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 .
Dans ce papier, nous initions l’etude des proprietes analytiques du noyau de la chaleur de Liouville. En particulier, nous etablissons des estimees de regularite pour le noyau et nous l’encadrons par des bornes inferieures et superieures non triviales.  This book is an introduction to the modern approach to the theory of Markov chains. The main goal of this approach is to determine the rate of convergence of a Markov chain to the stationary distribution as a function of the size and geometry of the state space. The authors develop the key tools for estimating convergence times, including coupling, strong stationary times, and spectral methods. Whenever possible, probabilistic methods are emphasized. The book includes many examples and provides brief introductions to some central models of statistical mechanics. Also provided are accounts of random walks on networks, including hitting and cover times, and analyses of several methods of shuffling cards. As a prerequisite, the authors assume a modest understanding of probability theory and linear algebra at an undergraduate level. ""Markov Chains and Mixing Times"" is meant to bring the excitement of this active area of research to a wide audience.