We consider shock measures in a class of conserving stochastic particle systems on Z. These shock measures have a product structure with a step-like density profile and include a second class particle at the shock position. We show for the asymmetric simple exclusion process, for the exponential bricklayers' process, and for a generalized zero range process, that under certain conditions these shocks, and therefore the second class particles, perform a simple random walk. Some previous results, including random walks of product shock measures and stationary shock measures seen from a second class particle, are direct consequences of our more general theorem. Multiple shocks can also be handled easily in this framework. Similar shock structure is also found in a nonconserving model, the branching coalescing random walk, where the role of the second class particle is played by the rightmost (or leftmost) particle.
... but no other surprises in the zero range world. We check all nearest neighbour 1-dimensional asymmetric zero range processes for random walking product shock measures as demonstrated already for a few cases in the literature. We find the totally asymmetric version of the celebrated q-zero range process as the only new example besides an already known model of doubly infinite occupation numbers and exponentially increasing jump rates. We also examine the interaction of shocks, which appears somewhat more involved for q-zero range than in the already known cases.
We study the time evolution of the ASEP on a one-dimensional torus with $L$ sites, conditioned on an atypically low current up to a finite time $t$. For a certain one-parameter family of initial measures with a shock we prove that the shock position performs a biased random walk on the torus and that the measure seen from the shock position remains invariant. We compute explicitly the transition rates of the random walk. For the large scale behaviour this result suggests that there is an atypically low current such that the optimal density profile that realizes this current is a hyperbolic tan…
We derive Donsker-Vardhan type results for functionals of the occupation times when the underlying random walk on $\mathbb Z^d$ is in the domain of attraction of an operator-stable law on $\mathbb R^d$. Applications to random walks on wreath products (also known as lamplighter groups) are given.
In this article we consider transient random walks on free products of graphs. We prove that the asymptotic range of these random walks exists and is strictly positive. In particular, we show that the range varies real-analytically in terms of probability measures of constant support. Moreover, we prove a central limit theorem associated with the range of the random walk.
In this article, we consider products of random walks on finite groups with moderate growth and discuss their cutoffs in the total variation. Based on several comparison techniques, we are able to identify the total variation cutoff of discrete time lazy random walks with the Hellinger distance cutoff of continuous time random walks. Along with the cutoff criterion for Laplace transforms, we derive a series of equivalent conditions on the existence of cutoffs, including the existence of pre-cutoffs, Peres' product condition and a formula generated by the graph diameters. For illustration, we c…
By developing the entropy theory of random walks on equivalence relations and analyzing the asymptotic geometry of horospheric products we describe the Poisson boundary for random walks on random horospheric products of trees.
This paper is an appendix to the paper "Random walks on free products of cyclic groups" by J.Mairesse and F.Mathéus. It contains the details of the computations and the proofs of the results concerning the examples treated there.
Suppose we are given the free product V of a finite family of finite or countable sets. We consider a transient random walk on the free product arising naturally from a convex combination of random walks on the free factors. We prove the existence of the asymptotic entropy and present three different, equivalent formulas, which are derived by three different techniques. In particular, we will show that the entropy is the rate of escape with respect to the Greenian metric. Moreover, we link asymptotic entropy with the rate of escape and volume growth resulting in two inequalities.
We consider random walks on the set of all words over a finite alphabet such that in each step only the last two letters of the current word may be modified and only one letter may be adjoined or deleted. We assume that the transition probabilities depend only on the last two letters of the current word. Furthermore, we consider also the special case of random walks on free products by amalgamation of finite groups which arise in a natural way from random walks on the single factors. The aim of this paper is to compute several equivalent formulas for the rate of escape with respect to natural …
We prove existence of the large deviation principle, with a proper convex rate function, for the distribution of the renormalized distance from the origin of a random walk on a free product of finitely generated groups. As a consequence, we derive the same principle for nearest-neighbour random walks on regular trees.
We prove bounded rational ergodicity for some discrepancy skew products whose rotation number has bad rational approximation. This is done by considering the asymptotics of associated affine random walks.
We bound the rate of convergence to uniformity for certain random walks on the complete monomial groups G \wr S_n for any group G. These results provide rates of convergence for random walks on a number of groups of interest: the hyperoctahedral group Z_2 \wr S_n, the generalized symmetric group Z_m \wr S_n, and S_m \wr S_n. These results provide benchmarks to which many other random walks, modeling a wide range of phenomena, may be compared using the comparison technique, thereby yielding bounds on the rates of convergence to uniformity for previously intractable random walks.
We consider nearest-neighbor random walks on free products of finitely many copies of the integers with amalgamation over nontrivial subgroups. When all the subgroups have index two, we find the Green function of the random walks in terms of complete elliptic integrals. Our technique is to apply Voiculescu's operator--valued R--transform.
