Simple Random Walk Research Articles

Continuity for the asymptotic shape in the frog model with random initial configurations

We consider the so-called frog model with random initial configurations, which is described by the following evolution mechanism of simple random walks on the multidimensional cubic lattice: Some particles are randomly assigned to any site of the multidimensional cubic lattice. Initially, only particles at the origin are active and they independently perform simple random walks. The other particles are sleeping and do not move at first. When sleeping particles are hit by an active particle, they become active and start doing independent simple random walks. An interest of this model is how initial configurations affect the asymptotic shape of the set of all sites visited by active particles up to a certain time. Thus, in this paper, we prove continuity for the asymptotic shape in the law of the initial configuration.

The social network model on infinite graphs

Given an infinite connected regular graph $G=(V,E)$, place at each vertex $\operatorname{Poisson}(\lambda)$ walkers performing independent lazy simple random walks on $G$ simultaneously. When two walkers visit the same vertex at the same time they are declared to be acquainted. We show that when $G$ is vertex-transitive and amenable, for all $\lambda>0$ a.s. any pair of walkers will eventually have a path of acquaintances between them. In contrast, we show that when $G$ is nonamenable (not necessarily transitive) there is always a phase transition at some $\lambda_{\mathrm{c}}(G)>0$. We give general bounds on $\lambda_{\mathrm{c}}(G)$ and study the case that $G$ is the $d$-regular tree in more detail. Finally, we show that in the nonamenable setup, for every $\lambda$ there exists a finite time $t_{\lambda}(G)$ such that a.s. there exists an infinite set of walkers having a path of acquaintances between them by time $t_{\lambda}(G)$.

Localization for Random Walks Among Random Obstacles in a Single Euclidean Ball

Place an obstacle with probability $1-p$ independently at each vertex of $\mathbb Z^d$, and run a simple random walk until hitting one of the obstacles. For $d\geq 2$ and $p$ strictly above the critical threshold for site percolation, we condition on the environment where the origin is contained in an infinite connected component free of obstacles, and we show that for environments with probability tending to one as $n\to \infty$ there exists a unique discrete Euclidean ball of volume $d \log_{1/p} n$ asymptotically such that the following holds: conditioned on survival up to time $n$ we have that at any time $t \in [o(n),n]$ with probability tending to one the simple random walk is in this ball. This work relies on and substantially improves a previous result of the authors on localization in a region of volume poly-logarithmic in $n$ for the same problem.

Closed Walk Sampler: An Efficient Method for Estimating Eigenvalues of Large Graphs

Eigenvalues of a graph are of high interest in graph analytics for Big Data due to their relevance to many important properties of the graph including network resilience, community detection and the speed of viral propagation. Accurate computation of eigenvalues of extremely large graphs is usually not feasible due to the prohibitive computational and storage costs and also because full access to many social network graphs is often restricted to most researchers. In this paper, we present a series of new sampling algorithms which solve both of the above-mentioned problems and estimate the two largest eigenvalues of a large graph efficiently and with high accuracy. Unlike previous methods which try to extract a subgraph with the most influential nodes, our algorithms sample only a small portion of the large graph via a simple random walk, and arrive at estimates of the two largest eigenvalues by estimating the number of closed walks of a certain length. Our experimental results using real graphs show that our algorithms are substantially faster while also achieving significantly better accuracy on most graphs than the current state-of-the-art algorithms.

The cover time of deterministic random walks for general transition probabilities

A deterministic random walk is a deterministic process analogous to a random walk. While there are some results on the cover time of the rotor-router model, which is a deterministic random walk corresponding to a simple random walk, nothing is known about the cover time of deterministic random walks emulating general transition probabilities. This paper is concerned with the shortest remaining time (SRT)-router model with multiple tokens, which is a deterministic process coping with general transition probabilities possibly containing irrational numbers. For the model, we give an upper bound of the cover time, which is the first result on the cover time of deterministic random walks for general transition probabilities. Our upper bound also improves the existing bounds for the rotor-router model in some cases.

Record statistics for a discrete-time random walk with correlated steps

The characterization of record events is considered for a discrete-time random walk model with long-term memory arising from correlations between successive steps. An important feature is that the correlations are strong enough to give rise to super-diffusivity and transience. Various quantities related to record statistics are calculated exactly, highlighting important differences in behaviour from the simple random walk with independent steps.

