Our paper gives bounds for the rate of convergence for a class of random walks on the d-dimensional torus generated by a set of n vectors in R^d/Z^d. We give bounds on the discrepancy distance from Haar measure; our lower bound holds for all such walks, and if the generators arise from the rows of a "badly approximable" matrix, then there is a corresponding upper bound. The bounds are sharp for walks on the circle.
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 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 consider one or more independent random walks on the $d\ge 3$ dimensional discrete torus. The walks start from vertices chosen independently and uniformly at random. We analyze the fluctuation behavior of the size of some random sets arising from the trajectories of the random walks at a time proportional to the size of the torus. Examples include vacant sets and the intersection of ranges. The proof relies on a refined analysis of tail estimates for hitting time and can be applied for other vertex-transitive graphs.
We study the mixing time of a random walk on the torus, alternated with a Lebesgue measure preserving Bernoulli map. Without the Bernoulli map, the mixing time of the random walk alone is $O(1/ε^2)$, where $ε$ is the step size. Our main results show that for a class of Bernoulli maps, when the random walk is alternated with the Bernoulli map $φ$ the mixing time becomes $O(|\ln ε|)$. We also study the \emph{dissipation time} of this process, and obtain $O(|\ln ε|)$ upper and lower bounds with explicit constants.
For $d\ge 3$ we construct a new coupling of the trace left by a random walk on a large $d$-dimensional discrete torus with the random interlacements on $\mathbb Z^d$. This coupling has the advantage of working up to macroscopic subsets of the torus. As an application, we show a sharp phase transition for the diameter of the component of the vacant set on the torus containing a given point. The threshold where this phase transition takes place coincides with the critical value $u_*(d)$ of random interlacements on $\mathbb Z^d$. Our main tool is a variant of the soft-local time coupling techniqu…
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.
We investigate the relation between the local picture left by the trajectory of a simple random walk on the torus (Z/NZ)^d, d >= 3, until u N^d time steps, u > 0, and the model of random interlacements recently introduced by Sznitman. In particular, we show that for large N, the joint distribution of the local pictures in the neighborhoods of finitely many distant points left by the walk up to time u N^d converges to independent copies of the random interlacement at level u.
Let a simple random walk run inside a torus of dimension three or higher for a number of steps which is a constant proportion of the volume. We examine geometric properties of the range, the random subgraph induced by the set of vertices visited by the walk. Distance and mixing bounds for the typical range are proven that are a $k$-iterated log factor from those on the full torus for arbitrary $k$. The proof uses hierarchical renormalization and techniques that can possibly be applied to other random processes in the Euclidean lattice. We use the same technique to bound the heat kernel of a ra…
Let x and y be points chosen uniformly at random from $\Z_n^4$, the four-dimensional discrete torus with side length n. We show that the length of the loop-erased random walk from x to y is of order $n^2 (\log n)^{1/6}$, resolving a conjecture of Benjamini and Kozma. We also show that the scaling limit of the uniform spanning tree on $\Z_n^4$ is the Brownian continuum random tree of Aldous. Our proofs use the techniques developed by Peres and Revelle, who studied the scaling limits of the uniform spanning tree on a large class of finite graphs that includes the d-dimensional discrete torus for…
Consider the subgraph of the discrete $d$-dimensional torus of size length $N$, $d\ge3$, induced by the range of the simple random walk on the torus run until the time $uN^d$. We prove that for all $d\ge 3$ and $u>0$, the mixing time for the random walk on this subgraph is of order $N^2$ with probability at least $1 - Ce^{-(\log N)^2}$.
Let $ρ$ be a probability measure on $\mathrm{SL}\_d(\mathbb{Z})$ and consider the random walk defined by $ρ$ on the torus $\mathbb{T}^d = \mathbb{R}^d/\mathbb{Z}^d$. Bourgain, Furmann, Lindenstrauss and Mozes proved that under an assumption on the group generated by the support of $$ρ$$, the random walk starting at any irrational point equidistributes in the torus. In this article, we study the central limit theorem and the law of the iterated logarithm for this walk starting at some point having good diophantine properties.