Formulas are derived for counting walks in the Kronecker product of graphs, and the associated spectral distributions are obtained by the Mellin convolution of probability distributions. Two-dimensional restricted lattices admitting the Kronecker product structure are listed, and their spectral distributions are calculated in terms of elliptic integrals.
Suppose we are given the free product $V$ of a finite family of finite or countable sets $(V_i)_{i\in\mathcal{I}}$ and probability measures on each $V_i$, which govern random walks on it. We consider a transient random walk on the free product arising naturally from the random walks on the $V_i$. We prove the existence of the rate of escape with respect to the block length, that is, the speed, at which the random walk escapes to infinity, and furthermore we compute formulas for it. For this purpose, we present three different techniques providing three different, equivalent formulas.
We find uniform lower bounds on the drift for a large family of random walks on graph products, of the form $ \mathbb{P} (|Z_{n}| \leq κn) \leq e ^{-κn} $ for $ κ> 0 $. This includes the simple random walk for a right-angled Artin group with a sparse defining graph. This is done by extending an argument of Gouëzel, along with the combinatorial notion of a piling introduced by Crisp, Godelle, and Wiest. We do not use any moment conditions, instead considering random walks which alternate between one measure uniformly distributed on vertex groups, and another measure over which we make almost no…
The connective constant $μ(G)$ of a graph $G$ is the asymptotic growth rate of the number $σ_{n}$ of self-avoiding walks of length $n$ in $G$ from a given vertex. We prove a formula for the connective constant for free products of quasi-transitive graphs and show that $σ_{n}\sim A_{G} μ(G)^{n}$ for some constant $A_{G}$ that depends on $G$. In the case of finite products $μ(G)$ can be calculated explicitly and is shown to be an algebraic number.
Suppose we are given finitely generated groups $Γ_1,...,Γ_m$ equipped with irreducible random walks. Thereby we assume that the expansions of the corresponding Green functions at their radii of convergence contain only logarithmic or algebraic terms as singular terms up to sufficiently large order (except for some degenerate cases). We consider transient random walks on the free product {$Γ_1 \ast ... \astΓ_m$} and give a complete classification of the possible asymptotic behaviour of the corresponding $n$-step return probabilities. They either inherit a law of the form $\varrho^{nδ} n^{-λ_i} …
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…
Suppose that G and H are finite, connected graphs, G regular, X is a lazy random walk on G and Z is a reversible ergodic Markov chain on H. The generalized lamplighter chain X* associated with X and Z is the random walk on the wreath product H\wr G, the graph whose vertices consist of pairs (f,x) where f=(f_v)_{v\in V(G)} is a labeling of the vertices of G by elements of H and x is a vertex in G. In each step, X* moves from a configuration (f,x) by updating x to y using the transition rule of X and then independently updating both f_x and f_y according to the transition probabilities on H; f_z…
Wildberger's construction enables us to obtain a hypergroup from a special graph via random walks. We will give a probability theoretic interpretation to products on the hypergroup. The hypergroup can be identified with a commutative algebra whose basis is transition matrices. We will estimate the operator norm of such a transition matrix and clarify a relationship between their matrix products and random walks.
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…
Let $A_kA_{k-1}\cdots A_1$ be product of some nonnegative 2-by-2 matrices. In general, its elements are hard to evaluate. Under some conditions, we show that $\forall i,j\in\{1,2\},$ $(A_kA_{k-1}\cdots A_1)_{i,j}\sim c\varrho(A_k)\varrho(A_{k-1})\cdots \varrho(A_1)$ as $k\rightarrow\infty,$ where $\varrho(A_n)$ is the spectral radius of the matrix $A_n$ and $c\in(0,\infty)$ is some constant, so that the elements of $A_kA_{k-1}\cdots A_1$ can be estimated. As applications, consider the maxima of certain excursions of (2,1) and (1,2) random walks with asymptotically zero drifts. We get some deli…
We consider random walk $(X_n)_{n\geq0}$ on $\mathbb{Z}^d$ in a space--time product environment $ω\inΩ$. We take the point of view of the particle and focus on the environment Markov chain $(T_{n,X_n}ω)_{n\geq0}$ where $T$ denotes the shift on $Ω$. Conditioned on the particle having asymptotic mean velocity equal to any given $ξ$, we show that the empirical process of the environment Markov chain converges to a stationary process $μ_ξ^{\infty}$ under the averaged measure. When $d\geq3$ and $ξ$ is sufficiently close to the typical velocity, we prove that averaged and quenched large deviations a…
In this paper, a complex-valued measure of bi-product path space induced by quantum walk is presented. In particular, we consider three types of conditional return paths in a power set of the bi-product path space (1) $Λ\times Λ$, (2) $Λ\times Λ'$ and (3) $Λ'\times Λ'$, where $Λ$ is the set of all $2n$-length ($n\in \mathbb{N}$) return paths and $Λ'(\subseteq Λ)$ is the set of all $2n$-length return paths going through $nx$ ($x\in [-1,1]$) at time $n$. We obtain asymptotic behaviors of the complex-valued measures for the situations (1)-(3) which imply two kinds of weak convergence theorems (Th…
In bricklayers' model, which is a generalization of the misanthrope processes, we show that a nontrivial class of product distributions is closed under the time-evolution of the process. This class also includes measures fitting to shock data of the limiting PDE. In particular, we show that shocks of this type with discontinuity of size one perform ordinary nearest neighbor random walks only interacting, in an attractive way, via their jump rates. Our results are related to those of Belitsky and Schuetz on the simple exclusion process, although we do not use quantum formalism as they do. The s…
We consider a non-nestling random walk in a product random environment. We assume an exponential moment for the step of the walk, uniformly in the environment. We prove an invariance principle (functional central limit theorem) under almost every environment for the centered and diffusively scaled walk. The main point behind the invariance principle is that the quenched mean of the walk behaves subdiffusively.
