular we evaluate by combinatorial methods the asymptotic behaviour of the expected value of the distance from the origin, the maximal deviation and the maximal span in $n$ steps, proving that for all these quantities the order is $n^{1/4}$ for the horizontal projection and $n^{1/2}$ for the vertical one (the exact constants are determined). Then we rescale the two projections of the random walk dividing by $n^{1/4}$ and $n^{1/2}$ the horizontal and vertical…
Fluctuations for diameter and perimeter of convex hulls of multiple random walks
Institute of Mathematics, University of Wrocław（弗罗茨瓦夫大学数学研究所）；University of Zagreb Faculty of Science（萨格勒布大学理学院）；University of Zagreb Faculty of Electrical Engineering and Computing（萨格勒布大学电气工程和计算学院）；Department of Mathematical Sciences, Durham University（杜伦大学数学科学系）；Department of Mathematics, The University of Hong Kong（香港大学数学系）
We analyze random walks on a class of semigroups called ``left-regular bands''. These walks include the hyperplane chamber walks of Bidigare, Hanlon, and Rockmore. Using methods of ring theory, we show that the transition matrices are diagonalizable and we calculate the eigenvalues and multiplicities. The methods lead to explicit formulas for the projections onto the eigenspaces. As examples of these semigroup walks, we construct a random walk on the maximal chains of any distributive lattice, as well as two random walks associated with any matroid. The examples include a q-analogue of the Tse…
Any Zariski dense countable subgroup of $SL(d,R)$ is shown to carry a non-degenerate finitely supported symmetric random walk such that its harmonic measure on the flag space is singular. The main ingredients of the proof are: (1) a new upper estimate for the Hausdorff dimension of the projections of the harmonic measure onto Grassmannians in $R^d$ in terms of the associated differential entropies and differences between the Lyapunov exponents; (2) an explicit construction of random walks with uniformly bounded entropy and Lyapunov exponents going to infinity.
This paper introduces the Attracting Random Walks model, which describes the dynamics of a system of particles on a graph with $n$ vertices. At each step, a single particle moves to an adjacent vertex (or stays at the current one) with probability proportional to the exponent of the number of other particles at a vertex. From an applied standpoint, the model captures the rich get richer phenomenon. We show that the Markov chain exhibits a phase transition in mixing time, as the parameter governing the attraction is varied. Namely, mixing time is $O(n\log n)$ when the temperature is sufficientl…
In this paper, we give explicit rates in the central limit theorem and in the almost sure invariance principle for general R d-valued cocycles that appear in the study of the left random walk on linear groups. Our method of proof lies on a suitable martingale approximation and on a careful estimation of some coupling coefficients linked with the underlying Markov structure. Concerning the martingale part, the available results in the literature are not accurate enough to give almost optimal rates whether in the central limit theorem for the Wasserstein distance, or in the strong approximation.…
Effective equidistribution of random walks on simple homogeneous spaces
CNRS – LAGA, Université Sorbonne Paris Nord（法国国家科学研究中心拉格朗日分析几何实验室，索邦巴黎北大学）；State Key Laboratory of Mathematical Sciences, Academy of Mathematics and System Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学科学重点实验室）
In this work, we investigate a novel setting of Markovian loop measures and introduce a new class of loop measures called Bosonic loop measures. Namely, we consider loop soups with varying intensity $ μ\le 0 $ (chemical potential in physics terms), and secondly, we study Markovian loop measures on graphs with an additional "time" dimension leading to so-called space-time random walks and their loop measures and Poisson point loop processes. Interesting phenomena appear when the additional coordinate of the space-time process is on a discrete torus with non-symmetric jump rates. The projection …
Let $V=\mathbb R^d$ be the Euclidean $d$-dimensional space, $μ$ (resp $λ$) a probability measure on the linear (resp affine) group $G=G L (V)$ (resp $H= \Aff (V))$ and assume that $μ$ is the projection of $λ$ on $G$. We study asymptotic properties of the iterated convolutions $μ^n *δ\_{v}$ (resp $λ^n*δ\_{v})$ if $v\in V$, i.e asymptotics of the random walk on $V$ defined by $μ$ (resp $λ$), if the subsemigroup $T\subset G$ (resp.\ $Σ\subset H$) generated by the support of $μ$ (resp $λ$) is "large". We show spectral gap properties for the convolution operator defined by $μ$ on spaces of homogene…
Combinatorial Levy processes evolve on general state spaces of countable combinatorial structures. In this setting, the usual Levy process properties of stationary, independent increments are defined in an unconventional way in terms of the symmetric difference operation on sets. In discrete time, the description of combinatorial Levy processes gives rise to the notion of combinatorial random walks. These processes behave differently than random walks and Levy processes on other state spaces. Standard examples include processes on sets, graphs, and n-ary relations, but the framework permits fa…
This paper is dedicated to the investigation of a $1+1$ dimensional self-interacting and partially directed self-avoiding walk, usually referred to by the acronym IPDSAW and introduced in \cite{ZL68} by Zwanzig and Lauritzen to study the collapse transition of an homopolymer dipped in a poor solvant. In \cite{POBG93}, physicists displayed numerical results concerning the typical growth rate of some geometric features of the path as its length $L$ diverges. From this perspective the quantities of interest are the projections of the path onto the horizontal axis (also called horizontal extension…
Ballistic Speed and Potential-Theoretic Recurrence for Homogeneous Open Quantum Random Walks
Politecnico di Milano（米兰理工大学）；University College, Yonsei University（延世大学校）；United Arab Emirates University（阿联酋大学）；Hankyong National University（韩国汉阳女子大学）
Let $ξ_1,ξ_2,\ldots$ be a sequence of independent copies of a random vector in $\mathbb R^d$ having an absolutely continuous distribution. Consider a random walk $S_i:=ξ_1+\cdots+ξ_i$, and let $C_{n,d}:=\text{conv}(0,S_1,S_2,\ldots,S_n)$ be the convex hull of the first $n+1$ points it has visited. The polytope $C_{n,d}$ is called $k$-neighborly if for every indices $0\leq i_0 <\cdots < i_k\leq n$ the convex hull of the $k+1$ points $S_{i_0},\ldots, S_{i_k}$ is a $k$-dimensional face of $C_{n,d}$. We study the probability that $C_{n,d}$ is $k$-neighborly in various high-dimensional asymptotic r…