We prove that the scaling limit of the weakly self-avoiding walk on a $d$-dimensional discrete torus is Brownian motion on the continuum torus if the length of the rescaled walk is $o(V^{1/2})$ where $V$ is the volume (number of points) of the torus and if $d>4$. We also prove that the diffusion constant of the resulting torus Brownian motion is the same as the diffusion constant of the scaling limit of the usual weakly self-avoiding walk on $\mathbb{Z}^d$. This provides further manifestation of the fact that the weakly self-avoiding walk model on the torus does not feel that it is on the toru…
We introduce a model of branching interlacements for general critical offspring distributions. It consists of a countable collection of infinite tree-indexed random walk trajectories on $Z^d,d\geq5$. We show that this model turns out to be the local limit of the tree-indexed random walk in a discrete torus, conditioned on the size proportional to the volume of the torus. This generalizes the previous results of Angel, Ráth and the author, for the critical geometric offspring distribution. Our model also includes the model of random interlacements introduced by Sznitman as a degenerate case. To…
This work continues the investigation, initiated in a recent work by Benjamini and Sznitman, of percolative properties of the set of points not visited by a random walk on the discrete torus (Z/NZ)^d up to time uN^d in high dimension d. If u>0 is chosen sufficiently small it has been shown that with overwhelming probability this vacant set contains a unique giant component containing segments of length c_0 log N for some constant c_0 > 0, and this component occupies a non-degenerate fraction of the total volume as N tends to infinity. Within the same setup, we investigate here the complement o…
Sharp estimates for the length of loop erased random walk between two vertices on the [n]^d -torus, d > 4, are established. The mean length is order n^{d/2} . In dimension 4 we have only an upper bound.
A small world is obtained from the $d$-dimensional torus of size 2L adding randomly chosen connections between sites, in a way such that each site has exactly one random neighbour in addition to its deterministic neighbours. We study the asymptotic behaviour of the meeting time $T_L$ of two random walks moving on this small world and compare it with the result on the torus. On the torus, in order to have convergence, we have to rescale $T_L$ by a factor $C_1L^2$ if $d=1$, by $C_2L^2\log L$ if $d=2$ and $C_dL^d$ if $d\ge3$. We prove that on the small world the rescaling factor is $C^\prime_dL^d…
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…
This work proves that the fluctuations of the cover time of simple random walk in the discrete torus of dimension at least three with large side-length are governed by the Gumbel extreme value distribution. This result was conjectured for example in the book by Aldous & Fill. We also derive some corollaries which qualitatively describe "how" covering happens. In addition, we develop a new and stronger coupling of the model of random interlacements, introduced by Sznitman, and random walk in the torus. This coupling is used to prove the cover time result and is also of independent interest.
The main result of this paper is a decomposition theorem for a measure on the one-dimensional torus. Given a sufficiently large subset $S$ of the positive integers, an arbitrary measure on the torus is decomposed as the sum of two measures. The first one $μ_1$ has the property that the random walk with initial distribution $μ_1$ evolved by the action of $S$ equidistributes very fast. The second measure $μ_2$ in the decomposition is concentrated on very small neighborhoods of a small number of points.
