Exceptional times for the dynamical discrete web

Superdiffusive and subdiffusive exceptional times in the dynamical discrete web

Consider a stochastic process that behaves as a d-dimensional simple and symmetric random walk, except that, with a certain fixed probability, at each step, it chooses instead to jump to a given site with probability proportional to the time it has already spent there. This process has been analyzed in physics literature under the name random walk with preferential relocations, where it is argued that the position of the walker after n steps, scaled by , converges to a Gaussian random variable; because of the spatial scaling, the process is said to undergo a slow diffusion.In this paper, we generalize this model by allowing the underlying random walk to be any Markov process and the random run-lengths (time between two relocations) to be i.i.d.-distributed. We also allow the memory of the walker to fade with time, meaning that when a relocations occurs, the walker is more likely to go back to a place it has visited more recently.We rigorously prove that the central limit theorem described above (plus a local limit theorem and the convergence of the weighted occupation measure) by associating to the process a growing family of vertex-weighted random recursive trees and a Markov chain indexed by this tree. The spatial scaling of our relocated random walk is related to the height of a ‘typical’ vertex in the random tree. This typical height can range from doubly-logarithmic to logarithmic or even a power of the number of nodes of the tree, depending on the form of the memory.

For a symmetric homogeneous and irreducible random walk on the d-dimensional integer lattice, which have a finite variance of jumps, we study passage times (taking values in [0,∞]) determined by a starting point x, a hitting state y, and a taboo state z. We find the probability that these passage times are finite, and study the distribution tail. In particular, it turns out that, for the above-mentioned random walks on ℤ d except for a simple random walk on ℤ, the order of the distribution tail decrease is specified by dimension d only. In contrast, for a simple random walk on ℤ, the asymptotic properties of hitting times with taboo essentially depend on mutual location of the points x, y, and z. These problems originated in recent study of a branching random walk on ℤ d with a single source of branching.

We derive a local limit theorem for normal, moderate, and large deviations for symmetric simple random walk on the square lattice in dimensions one and two that is an improvement of existing results for points that are particularly distant from the walk's starting point. More specifically, we give explicit asymptotic expressions in terms of $n$ and $x$, where $x$ is thought of as dependent on $n$, in dimensions one and two for $P(S_n=x)$, the probability that symmetric simple random walk $S$ started at the origin is at some point $x$ at time $n$, that are valid for all $x$. We also show that the behavior of planar symmetric simple random walk differs radically from that of planar standard Brownian motion outside of the disk of radius $n^{3/4}$, where the random walk ceases to be approximately rotationally symmetric. Indeed, if $n^{3/4}=o(|S_n|)$, $S_n$ is more likely to be found along the coordinate axes. In this paper, we give a description of how the transition from approximate rotational symmetry to complete concentration of $S$ along the coordinate axes occurs.

Polysaccharides, such as cellulose, are often processed by dissolution in solvent mixtures, e.g. an ionic liquid (IL) combined with a dipolar aprotic co-solvent (CS) that the polymer does not dissolve in. A multi-walker, discrete-time, discrete-space 1-dimensional random walk can be applied to model solvation of a polymer in a multi-component solvent mixture. The number of IL pairs in a solvent mixture and the number of solvent shells formable, x, is associated with n, the model time-step, and N, the number of random walkers. The mean number of distinct sites visited is proportional to the amount of polymer soluble in a solution. By also fitting a polynomial regression model to the data, we can associate the random walk terms with chemical interactions between components and probe where the system deviates from a 1-D random walk. The 'frustration' between solvents shells is given as ln x in the random walk model and as a negative IL:IL interaction term in the regression model. This frustration appears in regime II of the random walk model (high volume fractions of IL) where walkers interfere with each other, and the system tends to its limiting behaviour. In the low concentration regime, (regime I) the solvent shells do not interact, and the system depends only on IL and CS terms. In both models (and both regimes), the system is almost entirely controlled by the volume available to solvation shells, and thus is a counting/space-filling problem, where the molar volume of the CS is important. Small deviations are observed when there is an IL-CS interaction. The use of two models, built on separate approaches, confirm these findings, demonstrating that this is a real effect and offering a route to identifying such systems. Specifically, the majority of CSs - such as dimethylformide - follow the random walk model, whilst 1-methylimidazole, dimethyl sulfoxide, 1,3-dimethyl-2-imidazolidinone and tetramethylurea offer a CS-mediated improvement and propylene carbonate results in a CS-mediated hindrance. It is shown here that systems, which are very complex at a molecular level, may, nonetheless, be effectively modelled as a simple random walk in phase-space. The 1-D random walk model allows prediction of the ability of solvent mixtures to dissolve cellulose based on only two dissolution measurements (one in neat IL) and molar volume.

