Given any @math and for @math denoting a sample of the two-dimensional discrete Gaussian free field on @math pinned at the origin, we consider the random walk on @math among random conductances where the conductance of edge @math is given by @math . We show that, for almost every @math , this random walk is recurrent and that, with probability tending to 1 as @math , the return probability at time @math decays as @math . In addition, we prove a version of subdiffusive behavior by showing that the expected exit time from a ball of radius @math scales as @math with @math for all @math . Our results rely on delicate control of the effective resistance for this random network. In particular, we show that the effective resistance between two vertices at Euclidean distance @math behaves as @math .
A random walk naturally associated with LBM is the continuous time simple symmetric random walk with exponential holding time at @math having parameter @math where, in our notation, @math . A more natural (albeit qualitatively similar, as far as long-time behavior is concerned) modification is to use @math (see ) in instead of @math ; we will refer to the associated process as the Liouville Random Walk (LRW) below. Formally, this process is a continuous-time Markov chain on @math with generator The nature of the transition rates of the LRW precludes formulation using conductances and, no surprise, our analysis is thus quite different from those mentioned above. For instance, unlike for the LRW, our random walk moves preferably towards neighbors with a higher potential, emphasizing the trapping effects of the random environment; see Fig. . The off-diagonal heat kernel computation in @cite_24 is also of a different flavor: Our control of the return probability relies crucially on the electric-resistance metric while the off-diagonal LBM heat kernel is expected to be related to the Liouville first passage (Liouville FPP) percolation metric (see @cite_36 @cite_32 ).
@cite_24 @cite_32 @cite_36
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We consider a finite random walk on a weighted graph G; we show that the fraction of time spent in a set of vertices A converges to the stationary probability @math with error probability exponentially small in the length of the random walk and the square of the size of the deviation from @math . The exponential bound is in terms of the expansion of G and improves previous results of [D. Aldous, Probab. Engrg. Inform. Sci., 1 (1987), pp. 33--46], [L. Lovasz and M. Simonovits, Random Structures Algorithms , 4 (1993), pp. 359--412], [M. Ajtai, J. Komlos, and E. Szemeredi, Deterministic simulation of logspace, in Proc. 19th ACM Symp. on Theory of Computing, 1987]. We show that taking the sample average from one trajectory gives a more efficient estimate of @math than the standard method of generating independent sample points from several trajectories. Using this more efficient sampling method, we improve the algorithms of Jerrum and Sinclair for approximating the number of perfect matchings in a dense graph and for approximating the partition function of a ferromagnetic Ising system, and we give an efficient algorithm to estimate the entropy of a random walk on an unweighted graph. Abstract We consider random walks in dynamic random environments given by Markovian dynamics on Z d . We assume that the environment has a stationary distribution μ and satisfies the Poincare inequality w.r.t. μ . The random walk is a perturbation of another random walk (called “unperturbed”). We assume that also the environment viewed from the unperturbed random walk has stationary distribution μ . Both perturbed and unperturbed random walks can depend heavily on the environment and are not assumed to be finite-range. We derive a law of large numbers, an averaged invariance principle for the position of the walker and a series expansion for the asymptotic speed. We also provide a condition for non-degeneracy of the diffusion, and describe in some details equilibrium and convergence properties of the environment seen by the walker. All these results are based on a more general perturbative analysis of operators that we derive in the context of L 2 - bounded perturbations of Markov processes by means of the so-called Dyson–Phillips expansion. In this paper, we derive upper bounds for the heat kernel of the simple random walk on the infinite cluster of a supercritical long range percolation process. For any @math and for any exponent @math giving the rate of decay of the percolation process, we show that the return probability decays like @math up to logarithmic corrections, where @math denotes the time the walk is run. Moreover, our methods also yield generalized bounds on the spectral gap of the dynamics and on the diameter of the largest component in a box. Besides its intrinsic interest, the main result is needed for a companion paper studying the scaling limit of simple random walk on the infinite cluster. Invariance properties of random walks in bounded domains are a topic of growing interest since they contribute to improving our understanding of diffusion in confined geometries. Recently, limited to Pearson random walks with exponentially distributed straight paths, it has been shown that under isotropic uniform incidence, the average length of the trajectories through the domain is independent of the random walk characteristic and depends only on the ratio of the volume's domain over its surface. In this paper, thanks to arguments of integral geometry, we generalize this property to any isotropic bounded stochastic process and we give the conditions of its validity for isotropic unbounded stochastic processes. The analytical form for the traveled distance from the boundary to the first scattering event that ensures the validity of the Cauchy formula is also derived. The generalization of the Cauchy formula is an analytical constraint that thus concerns a very wide range of stochastic processes, from the original Pearson random walk to a Rayleigh distribution of the displacements, covering many situations of physical importance.