Critical behaviour of self-avoiding walks: that cross a square

Abstract

Consider the set of all self-avoiding walks in the square lattice which start at (0, 0), end at (L, L), and are entirely contained in the square (0, L)*(0, L). Associate a fugacity x with each step of the walk. Whittington and Guttmann (1990) showed that the dominant walks have O(L) steps when x is small and O(L2) steps when x is large, and they conjectured that there is a single transition point at x= mu -1 where mu is the inverse of the connective constant for (unconstrained) self-avoiding walks. We present a rigorous proof of this conjecture (and its analogue in higher dimensions). We also discuss what can be said rigorously about two scaling exponents associated with this phase transition, and compare this with analogous results that have been obtained exactly (and rigorously) on the discrete Sierpinski gasket by Hattori, Hattori and Kusuoka (1990).