We discuss recurrence and ergodicity properties of random walks and associated skew products for large classes of locally compact groups and homogeneous spaces. In particular we show that a closed subgroup of a product of finitely many linear groups over local fields supports a recurrent random walk if and only if it has at most quadratic growth. We give also a detailed analysis of ergodicity properties for special classes of random walks on homogeneous spaces. The structure of closed subgroups of linear groups over local fields and the properties of group actions with respect to stationary me…
Crystal lattices are known to be one of the generalizations of classical periodic lattices which can be embedded into some Euclidean spaces properly. As to make a wide range of multidimensional discrete distributions on Euclidean spaces more treatable, multidimensional Euler products and multidimensional Shintani zeta functions on crystal lattices are introduced. They are completely different from existing Ihara zeta functions on graphs in that our zeta functions are defined on crystal lattices directly. Via a concept of periodic realizations of crystal lattices, we make it possible to provide…
We study the neighborhoods of a typical point $Z_n$ visited at $n$-th step of a random walk, determined by the condition that the transition probabilities stay close to $μ^{*n}(Z_n)$. If such neighborhood contains a ball of radius $C \sqrt{n}$, we say that the random walk has almost invariant transition probabilities. We prove that simple random walks on wreath products of $\mathbb{Z}$ with finite groups have almost invariant distributions. A weaker version of almost invariance implies a necessary condition of Ozawa's criterion for the property $H_{\rm FD}$. We define and study the radius of a…
Considering the hydrodynamical limit of some interacting particle systems leads to hyperbolic differential equation for the conserved quantities, e.g. the inviscid Burgers equation for the simple exclusion process. The physical solutions of these partial differential equations develop discontinuities, called shocks. The microscopic structure of these shocks is of much interest and far from being well understood. We introduce a domain growth model in which we find a stationary (in time) product measure for the model, as seen from a defect tracer or second class particle, travelling with the sho…
Given a probability measure on a finitely generated group, its Martin boundary is a way to compactify the group using the Green function of the corresponding random walk. It is known from the work of W. Woess that when a finitely supported random walk on a free product of abelian groups is adapted to the free product structure, the Martin boundary coincides with the geometric boundary. The main goal of this paper is to deal with non-adapted finitely supported random walks, for which there is no explicit formula for the Green function. Nevertheless, we show that the Martin boundary still coinci…
Consider a random walk among random conductances on $\mathbb{Z}^d$ with $d\geq 2$. We study the quenched limit law under the usual diffusive scaling of the random walk conditioned to have its first coordinate positive. We show that the conditional limit law is the product of a Brownian meander and a $(d-1)$-dimensional Brownian motion.
We prove a Law of Iterated Logarithm for random walks on a family of diagonal products constructed by Brieussel and Zheng (2021). This provides a wide variety of new examples of Law of Iterated Logarithm behaviours for random walks on groups. In particular, it follows that for any $\frac{1}{2}\leq β\leq 1$ there is a group $G$ and random walk $W_n$ on $G$ with $\mathbb{E}|W_n|\simeq n^β$ such that $$0<\limsup \frac{|W_n|}{n^β(\log\log n)^{1-β}}<\infty$$ and $$0<\liminf \frac{|W_n|(\log\log n)^{1-β}}{n^β}<\infty.$$
We study the isoperimetric profiles of certain families of finitely generated groups defined via marked Schreier graphs and permutation wreath products. The groups we study are among the "simplest" examples within a much larger class of groups, all defined via marked Schreier graphs and/or action on rooted trees, which includes such examples as the long range group, Grigorchuck group and the basillica group. The highly non-linear structure of these groups make them both interesting and difficult to study. Because of the relative simplicity of the Schreier graphs that define the groups we study…
We study certain phase transitions of branching random walks (BRW) on Cayley graphs of free products. The aim of this paper is to compare the size and structural properties of the trace, i.e., the subgraph that consists of all edges and vertices that were visited by some particle, with those of the original Cayley graph. We investigate the phase when the growth parameter $λ$ is small enough such that the process survives but the trace is not the original graph. A first result is that the box-counting dimension of the boundary of the trace exists, is almost surely constant and equals the Hausdo…
We determine all positive harmonic functions for a large class of "semi-isotropic" random walks on the lamplighter group, i.e., the wreath product of the cyclic group of order q with the infinite cyclic group. This is possible via the geometric realization of a Cayley graph of that group as the Diestel-Leader graph DL(q,q). More generally, DL(q,r) is the horocyclic product of two homogeneous trees with respective degrees $q+1$ and $r+1$, and our result applies to all DL-graphs. This is based on a careful study of the minimal harmonic functions for semi-isotropic walks on trees.
