Weakly self-avoiding walk on a high-dimensional torus

TL;DR

This paper investigates the length scale at which weakly self-avoiding walks on high-dimensional tori deviate from their behavior on infinite lattices, showing they behave similarly until reaching a length proportional to the square root of the volume.

Contribution

It proves that the partition function's asymptotic form on a high-dimensional torus matches that on z^d until the walk length reaches order V^{1/2}, supporting a conjecture about phase transition.

Findings

Partition function asymptotics match z^d until n  V^{1/2}

Walks of length  V^{1/2} feel the torus effects

Supports conjecture of a phase transition at n  V^{1/2}

Abstract

How long does a self-avoiding walk on a discrete $d$-dimensional torus have to be before it begins to behave differently from a self-avoiding walk on $\mathbb{Z}^d$? We consider a version of this question for weakly self-avoiding walk on a torus in dimensions $d>4$. On $\mathbb{Z}^d$ for $d>4$, the partition function for $n$-step weakly self-avoiding walk is known to be asymptotically purely exponential, of the form $A\mu^n$, where $\mu$ is the growth constant for weakly self-avoiding walk on $\mathbb{Z}^d$. We prove the identical asymptotic behaviour $A\mu^n$ on the torus (with the same $A$ and $\mu$ as on $\mathbb{Z}^d$) until $n$ reaches order $V^{1/2}$, where $V$ is the number of vertices in the torus. This shows that the walk must have length of order at least $V^{1/2}$ before it "feels" the torus in its leading asymptotics. Our results support the conjecture that the behaviour of the partition function does change once $n$ reaches $V^{1/2}$, and we relate this to a conjectural critical scaling window which separates the dilute phase $n \ll V^{1/2}$ from the dense phase $n \gg V^{1/2}$. To prove the conjecture and to establish the existence of the scaling window remains a challenging open problem. The proof uses a novel lace expansion analysis based on the "plateau" for the torus two-point function obtained in previous work.