Let $\mathbb{T}^d_N$, $d\ge 2$, be the discrete $d$-dimensional torus with $N^d$ points. Place a particle at each site of $\mathbb{T}^d_N$ and let them evolve as independent, nearest-neighbor, symmetric, continuous-time random walks. Each time two particles meet, they coalesce into one. Denote by $C_N$ the first time the set of particles is reduced to a singleton. Cox [6] proved the existence of a time-scale $θ_N$ for which $C_N/θ_N$ converges to the sum of independent exponential random variables. Denote by $Z^N_t$ the total number of particles at time $t$. We prove that the sequence of Marko…
We use the lace expansion to study the long-distance decay of the two-point function of weakly self-avoiding walk on the integer lattice $\mathbb{Z}^d$ in dimensions $d>4$, in the vicinity of the critical point, and prove an upper bound $|x|^{-(d-2)}\exp[-c|x|/ξ]$, where the correlation length $ξ$ has a square root divergence at the critical point. As an application, we prove that the two-point function for weakly self-avoiding walk on a discrete torus in dimensions $d>4$ has a "plateau." We also discuss the significance and consequences of the plateau for the analysis of critical behaviour on…
Let $\Cal S$ be an abelian group of automorphisms of a probability space $(X, {\Cal A}, μ)$ with a finite system of generators $(A_1, ..., A_d)$. Let $A^{\el}$ denote $A_1^{\ell_1} ... A_d^{\ell_d}$, for ${\el}= (\ell_1, ..., \ell_d)$. If $(Z_k)$ is a random walk on $\Z^d$, one can study the asymptotic distribution of the sums $\sum_{k=0}^{n-1} \, f \circ A^{\,{Z_k(ω)}}$ and $\sum_{\el \in \Z^d} \PP(Z_n= \el) \, A^\el f$, for a function $f$ on $X$. In particular, given a random walk on commuting matrices in $SL(ρ, \Z)$ or in ${\Cal M}^*(ρ, \Z)$ acting on the torus $\T^ρ$, $ρ\geq 1$, what is th…
We consider random walk on a discrete torus E of side-length N, in sufficiently high dimension d. We investigate the percolative properties of the vacant set corresponding to the collection of sites which have not been visited by the walk up to time uN^d. We show that when u is chosen small, as N tends to infinity, there is with overwhelming probability a unique connected component in the vacant set which contains segments of length const log N. Moreover, this connected component occupies a non-degenerate fraction of the total number of sites N^d of E, and any point of E lies within distance a…
We consider a random walk $S_k$ with i.i.d. steps on a compact group equipped with a bi-invariant metric. We prove quantitative ergodic theorems for the sum $\sum_{k=1}^N f(S_k)$ with Hölder continuous test functions $f$, including the central limit theorem, the law of the iterated logarithm and an almost sure approximation by a Wiener process, provided the distribution of $S_k$ converges to the Haar measure in the $p$-Wasserstein metric fast enough. As an example we construct discrete random walks on an irrational lattice on the torus $\mathbb{R}^d/\mathbb{Z}^d$, and find their precise rate o…
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 …
In this paper, following the recent paper on Walk/Zeta Correspondence by the first author and his coworkers, we compute the zeta function for the three- and four-state quantum walk and correlated random walk, and the multi-state random walk on the one-dimensional torus by using the Fourier analysis. We deal with also the four-state quantum walk and correlated random walk on the two-dimensional torus. In addition, we introduce a new class of models determined by the generalized Grover matrix bridging the gap between the Grover matrix and the positive-support of the Grover matrix. Finally, we gi…
We consider a variant of random walks on finite groups. At each step, we choose an element from a set of generators ("directions") uniformly, and an integer from a power law ("speed") distribution associated with the chosen direction. We show that if the finite group is nilpotent, the mixing time of this walk is of the same order of magnitude as the diameter of a suitable pseudo-metric, D(S, a), on the group, which depends only on the generators and speeds. Additionally, we give sharp bounds on the $\ell^2$-distance between the distribution of the position of the walker and the stationary dist…
We study the mixing time of a random walker who moves inside a dynamical random cluster model on the d-dimensional torus of side-length n. In this model, edges switch at rate μbetween open and closed, following a Glauber dynamics for the random cluster model with parameters p,q. At the same time, the walker jumps at rate 1 as a simple random walk on the torus, but is only allowed to traverse open edges. We show that for small enough p the mixing time of the random walker is of order n^2/μ. In our proof we construct of a non-Markovian coupling through a multi-scale analysis of the environment, …
We consider random walk on dynamical percolation on the discrete torus $\mathbb{Z}_n^d$. In previous work, mixing times of this process for $p<p_c(\mathbb{Z}^d)$ were obtained in the annealed setting where one averages over the dynamical percolation environment. Here we study exit times in the quenched setting, where we condition on a typical dynamical percolation environment. We obtain an upper bound for all $p$ which for $p<p_c$ matches the known lower bound.
