In this paper we show that on bounded degree graphs and general trees, the cover time of the simple random walk is asymptotically equal to the product of the number of edges and the square of the expected supremum of the Gaussian free field on the graph, assuming that the maximal hitting time is significantly smaller than the cover time. Previously, this was only proved for regular trees and the 2D lattice. Furthermore, for general trees, we derive exponential concentration for the cover time, which implies that the standard deviation of the cover time is bounded by the geometric mean of the cover time and the maximal hitting time.
We consider the maximum of the discrete two dimensional Gaussian free field in a box, and prove the existence of a (dense) deterministic subsequence along which the maximum, centered at its mean, is tight. The method of proof relies on an argument developed by Dekking and Host for branching random walks with bounded increments and on comparison results specific to Gaussian fields. We present a deterministic algorithm that given a tree T with n vertices, a starting vertex v and a slackness parameter epsilon > 0, estimates within an additive error of epsilon the cover and return time, namely, the expected time it takes a simple random walk that starts at v to visit all vertices of T and return to v. The running time of our algorithm is polynomial in n epsilon, and hence remains polynomial in n also for epsilon = 1 n^ O(1) . We also show how the algorithm can be extended to estimate the expected cover (without return) time on trees. We consider the maximum of the discrete two-dimensional Gaussian free field (GFF) in a box and prove that its maximum, centered at its mean, is tight, settling a longstanding conjecture. The proof combines a recent observation by Bolthausen, Deuschel, and Zeitouni with elements from Bramson's results on branching Brownian motion and comparison theorems for Gaussian fields. An essential part of the argument is the precise evaluation, up to an error of order 1, of the expected value of the maximum of the GFF in a box. Related Gaussian fields, such as the GFF on a two-dimensional torus, are also discussed. © 2011 Wiley Periodicals, Inc. Disordered systems are an important class of models in sta- tistical mechanics, having the defining characteristic that the energy landscape is a fixed realization of a random field. Examples include var- ious models of glasses and polymers. They also arise in other areas, like fitness models in evolutionary biology. The ground state of a disordered system is the state with minimum energy. The system is said to be chaotic if a small perturbation of the energy landscape causes a dras- tic shift of the ground state. We present a rigorous theory of chaos in disordered systems that confirms long-standing physics intuition about connections between chaos, anomalous fluctuations of the ground state energy, and the existence of multiple valleys in the energy landscape. Combining these results with mathematical tools like hypercontractiv- ity, we establish the existence of the above phenomena in eigenvectors of GUE matrices, the Kauffman-Levin model of evolutionary biology, directed polymers in random environment, a subclass of the generalized Sherrington-Kirkpatrick model of spin glasses, the discrete Gaussian free field, and continuous Gaussian fields on Euclidean spaces. We also list several open questions. In this paper, we study the tightness of solutions for a fam- ily of recursive equations. These equations arise naturally in the study of random walks on tree-like structures. Examples include the maxi- mal displacement of branching random walk in one dimension, and the cover time of symmetric simple random walk on regular binary trees. Recursion equations associated with the distribution functions of these quantities have been used to establish weak laws of large numbers. Here, we use these recursion equations to establish the tightness of the corre- sponding sequences of distribution functions after appropriate centering. We phrase our results in a fairly general context, which we hope will fa- cilitate their application in other settings.