On the range of simple symmetric random walks on the line

The cutoff phenomenon describes a sharp transition in the convergence of a family of ergodic finite Markov chains to equilibrium. Many natural families of chains are believed to exhibit cutoff, and yet establishing this fact is often extremely challenging. An important such family of chains is the random walk on G(n,d), a random d-regular graph on n vertices. It is well known that almost every such graph for d≥3 is an expander, and even essentially Ramanujan, implying a mixing time of O(logn). According to a conjecture of Peres, the simple random walk on G(n,d) for such d should then exhibit cutoff with high probability (whp). As a special case of this, Durrett conjectured that the mixing time of the lazy random walk on a random 3-regular graph is whp (6+o(1))log2n. In this work we confirm the above conjectures and establish cutoff in total-variation, its location, and its optimal window, both for simple and for non-backtracking random walks on G(n,d). Namely, for any fixed d≥3, the simple random walk on G(n,d) whp has cutoff at (d/(d−2))logd−1n with window order logn. Surprisingly, the non-backtracking random walk on G(n,d) whp has cutoff already at logd−1n with constant window order. We further extend these results to G(n,d) for any d=no(1) that grows with n (beyond which the mixing time is O(1)), where we establish concentration of the mixing time on one of two consecutive integers

Mathematical modelling of the movement of animals, micro-organisms and cells is of great relevance in the fields of biology, ecology and medicine. Movement models can take many different forms, but the most widely used are based on the extensions of simple random walk processes. In this review paper, our aim is twofold: to introduce the mathematics behind random walks in a straightforward manner and to explain how such models can be used to aid our understanding of biological processes. We introduce the mathematical theory behind the simple random walk and explain how this relates to Brownian motion and diffusive processes in general. We demonstrate how these simple models can be extended to include drift and waiting times or be used to calculate first passage times. We discuss biased random walks and show how hyperbolic models can be used to generate correlated random walks. We cover two main applications of the random walk model. Firstly, we review models and results relating to the movement, dispersal and population redistribution of animals and micro-organisms. This includes direct calculation of mean squared displacement, mean dispersal distance, tortuosity measures, as well as possible limitations of these model approaches. Secondly, oriented movement and chemotaxis models are reviewed. General hyperbolic models based on the linear transport equation are introduced and we show how a reinforced random walk can be used to model movement where the individual changes its environment. We discuss the applications of these models in the context of cell migration leading to blood vessel growth (angiogenesis). Finally, we discuss how the various random walk models and approaches are related and the connections that underpin many of the key processes involved.

A random walk in a sparse random environment is a model introduced by Matzavinos et al. [Electron. J. Probab. 21, paper no. 72: 2016] as a generalization of both a simple symmetric random walk and a classical random walk in a random environment. A random walk in a sparse random environment is a nearest neighbor random walk on that jumps to the left or to the right with probability 1/2 from every point of and jumps to the right (left) with the random probability λ k+1 (1 - λ k+1) from the point S k , . Assuming that are independent copies of a random vector and the mean is finite (moderate sparsity) we obtain stable limit laws for X n , properly normalized and centered, as n → ∞. While the case ξ ≤ M a.s. for some deterministic M > 0 (weak sparsity) was analyzed by Matzavinos et al., the case (strong sparsity) will be analyzed in a forthcoming paper.

We show that the ‘erasing-larger-loops-first’ (ELLF) method, which was first introduced for erasing loops from the simple random walk on the Sierpiński gasket, does work also for non-Markov random walks, in particular, self-repelling walks to construct a new family of self-avoiding walks on the Sierpiński gasket. The one-parameter family constructed in this method continuously connects the loop-erased random walk and a self-avoiding walk which has the same asymptotic behavior as the ‘standard’ self-avoiding walk. We prove the existence of the scaling limit and study some path properties: The exponent $ν$ governing the short-time behavior of the scaling limit varies continuously in the parameter. The limit process is almost surely self-avoiding, while it has path Hausdorff dimension $1/ν $, which is strictly greater than $1$.

