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.

Keywords

Mathematical Subject Classification

Milestones

Authors