Exponential tail bounds for loop-erased random walk in two dimensions

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 $1$.

In this paper we extend the loop‐erased random walk (LERW) to the directed hypergraph setting. We then generalize Wilson's algorithm for uniform sampling of spanning trees to directed hypergraphs. In several special cases, this algorithm perfectly samples spanning hypertrees in expected polynomial time. Our main application is to the reachability problem, also known as the directed all‐terminal network reliability problem. This classical problem is known to be # P‐complete, hence is most likely intractable (Ball and Provan, SIAM J Comput 12 (1983) 777–788). We show that in the case of bi‐directed graphs, a conjectured polynomial bound for the expected running time of the generalized Wilson algorithm implies a FPRAS for approximating reachability. Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 201‐223, 2014

The loop-erased random walk (LERW) in $\mathbb{Z}^{4}$ is the process obtained by erasing loops chronologically for a simple random walk. We prove that the escape probability of the LERW renormalized by $(\log n)^{\frac{1}{3}}$ converges almost surely and in $L^{p}$ for all $p>0$. Along the way, we extend previous results by the first author building on slowly recurrent sets. We provide two applications for the escape probability. We construct the two-sided LERW, and we construct a $\pm 1$ spin model coupled with the wired spanning forests on $\mathbb{Z}^{4}$ with the bi-Laplacian Gaussian field on $\mathbb{R}^{4}$ as its scaling limit.

We study an infinite system of independent symmetric random walks on a hierarchical group, in particular, the <em>c</em>-random walks. Such walks are used, e.g., in mathematical physics and population biology. The number variance problem consists in investigating if the variance of the number of “particles” $N_n(L)$ lying in the ball of radius $L$<em></em> at a given step $n$<em></em> remains bounded, or even better, converges to a finite limit, as $L\\to\\infty$. We give a necessary and sufficient condition and discuss its relationship to transience/recurrence property of the walk. Next we consider normalized fluctuations of $N_n(L)$ around the mean as $n\\to\\infty$ and $L$<em></em> is increased in an appropriate way. We prove convergence of finite dimensional distributions to a Gaussian process whose properties are discussed. As the $c$<em></em>-random walks mimic symmetric stable processes on $\\mathbb{R}$, we compare our results with those obtained by Hambly and Jones (2007, 2009), who studied the number variance problem for an infinite system of independent symmetric stable processes on $\\mathbb{R}$. Since the hierarchical group is an ultrametric space, corresponding results for symmetric stable processes and hierarchical random walks may be analogous or quite different, as has been observed in other contexts. An example of a difference in the present context is that for the stable processes a fluctuation limit process is a Gaussian process which is not Markovian and has long range dependent stationary increments, but the counterpart for hierarchical random walks is Markovian, and in a special case it has independent increments.

The set of visited sites and the number of visited sites are two basic properties of the random walk trajectory. We consider two independent random walks on a hyper-cubic lattice and study ordering probabilities associated with these characteristics. The first is the probability that during the time interval (0,t), the number of sites visited by a walker never exceeds that of another walker. The second is the probability that the sites visited by a walker remain a subset of the sites visited by another walker. Using numerical simulations, we investigate the leading asymptotic behaviors of the ordering probabilities in spatial dimensions d=1,2,3,4. We also study the evolution of the number of ties between the number of visited sites. We show analytically that the average number of ties increases as $a_1\ln t$ with $a_1=0.970508$ in one dimension and as $(\ln t)^2$ in two dimensions.

Comments that the Laplacian random walk with eta =1, recently introduced by Lyklema and Evertsz (1986), is the same process as the loop-erased self-avoiding random walk analysed previously by the author (1980). Some rigorous results about this process are reviewed.

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.

We consider the infinite particle system in Z d known commonly as symmetric random walk with simple exclusion. This consists of particles in Z d , each trying to execute a more or less independent random walk with jump probabilities p(x). We assume that p(x) is symmetric, i.e., p(x) = p(−x), and that it has finite range. If d = 1, we assume, in addition, that p(1) +p(−1) < 1, so that only the one-dimensional nearest neighbor random walk is excluded. Particles wait for an exponential time, pick a random jump size x with probability p(x), and try to jump to a new site defined by a jump equal to x. If the site is free, the jump is completed and things start all over again. Otherwise the jump is disallowed and things start all over again with the particle remaining in the old site. All the particles are doing this simultaneously and independently of each other. See [3] for a discussion of the model.