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 $$\mu ^{*n}(Z_n)$$ . 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 $$H_\mathrm{FD}$$ . We define and study the radius of almost invariance. We estimate this radius for random walks on iterated wreath products and show this radius can be asymptotically strictly smaller than n / L(n), where L(n) denotes the drift function of the random walk. We show that the radius of individual almost invariance of a simple random walk on the wreath product of $$\mathbb {Z}^2$$ with a finite group is asymptotically strictly larger than n / L(n). Finally, we show the existence of groups such that the radius of almost invariance is smaller than a given function, but remains unbounded. We also discuss possible limiting distribution of ratios of transition probabilities on non almost invariant scales.

The aim of this chapter is to give a review of the author’s results on random walks on groups. In Section 2 we consider some useful properties of transition functions of random walks (see [7; part I]). Section 3 is devoted to a special decomposition of harmonic functions of random walks [8]. Finally, in Section 4 we deal with strong ratio limit theorems for random walks (see [7; part II] and [10]). Theorem 5 of Section 4 generalizes the well-known result by Avez [1] concerning symmetric random walks on discrete groups. Our list of references does not pretend to be complete.

Let W be an integer-valued random variable satisfying \(E[W] =: \delta \ge 0\) and \(P(W 0\), and consider a self-interacting random walk that behaves like a simple symmetric random walk with the exception that on the first visit to any integer \(x\in \mathbb Z\), the size of the next step is an independent random variable with the same distribution as W. We show that this self-interacting random walk is recurrent if \(\delta \le 1\) and transient if \(\delta >1\). This is a special case of our main result which concerns the recurrence and transience of excited random walks (or cookie random walks) with non-nearest neighbor jumps.

In nature, search processes that use randomly oriented steps of different lengths have been observed at both the microscopic and the macroscopic scales. Physicists have analyzed in depth two such processes on grid topologies: Intermittent Search, which uses two step lengths, and Levy Walk, which uses many. Taking a computational perspective, this paper considers the number of distinct step lengths k as a complexity measure of the considered process. Our goal is to understand what is the optimal achievable time needed to cover the whole terrain, for any given value of k. Attention is restricted to dimension one, since on higher dimensions, the simple random walk already displays a quasi linear cover time.

Une marche aleatoire simple $\mathcal{W}$ sur $\mathbb{Z}^{2}$ induit pour chaque sous-graphe couvrant $G\subset \mathbb{Z}^{2}$ une marche $\mathcal{W}_{G}$ : notamment, si on regarde $\mathcal{W}$ comme un mot infini uniforme sur l’alphabet $\{\mathbf{x},-\mathbf{x},\mathbf{y},-\mathbf{y}\}$, alors la marche $\mathcal{W}_{G}$ commence a l’origine et suit les directions definis par $\mathcal{W}$, en acceptant seulement les pas de $\mathcal{W}$ le long desquels $\mathcal{W}_{G}$ reste sur $G$. Pour chaque $G\subset \mathbb{Z}^{2}$ fixe, la marche $\mathcal{W}_{G}$ a la loi d’une marche aleatoire simple sur $G$ et alors $\mathcal{W}_{G}$ est presque surement recurrent dans le sens que $\mathcal{W}_{G}$ visite chaque sommet de $G$ connexe a l’origine un nombre infini de fois. Alors, une question naturelle surgit : est-ce que presque surement pour $\mathcal{W}$, les marches $\mathcal{W}_{G}$ sont recurrents pour tous $G\subset \mathbb{Z}^{2}$ a la fois? Dans cet article, on repond a cette question d’une maniere negative, en montrent l’existence des graphes exceptionnels sur lesquels la marche est transitoire. En fait, on montre que meme si on considere un nombre denombrable des marches aleatoires simples independantes, alors des graphes exceptionnels existent. D’autre cote, on montre que il n’existe pas des graphes exceptionnels pour la marche aleatoire branchante.