Let $\mathcal{T}_n$ be the cover time of two-dimensional discrete torus $\mathbb{Z}^2_n=\mathbb{Z}^2/n\mathbb{Z}^2$. We prove that $\mathbb{P}[\mathcal{T}_n\leq \frac{4}πγn^2\ln^2 n]=\exp(-n^{2(1-\sqrtγ)+o(1)})$ for $γ\in (0,1)$. One of the main methods used in the proofs is the decoupling of the walker's trace into independent excursions by means of soft local times.
We obtain the leading orders of the maximum and the minimum of local times for the simple random walk on the two-dimensional torus at time proportional to the cover time. We also estimate the number of points with large (or small) values of the local times. These are analogues of estimates on the two-dimensional Gaussian free fields by Bolthausen, Deuschel, and Giacomin [Ann. Probab., 29 (2001)] and Daviaud [Ann. Probab., 34 (2006)], but we have different exponents from the case of the Gaussian free field.
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…
We consider the cover time for a simple random walk on the two-dimensional discrete torus of side length $n$. Dembo, Peres, Rosen, and Zeitouni [Ann. Math. 160:433-464, 2004] identified the leading term in the asymptotics for the cover time as $n$ goes to infinity. In this paper, we study the exact second order term. This is a discrete analogue of the work on the cover time for planar Brownian motion by Belius and Kistler [Probab. Theory Relat Fields. 167:461-552, 2017].
We consider dynamical percolation on the $d$-dimensional discrete torus of side length $n$, $\mathbb{Z}_n^d$, where each edge refreshes its status at rate $μ=μ_n\le 1/2$ to be open with probability $p$. We study random walk on the torus, where the walker moves at rate $1/(2d)$ along each open edge. In earlier work of two of the authors with A. Stauffer, it was shown that in the subcritical case $p<p_c(\mathbb{Z}^d)$, the (annealed) mixing time of the walk is $Θ(n^2/μ)$, and it was conjectured that in the supercritical case $p>p_c(\mathbb{Z}^d)$, the mixing time is $Θ(n^2+1/μ)$; here the implie…
Using the generators, we establish a connection between the Sinai's random walk and the so-called Brox process. We first find the Dirichlet form of the Brox diffusion, and then prove that it is the limit of the Dirichlet form of the Sinai's random walk. This also gives a natural way to connect between the Brox diffusion and the Brownian motion.
We study the behavior of random walk on dynamical percolation. In this model, the edges of a graph G are either open or closed and refresh their status at rate μ while at the same time a random walker moves on G at rate 1 but only along edges which are open. On the d-dimensional torus with side length n, we prove that in the subcritical regime, the mixing times for both the full system and the random walker are n^2/μ up to constants. We also obtain results concerning mean squared displacement and hitting times. Finally, we show that the usual recurrence transience dichotomy for the lattice Z^d…
We provide an analysis of the expected meeting time of two independent random walks on a regular graph. For 1-D circle and 2-D torus graphs, we show that the expected meeting time can be expressed as the sum of the inverse of non-zero eigenvalues of a suitably defined Laplacian matrix. We also conjecture based on empirical evidence that this result holds more generally for simple random walks on arbitrary regular graphs. Further, we show that the expected meeting time for the 1-D circle of size $N$ is $Θ(N^2)$, and for a 2-D $N \times N$ torus it is $Θ(N^2 log N)$.
This paper focuses on the problem of modeling for small world effect on complex networks. Let's consider the supercritical Poisson continuous percolation on $d$-dimensional torus $T^d_n$ with volume $n^d$. By adding "long edges (short cuts)" randomly to the largest percolation cluster, we obtain a random graph $\mathscr G_n$. In the present paper, we first prove that the diameter of $\mathscr G_n$ grows at most polynomially fast in $\ln n$ and we call it the Poisson Geometry Small World. Secondly, we prove that the random walk on $\mathscr G_n$ possesses the rapid mixing property, namely, the …
Let x and y be chosen uniformly in a graph G. We find the limiting distribution of the length of a loop-erased random walk from x to y on a large class of graphs that include the discrete torus in dimensions 5 and above. Moreover, on this family of graphs we show that a suitably normalized finite-dimensional scaling limit of the uniform spanning tree is a Brownian continuum random tree.
