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Publication# Crossing velocities for an annealed random walk in a random potential

Abstract

We consider a random walk in an i.i.d. non-negative potential on the d-dimensional integer lattice. The walk starts at the origin and is conditioned to hit a remote location y on the lattice. We prove that the expected time under the annealed path measure needed by the random walk to reach y grows only linearly in the distance from y to the origin. In dimension 1 we show the existence of the asymptotic positive speed. (C) 2011 Elsevier B.V. All rights reserved.

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Related concepts (26)

Related publications (26)

Random walk

In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space. An elementary example of a random walk is the random walk on the integer number line which starts at 0, and at each step moves +1 or −1 with equal probability. Other examples include the path traced by a molecule as it travels in a liquid or a gas (see Brownian motion), the search path of a foraging animal, or the price of a fluctuating stock and the financial status of a gambler.

Simulated annealing

Simulated annealing (SA) is a probabilistic technique for approximating the global optimum of a given function. Specifically, it is a metaheuristic to approximate global optimization in a large search space for an optimization problem. For large numbers of local optima, SA can find the global optima. It is often used when the search space is discrete (for example the traveling salesman problem, the boolean satisfiability problem, protein structure prediction, and job-shop scheduling).

Hill climbing

In numerical analysis, hill climbing is a mathematical optimization technique which belongs to the family of local search. It is an iterative algorithm that starts with an arbitrary solution to a problem, then attempts to find a better solution by making an incremental change to the solution. If the change produces a better solution, another incremental change is made to the new solution, and so on until no further improvements can be found. For example, hill climbing can be applied to the travelling salesman problem.

An integer program (IP) is a problem of the form $\min \{f(x) : \, Ax = b, \ l \leq x \leq u, \ x \in \Z^n\}$, where $A \in \Z^{m \times n}$, $b \in \Z^m$, $l,u \in \Z^n$, and $f: \Z^n \rightarrow \Z$ is a separable convex objective function.
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