The Self-Avoiding Walk pp 229-255 | Cite as

# 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)].

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