Self-avoiding walk on the complete graph

Abstract

There is an extensive literature concerning self-avoiding walk on infinite graphs, but the subject is relatively undeveloped on finite graphs. The purpose of this paper is to elucidate the phase transition for self-avoiding walk on the simplest finite graph: the complete graph. We make the elementary observation that the susceptibility of the self-avoiding walk on the complete graph is given exactly in terms of the incomplete gamma function. The known asymptotic behaviour of the incomplete gamma function then yields a complete description of the finite-size scaling of the self-avoiding walk on the complete graph. As a basic example, we compute the limiting distribution of the length of a self-avoiding walk on the complete graph, in subcritical, critical, and supercritical regimes. This provides a prototype for more complex unsolved problems such as the self-avoiding walk on the hypercube or on a highdimensional torus.

The length of self-avoiding walks on the complete graph

“…In addition, the limit theorems given in Theorem 1.2 in Cases (v) and (vi) agree with Eqs. (1.27) and (1.26) from [27]. In Case (iii), our limiting distribution expressed in terms of a conditioned normal variable appears superficially different to [27,Eq.…”

“…(1.27) and (1.26) from [27]. In Case (iii), our limiting distribution expressed in terms of a conditioned normal variable appears superficially different to [27,Eq. (1.28)], which is characterised in terms of a moment generating function, however it can be easily verified that the results are equivalent.…”

“…In Cases (i) and (ii), our standardisation of L n yields non-degenerate limits which are not equivalent to the limits presented in Eqs. (1.30) and (1.29) of [27].…”

“…The main technical tool used in the proof of Theorems 1.1 and 1.2 is a uniform asymptotic expansion of the incomplete gamma function [26]. During the final stages of preparation of this article, we became aware of the very recent manuscript [27], which also studies SAW on the complete graph. The asymptotic expressions for the mean given in Theorem 1.1 are also presented in [27].…”

“…During the final stages of preparation of this article, we became aware of the very recent manuscript [27], which also studies SAW on the complete graph. The asymptotic expressions for the mean given in Theorem 1.1 are also presented in [27]. In addition, the limit theorems given in Theorem 1.2 in Cases (v) and (vi) agree with Eqs.…”