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Testable Algorithms for SelfAvoiding Walks
, 1994
"... We present a polynomial time Monte Carlo algorithm for almost uniformly generating and approximately counting selfavoiding walks in rectangular lattices Z d . These are classical problems that arise, for example, in the study of long polymer chains. While there are a number of Monte Carlo algorit ..."
Abstract

Cited by 17 (3 self)
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We present a polynomial time Monte Carlo algorithm for almost uniformly generating and approximately counting selfavoiding walks in rectangular lattices Z d . These are classical problems that arise, for example, in the study of long polymer chains. While there are a number of Monte Carlo algorithms used to solve these problems in practice, these are heuristic and their correctness relies on unproven conjectures. In contrast, our algorithm depends on a single, widelybelieved conjecture that is weaker than preceding assumptions, and, more importantly, is one which the algorithm itself can test. Thus our algorithm is reliable, in the sense that it either outputs answers that are guaranteed, with high probability, to be correct, or finds a counterexample to the conjecture. 1 Summary 1.1 Background A selfavoiding walk in a graph is a walk which starts at a fixed origin and passes through each vertex at most once. This paper is concerned with selfavoiding walks in lattices, in par...
Unsolved problems concerning random walks on trees
 Classical and Modern Branching Processes, K. Athreya and P. Jagers (editors
, 1997
"... Abstract. We state some unsolved problems and describe relevant examples concerning random walks on trees. Most of the problems involve the behavior of random walks with drift: e.g., is the speed on GaltonWatson trees monotonic in the drift parameter? These random walks have been used in MonteCarl ..."
Abstract

Cited by 9 (1 self)
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Abstract. We state some unsolved problems and describe relevant examples concerning random walks on trees. Most of the problems involve the behavior of random walks with drift: e.g., is the speed on GaltonWatson trees monotonic in the drift parameter? These random walks have been used in MonteCarlo algorithms for sampling from the vertices of a tree; in general, their behavior reflects the size and regularity of the underlying tree. Random walks are related to conductance. The distribution function for the conductance of GaltonWatson trees satisfies an interesting functional equation; is this distribution function absolutely continuous? §1. Introduction. To explore the structure of irregular trees, we consider nearestneighbor random walks on them. The behavior of simple random walk gives some information about the structure, but more can be gleaned by considering the oneparameter family of random walks RWλ described below. That is, the behavior of such random walks on spherically symmetric