Pattern theorems

Abstract

In this chapter we shall prove a useful theorem due to Kesten (1963) about the occurrence of patterns on self-avoiding walks, and investigate a number of its applications. Briefly, a pattern is a (short) self-avoiding walk that occurs as part of a longer self-avoiding walk. Kesten’s Pattern Theorem says that if a given pattern can possibly occur several times on a self-avoiding walk, then it must occur at least a N times on almost all N-step self-avoiding walks, for some a > 0 (in this context, “almost all” means “except for an exponentially small fraction”). This can be viewed as a weak analogue of classical “large deviations” estimates for the strong law of large numbers, which say that certain events have exponentially small probabilities [see for example Chapter 1 of Ellis (1985)].KeywordsTriangular LatticeOuter PointStep WalkDimensional CubeUnit Outer Normal VectorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.