Let $(G_n)_{n=1}^{\infty}$ be a sequence of finite graphs, and let Y_t be the length of a loop-erased random walk on G_n after t steps. We show that for a large family of sequences of finite graphs, which includes the case in which G_n is the d-dimensional torus of size-length n for $d \geq 4$, the process $(Y_t)_{t=0}^{\infty}$, suitably normalized, converges to the Rayleigh process introduced by Evans, Pitman, and Winter. Our proof relies heavily on ideas of Peres and Revelle, who used loop-erased random walks to show that the uniform spanning tree on large finite graphs converges to the Bro…
We consider the model of random walk on dynamical percolation introduced by Peres, Stauffer and Steif (2015). We obtain comparison results for this model for hitting and mixing times and for the spectral-gap and log-Sobolev constant with the corresponding quantities for simple random walk on the underlying graph $G$, for general graphs. When $G$ is the torus $\mathbb{Z}_n^d$, we recover the results of Peres et al. and we also extend them to the critical case. We also obtain bounds in the cases where $G$ is a transitive graph of moderate growth and also when it is the hypercube.
We define the model of two-dimensional random interlacements using simple random walk trajectories conditioned on never hitting the origin, and then obtain some properties of this model. Also, for random walk on a large torus conditioned on not hitting the origin up to some time proportional to the mean cover time, we show that the law of the vacant set around the origin is close to that of random interlacements at the corresponding level. Thus, this new model provides a way to understand the structure of the set of late points of the covering process from a microscopic point of view.
For any finitely generated group G, let n ---> Φ_G(n) be the function that describes the rough asymptotic behavior of the probability of return to the identity element at time 2n of a symmetric simple random walk on G (this is an invariant of quasi-isometry). We determine this function when G is the free solvable group S_{d,r} of derived length d on r generators and some other related groups.
It is well known that adding "long edges (shortcuts)" to a regularly constructed graph will make the resulted model a small world. Recently, \cite{W} indicated that, among all long edges, those edges with length proportional to the diameter of the regularly constructed graph may play the key role. In this paper, we modify the original Newman-Watts small world by adding only long special edges to the $d$-dimensional lattice torus (with size $n^d$) according to \cite{W}, and show that the diameter of the modified model and the mixing time of random walk on it grow polynomially fast in $\ln n$.
Let G be a free product of a finite family of finite groups, with the set of generators being formed by the union of the finite groups. We consider a transient nearest-neighbour random walk on G. We give a new proof of the fact that the harmonic measure is a special Markovian measure entirely determined by a finite set of polynomial equations. We show that in several simple cases of interest, the polynomial equations can be explicitely solved, to get closed form formulas for the drift. The examples considered are the modular group Z/2Z*Z/3Z, Z/3Z*Z/3Z, Z/kZ*Z/kZ, and the Hecke groups Z/2Z*Z/kZ…
We introduce a new class of asymmetric random walks on the one-dimensional infinite lattice. In this walk the direction of the jumps (positive or negative) is determined by a discrete-time renewal process which is independent of the jumps. We call this discrete-time counting process the `it generator process' of the walk. We refer the so defined walk to as `Asymmetric Discrete-Time Random Walk' (ADTRW). We highlight connections of the waiting-time density generating functions with Bell polynomials. We derive the discrete-time renewal equations governing the time-evolution of the ADTRW and anal…
We show that for the simple random walk on a Coxeter group generated by the Coxeter generators and identity, the likelihoods of being at any pair of states respect the weak Bruhat order. That is, after any number of steps, the most likely element is the identity, probabilities decrease along any geodesic from the identity, and the least likely element is the longest element, if the group is finite. The result remains true when different generators have different probabilities, so long as the identity is at least as likely as any other.
We consider a 1-dimensional Brownian motion whose diffusion coefficient varies when it crosses the origin. We study the long time behavior and we establish different regimes, depending on the variations of the diffusion coefficient: emergence of a non-Gaussian mul-tipeaked probability distribution and a dynamical transition to an absorbing static state. We compute the generator and we study the partial differential equation which involves its adjoint. We discuss global existence and blow-up of the solution to this latter equation.