Recurrence of Markov chain traces

Nous montrons que les graphes pour lesquels la marche aléatoire simple est transiente n’admettent pas de chaîne de Markov transiente aux plus proches voisins (même non réversible) visitant toutes les arêtes avec probabilité positive, tandis qu’il en existe une pour le réseau carré $\mathbb{Z}^{2}$. En particulier, le réseau $d$-dimensionnel $\mathbb{Z}^{d}$ admet une telle chaîne de Markov seulement lorsque $d=2$. Lorsque $d=2$ nous présentons un exemple de Gady Kozma, et le résultat général est obtenu en prouvant que la trace de toute chaîne de Markov sur un espace d’état dénombrable qui admet une mesure stationnaire est presque sûrement récurrente pour la marche simple. Le cas où la chaîne est réversible a été traité par Gurel-Gurevich, Lyons et le premier auteur. Nous exploitons des résultats récents de théorie du potentiel sur les chaînes de Markov non réversibles pour étendre leur résultat au cas non réversible.

Lower bounds on partial sums of expected hitting times

We consider a simple random walk on a connected undirected graph with N vertices. Palacios (2010) and Palacios and Renom (2012) investigated lower bounds on partial sums of expected hitting times of the random walk. In this paper, we show that Palacios’ (2010) lower bound is the best for all multigraphs. In addition, we show that the conjecture made by Palacios and Renom (2012) for simple graphs is true for N≤5, but it is not true for N≥6.

An entropic proof of cutoff on Ramanujan graphs

It is recently proved by Lubetzky and Peres that the simple random walk on a Ramanujan graph exhibits a cutoff phenomenon, that is to say, the total variation distance of the random walk distribution from the uniform distribution drops abruptly from near $1$ to near $0$. There are already a few alternative proofs of this fact. In this note, we give yet another proof based on functional analysis and entropic consideration.

The infinite two-sided loop-erased random walk

The loop-erased random walk (LERW) in $ {\mathbb {Z}}^{d}, d \geq 2$, is obtained by erasing loops chronologically from simple random walk. In this paper we show the existence of the two-sided LERW which can be considered as the distribution of the LERW as seen by a point in the “middle” of the path.

Chen–Stein method for the uncovered set of random walk on $\mathbb {Z}_{n}^{d}$ for $d \ge 3$

Let $X$ be a simple random walk on $\mathbb{Z}_n^d$ with $d\geq 3$ and let $t_{\rm{cov}}$ be the expected cover time. We consider the set of points $\mathcal{U}_\alpha$ of $\mathbb{Z}_n^d$ that have not been visited by the walk by time $\alpha t_{\rm{cov}}$ for $\alpha\in (0,1)$. It was shown in [MS17] that there exists $\alpha_1(d)\in (0,1)$ such that for all $\alpha>\alpha_1(d)$ the total variation distance between the law of the set $\mathcal{U}_\alpha$ and an i.i.d. sequence of Bernoulli random variables indexed by $\mathbb{Z}_n^d$ with success probability $n^{-\alpha d}$ tends to $0$ as $n \to \infty$. In [MS17] the constant $\alpha_1(d)$ converges to $1$ as $d\to\infty$. In this short note using the Chen--Stein method and a concentration result for Markov chains of Lezaud we greatly simplify the proof of [MS17] and find a constant $\alpha_1(d)$ which converges to $3/4$ as $d\to\infty$.

On covering monotonic paths with simple random walk

In this paper we study the probability that a $d$ dimensional simple random walk (or the first $L$ steps of it) covers each point in a nearest neighbor path connecting 0 and the boundary of an $L_{1}$ ball. We show that among all such paths, the one that maximizes the covering probability is the monotonic increasing one that stays within distance 1 from the diagonal. As a result, we can obtain an exponential upper bound on the decaying rate of covering probability of any such path when $d\geq 4$. The main tool is a general combinatorial inequality, that is interesting in its own right.

Transience of conditioned walks on the plane: encounters and speed of escape

We consider the two-dimensional simple random walk conditioned on never hitting the origin, which is,formally speaking, the Doob's $h$-transform of the simple random walk with respect to the potential kernel. We then study the behavior of the future minimum distance of the walk to the origin, and also prove that two independent copies of the conditioned walk, although both transient, will nevertheless meet infinitely many times a.s.

