The Einstein Relation for Random Walks on Graphs

Mechanics Research Communications, 2012

Preliminary but interesting and definite results are given on the application of graph theory concepts (random walk on graphs) to the double diffusivity theory proposed by Aifantis in the late 70s to model transport in media with high diffusivity paths such as metal polycrystals with a continuous distribution of grain boundaries possessing much higher diffusivity than the bulk, as well as in nanopolycrystals for which it has been shown recently that the double diffusivity model fits experimental observations. The new information provided by employing the graph theory tool is concerned with certain restrictions and relations that the phenomenological coefficients, entering in the coupled partial differential equations of double diffusivity, should satisfy depending on the topology and related details of the graph model adopted. © 2012 Elsevier Ltd. All rights reserved.

Journal of Statistical Physics, 2019

We consider random walks on the infinite cluster of a conditional bond percolation model on the infinite ladder graph. In a companion paper, we have shown that if the random walk is pulled to the right by a positive bias λ > 0, then its asymptotic linear speed v is continuous in the variable λ > 0 and differentiable for all sufficiently small λ > 0. In the paper at hand, we complement this result by proving that v is differentiable at λ = 0. Further, we show the Einstein relation for the model, i.e., that the derivative of the speed at λ = 0 equals the diffusivity of the unbiased walk.

SIAM Journal on Applied Mathematics, 2021

Graph-limit theory focuses on the convergence of sequences of increasingly large graphs, providing a framework for the study of dynamical systems on massive graphs, where classical methods would become computationally intractable. Through an approximation procedure, the standard ordinary differential equations are replaced by nonlocal evolution equations on the unit interval. In this work, we adopt this methodology to prove the validity of the continuum limit of random walks, a largely studied model for diffusion on graphs. We focus on two classes of processes on dense weighted graphs, in discrete and in continuous time, whose dynamics are encoded in the transition matrix of the associated Markov chain, or in the random-walk Laplacian. We further show that previous works on the discrete heat equation, associated to the combinatorial Laplacian, fall within the scope of our approach. Finally, we characterize the relaxation time of the process in the continuum limit.

Journal of Physics A: Mathematical and General, 2005

Random walks on graphs are widely used in all sciences to describe a great variety of phenomena where dynamical random processes are affected by topology. In recent years, relevant mathematical results have been obtained in this field, and new ideas have been introduced, which can be fruitfully extended to different areas and disciplines. Here we aim at giving a brief but comprehensive perspective of these progresses, with a particular emphasis on physical aspects. Contents 1 Introduction 2 Mathematical description of graphs 3 The random walk problem 4 The generating functions 5 Random walks on finite graphs 6 Infinite graphs 7 Random walks on infinite graphs 8 Recurrence and transience: the type problem 9 The local spectral dimension 10 Averages on infinite graphs 11 The type problem on the average 1 12 The average spectral dimension 21 13 A survey of analytical results on specific networks 23 13.1 Renormalization techniques. .

Physical Review Letters, 1987

Analysis of Monte Carlo enumerations for diAusion on the fractal structure generated by the random walk on a two-dimensional lattice allows us to predict a behavior &r)n "(1nn)' with v=0. 325~0.01 and a =0.35~0.03. This leads to the conjecture that v=a = -, ' . This value of v, and the presence of logarithmic corrections, are strongly supported by heuristic arguments based on Flory theory and on plausible assumptions.

Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering, 2009

The aim of this article is to discuss some of the notions and applications of random walks on finite graphs, especially as they apply to random graphs. In this section we give some basic definitions, in Section 2 we review applications of random walks in computer science, and in Section 3 we focus on walks in random graphs. Given a graph G = (V, E), let d G (v) denote the degree of vertex v for all v ∈ V. The random walk W v = (W v (t), t = 0, 1,. . .) is defined as follows: W v (0) = v and given x = W v (t), W v (t + 1) is a randomly chosen neighbour of x. When one thinks of a random walk, one often thinks of Polya's Classical result for a walk on the d-dimensional lattice Z d , d ≥ 1. In this graph two vertices x = (x 1 , x 2 ,. .. , x d) and y = (y 1 , y 2 ,. .. , y d) are adjacent iff there is an index i such that (i) x j = y j for j = i and (ii) |x i − y i | = 1. Polya [33] showed that if d ≤ 2 then a walk starting at the origin returns to the origin with probability 1 and that if d ≥ 3 then it returns with probability p(d) < 1. See also Doyle and Snell [22]. A random walk on a graph G defines a Markov chain on the vertices V. If G is a finite, connected and non-bipartite graph, then this chain has a stationary distribution π given by π v = d G (v)/(2|E|). Thus if P (t) v (w) = Pr(W v (t) = w), then lim t→∞ P (t) v (w) = π w , independent of the starting vertex v. In this paper we only consider finite graphs, and we will focus on two aspects of a random walk: The Mixing Time and the Cover Time.

Transactions of the American Mathematical Society, 1984

The difference Laplacian on a square lattice in Rn has been studied by many authors. In this paper an analogous difference operator is studied for an arbitrary graph. It is shown that many properties of the Laplacian in the continuous setting (e.g. the maximum principle, the Harnack inequality, and Cheeger's bound for the lowest eigenvalue) hold for this difference operator. The difference Laplacian governs the random walk on a graph, just as the Laplace operator governs the Brownian motion. As an application of the theory of the difference Laplacian, it is shown that the random walk on a class of graphs is transient.

2021

We show that for a uniformly irreducible random walk on a graph, with bounded range, there is a Floyd function for which the random walk converges to its corresponding Floyd boundary. Moreover if we add the assumptions, p(n)(v,w) ≤ Cρ, where ρ &lt; 1 is the spectral radius, then for any Floyd function f that satisfies ∑∞ n=1 nf(n) &lt; ∞, the Dirichlet problem with respect to the Floyd boundary is solvable.

2016

Two results on Asymptotic Behaviour of Random Walks in Random Environment Jeremy Voltz Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2016 In the first chapter of this thesis, we consider a model of directed polymer in 1 + 1 dimensions in a product-type random environment ω(t, x) = b(t)F (x), where the fields F and b are i.i.d., with F (x) continuous, symmetric and bounded, and b(t) = ±1 with probabilty 1/2. Thus ω can be viewed as the field F oscillating in time. We consider directed last-passage percolation through this random field; namely, we investigate the behavior of the length n polymer path with maximal action, where the action of a path is simply the sum of the environment variables it moves through. We prove a law of large numbers for the maximal action of the path from the origin to a fixed endpoint (n, bαnc), and investigate the limiting shape function a(α). We prove that this shape function is non-linear, and has a corner at α = 0, thus i...

Physical Review E, 2019

We study the rare fluctuations or large deviations of time-integrated functionals or observables of an unbiased random walk evolving on Erdös-Rényi random graphs, and construct a modified, biased random walk that explains how these fluctuations arise in the long-time limit. Two observables are considered: the sum of the degrees visited by the random walk and the sum of their logarithm, related to the trajectory entropy. The modified random walk is used for both quantities to explain how sudden changes in degree fluctuations, similar to dynamical phase transitions, are related to localization transitions. For the second quantity, we also establish links between the large deviations of the trajectory entropy and the maximum entropy random walk.