Kesten’s pattern theorem
2015
Extract Patterns A pattern is a finite subwalk in a self-avoiding walk. A pattern theorem proves that there exists a class of patterns (called proper patterns) which will occur with positive density on all, except exponentially few, walks of length n, if n is large. The definition of a pattern may be relaxed to include events, for example, the event that a walk fills a d-dimensional cube of side-length L and with its centre in a vertex of the walk. In this appendix, Kesten’s proof of a pattern theorem [348] for the self-avoiding walk is presented, using the approach of the presentation in reference [399]. Definition C.1A pattern P with vertices (p0,p1,…,pN) is said to occur at the j-th vertex of a walk ω=(ω0,ω1,…,ωn) if there is a vector v⃗∈Zd such that ωj+ℓ=pℓ+v⃗ for ℓ=0,1,…,N. The pattern P occurs k...