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)].
Unable to display preview. Download preview PDF.