TASEP speed process: an effective medium approach

We discuss the approximate phenomenological description of the motion of a single second-class particle in a two-species totally asymmetric simple exclusion process (TASEP) on a 1D lattice. Initially, the second class particle is located at the origin and to its left, all sites are occupied with first class particles while to its right, all sites are vacant. Ferrari and Kipnis proved that in any particular realization, the average velocity of the second class particle tends to a constant, but this mean value has a wide variation in different histories. We discuss this phenomena, here called the TASEP speed process, in an approximate effective medium description, in which the second class particle moves in a random background of the space-time dependent average density of the first class particles. We do this in three different approximations of increasing accuracy, treating the motion of the second-class particle, always in continuous time, first as a simple biassed random walk in a continuum Langevin equation description, then as a biassed Markovian random walk on a lattice with space and time dependent jump rates, and finally as a Non-Markovian biassed walk on a lattice, with a non-exponential distribution of waiting times between jumps. We find that, when the displacement at time T is x0, the conditional expectation of displacement, at time zT (z > 1) is zx0, and the variance of the displacement only varies as . We extend this approach to describe the trajectories of a tagged particle in the case of a finite lattice, where there are L classes of particles on an L-site line, initially placed in the order of increasing class number. Lastly, we discuss a variant of the problem in which the exchanges between adjacent particles happened at rates proportional to a power of the difference in their class numbers.

Sticky Brownian Motion and Its Numerical Solution

Sticky Brownian motion is the simplest example of a diffusion process that can spend finite time both in the interior of a domain and on its boundary. It arises in various applications such as in biology, materials science, and finance. This article spotlights the unusual behavior of sticky Brownian motions from the perspective of applied mathematics, and provides tools to efficiently simulate them. We show that a sticky Brownian motion arises naturally for a particle diffusing on $\mathbb{R}_+$ with a strong, short-ranged potential energy near the origin. This is a limit that accurately models mesoscale particles, those with diameters $\approx 100$nm-$10\mu$m, which form the building blocks for many common materials. We introduce a simple and intuitive sticky random walk to simulate sticky Brownian motion, that also gives insight into its unusual properties. In parameter regimes of practical interest, we show this sticky random walk is two to five orders of magnitude faster than alternative methods to simulate a sticky Brownian motion. We outline possible steps to extend this method towards simulating multi-dimensional sticky diffusions.

On transience of frogs on Galton–Watson trees

We consider an interacting particle system, known as the frog model, on infinite Galton–Watson trees allowing offspring 0 and 1. The system starts with one awake particle (frog) at the root of the tree and a random number of sleeping particles at the other vertices. Awake frogs move according to simple random walk on the tree and as soon as they encounter sleeping frogs, those will wake up and move independently according to simple random walk. The frog model is called transient if there are almost surely only finitely many particles returning to the root. In this paper we prove a 0–1-law for transience of the frog model and show the existence of a transient phase for certain classes of Galton–Watson trees.

Internal DLA on cylinder graphs: fluctuations and mixing

We use coupling ideas introduced in [13] to show that an IDLA process on a cylinder graph $G\times {\mathbb {Z}} $ forgets a typical initial profile in $\mathcal {O}( N\sqrt {\tau _{N}} (\log \! N)^{2} )$ steps for large $N$, where $N$ is the size of the base graph $G$, and $\tau _{N}$ is the total variation mixing time of a simple random walk on $G$. The main new ingredient is a maximal fluctuations bound for IDLA on $G\times \mathbb {Z}$ which only relies on the mixing properties of the base graph $G$ and the Abelian property.

Markovian explorations of random planar maps are roundish

The infinite discrete stable Boltzmann maps are "heavy-tailed" generalisations of the well-known Uniform Infinite Planar Quadrangulation. Very efficient tools to study these objects are Markovian step-by-step explorations of the lattice called peeling processes. Such a process depends on an algorithm which selects at each step the next edge where the exploration continues. We prove here that, whatever this algorithm, a peeling process always reveals about the same portion of the map, thus growing roughly metric balls. Applied to well-designed algorithms, this easily enables us to compare distances in the map and in its dual, as well as to control the so-called pioneer points of the simple random walk, both on the map and on its dual.

A rotor configuration with maximum escape rate

Rotor walk is a deterministic analogue of simple random walk. For any given graph, we construct a rotor configuration for which the escape rate of the corresponding rotor walk is equal to the escape rate of simple random walk, and thus answer a question of Florescu, Ganguly, Levine, and Peres (2014).

The frog model on non-amenable trees

We examine an interacting particle system on trees commonly referred to as the frog model. For its initial state, it begins with a single active particle at the root and i.i.d. $\mathrm{Poiss} (\lambda )$ many inactive particles at each non-root vertex. Active particles perform discrete time simple random walk and in the process activate any inactive particles they encounter. We show that for every non-amenable tree with bounded degree there exists a phase transition from transience to recurrence (with a non-trivial intermediate phase sometimes sandwiched in between) as $\lambda $ varies.