We consider a multidimensional random walk in a product random environment with bounded steps, transience in some spatial direction, and high enough moments on the regeneration time. We prove an invariance principle, or functional central limit theorem, under almost every environment for the diffusively scaled centered walk. The main point behind the invariance principle is that the quenched mean of the walk behaves subdiffusively.
We consider a random walk among a Poisson system of moving traps on ${\mathbb Z}$. In earlier work [DGRS12], the quenched and annealed survival probabilities of this random walk have been investigated. Here we study the path of the random walk conditioned on survival up to time $t$ in the annealed case and show that it is subdiffusive. As a by-product, we obtain an upper bound on the number of so-called thin points of a one-dimensional random walk, as well as a bound on the total volume of the holes in the random walk's range.
We prove the law of large numbers for the drift of random walks on the two-dimensional lamplighter group, under the assumption that the random walk has finite $(2+ε)$-moment. This result is in contrast with classical examples of abelian groups, where the displacement after $n$ steps, normalised by its mean, does not concentrate, and the limiting distribution of the normalised $n$-step displacement admits a density whose support is $[0,\infty)$. We study further examples of groups, some with random walks satisfying LLN for drift and other examples where such concentration phenomenon does not ho…
We determine the precise asymptotic behaviour (in space) of the Green kernel of simple random walk with drift on the Diestel-Leader graph $DL(q,r)$, where $q,r \ge 2$. The latter is the horocyclic product of two homogeneous trees with respective degrees $q+1$ and $r+1$. When $q=r$, it is the Cayley graph of the wreath product (lamplighter group) ${\mathbb Z}_q \wr {\mathbb Z}$ with respect to a natural set of generators. We describe the full Martin compactification of these random walks on $DL$-graphs and, in particular, lamplighter groups. This completes and provides a better approach to prev…
We give a simple and direct treatment of the strong convergence of quantum random walks to quantum stochastic operator cocycles, via the semigroup decomposition of such cocycles. Our approach also delivers convergence of the pointwise product of quantum random walks to the quantum stochastic Trotter product of the respective limit cocycles, thereby revealing the algebraic structure of the limiting procedure. The repeated quantum interactions model is shown to fit nicely into the convergence scheme described.
The lamplighter group over $\mathbb Z$ is the wreath product $\mathbb Z_q \wr \mathbb Z$. With respect to a natural generating set, its Cayley graph is the Diestel-Leader graph $DL(q,q)$. We study harmonic functions for the "simple" Laplacian on this graph, and more generally, for a class of random walks on $DL(q,r)$, where $q,r \ge 2$. The DL-graphs are horocyclic products of two trees, and we give a full description of all positive harmonic functions in terms of the boundaries of these two trees. In particular, we determine the minimal Martin boundary, that is, the set of minimal positive ha…
We study the cut-off phenomenon for random walks on free unitary quantum groups coming from quantum conjugacy classes of classical reflections. We obtain in particular a quantum analogue of the result of U. Porod concerning certain mixtures of such reflections. We also study random walks on quantum reflection groups and more generally free wreath products of finite group by quantum permutation groups.
Fix a finite set $S \subset {GL}(k,\mathbb{Z})$. Denote by $a_n$ the number of products of matrices in $S$ of length $n$ that are equal to 1. We show that the sequence $\{a_n\}$ is not always P-recursive. This answers a question of Kontsevich.
We show that the displacement and translation distance of non-elementary random walks on isometry groups of hyperbolic spaces satisfy large deviation principles with the same rate function $I$. Roughly, this means that there exists function $I(t)$ which accurately predicts the exponential decay rate of the probability that the translation distance of a random product of length $n$ is $tn$, and similarly for the displacement. This settles a special case of a conjecture concerning the large deviation principle for the spectral radius of random matrix products. In a second part, we give a charact…