Self-avoiding walks (SAWs) and loop-erased random walks (LERWs) are two ensembles of random paths with numerous applications in mathematics, statistical physics and quantum field theory. While SAWs are described by the n→0 limit of ϕ4-theory with O(n)-symmetry, LERWs have no obvious field-theoretic description. We analyse two candidates for a field theory of LERWs, and discover a connection between the corresponding and a priori unrelated theories. The first such candidate is the O(n)-symmetric ϕ4 theory at n=−2 whose link to LERWs was known in two dimensions due to conformal field theory. Here it is established in arbitrary dimension via a perturbation expansion in the coupling constant. The second candidate is a field theory for charge-density waves pinned by quenched disorder, whose relation to LERWs had been conjectured earlier using analogies with Abelian sandpiles. We explicitly show that both theories yield identical results to 4-loop order and give both a perturbative and a non-perturbative proof of their equivalence. This allows us to compute the fractal dimension of LERWs to order ε5 where ε=4−d. In particular, in d=3 our theory gives zLERW(d=3)=1.6243±0.001, in excellent agreement with the estimate z=1.62400±0.00005 of numerical simulations.

The rotor-router mechanism was introduced as a deterministic alternative to the random walk in undirected graphs. In this model, an agent is initially placed at one of the nodes of the graph. Each node maintains a cyclic ordering of its outgoing arcs, and during successive visits of the agent, propagates it along arcs chosen according to this ordering in round-robin fashion. The behavior of the rotor-router is fully deterministic but its performance characteristics (cover time, return time) closely resemble the expected values of the corresponding parameters of the random walk. In this work we consider the setting in which multiple, indistinguishable agents are deployed in parallel in the nodes of the graph, and move around the graph in synchronous rounds, interacting with a single rotor-router system. We propose new techniques which allow us to perform a theoretical analysis of the multi-agent rotor-router model, and to compare it to the scenario of parallel independent random walks in a graph. Our main results concern the n-node ring, and suggest a strong similarity between the performance characteristics of this deterministic model and random walks. We show that on the ring the rotor-router with k agents admits a cover time of between varTheta (n^2 / k^2) in the best case and varTheta (n^2 / log k) in the worst case, depending on the initial locations of the agents, and that both these bounds are tight. The corresponding expected value of the cover time for k random walks, depending on the initial locations of the walkers, is proven to belong to a similar range, namely between varTheta (n^2 / (k^2/log ^2 k)) and varTheta (n^2 / log k). Finally, we study the limit behavior of the rotor-router system. We show that, once the rotor-router system has stabilized, all the nodes of the ring are always visited by some agent every varTheta (n / k) steps, regardless of how the system was initialized. This asymptotic bound corresponds to the expected time between successive visits to a node in the case of k random walks. All our results hold up to a polynomially large number of agents (1 le k < n^{1/11}).

A two-person game $\Gm$ is considered which is specified by the following random walks. Let $x_n$ and $y_n$ be independent symmetric random walks on the set $E=\{0,1,\ldots,K\}$. Assume they start from the states a and b respectively $(1\le a y_{\sigma}$ then player~II pays player~I, say, \$1; if $x_{\tau} < y_{\sigma}$ then I pays II \$1; and if $x_{\tau}=y_{\sigma}$ then the outcome of the game is said to be a draw. The aim of each player is to maximize the expected value of hisincome. We find the equilibrium situation and the value of...

Mesh-pull based P2P live media streaming system has gained a lot of interests and successes, and collecting key metrics in time from millions of peers to monitor the system level quality is very important for its successful commercial deployment. But the real-time monitoring still faces great technology obstacles, especially considering the dynamic and large-scale scenario which is very common in p2p environment. In this article, we present the novel monitoring method of P2P live media streaming system based on random walks which sample the peers uniformly. The source node initializes many parallel independent random walks periodically to collect the key parameters overall the system and conduct the service quality estimation passively based on the random walk results. The accuracy of the monitoring is influenced by many factors like the number of parallel random walks, the hops of each random walk, the period of sampling and etc. We show the influence of different parameters and verify the effects and efficiency of the method by simulation at last.

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.

This paper discusses some convergence properties in the entropic sampling Monte Carlo methods with multiple random walkers, particularly in the Wang-Landau (WL) and 1/t algorithms. The classical algorithms are modified by the use of m-independent random walkers in the energy landscape to calculate the density of states (DOS). The Ising model is used to show the convergence properties in the calculation of the DOS, as well as the critical temperature, while the calculation of the number π by multiple dimensional integration is used in the continuum approximation. In each case, the error is obtained separately for each walker at a fixed time, t; then, the average over m walkers is performed. It is observed that the error goes as 1/sqrt[m]. However, if the number of walkers increases above a certain critical value m>m_{x}, the error reaches a constant value (i.e., it saturates). This occurs for both algorithms; however, it is shown that for a given system, the 1/t algorithm is more efficient and accurate than the similar version of the WL algorithm. It follows that it makes no sense to increase the number of walkers above a critical value m_{x}, since it does not reduce the error in the calculation. Therefore, the number of walkers does not guarantee convergence.

We determine the precise asymptotics of the logarithmic upper tail probability of the total intersection local time of p independent random walks in ℤd under the assumption p(d−2) > d. Our approach allows a direct treatment of the infinite time horizon.