Rate of convergence of the critical point of the memory-$\tau$ self-avoiding walk in dimensions $d>4$

TL;DR

This paper investigates how quickly the critical point of the memory-$\tau$ self-avoiding walk approaches that of the standard self-avoiding walk in high dimensions, establishing a convergence rate of $\tau^{-(d-2)/2}$ using the lace expansion.

Contribution

It provides the first quantitative rate of convergence for the critical point of the memory-$\tau$ walk to the self-avoiding walk in dimensions greater than four.

Findings

Critical point convergence rate is $\tau^{-(d-2)/2}$.

The lace expansion technique is used for the proof.

Convergence is monotonic as $\tau$ increases.

Abstract

We consider spread-out models of the self-avoiding walk and its finite-memory version, known as the memory-$\tau$ walk, which prohibits loops whose length is at most $\tau$, in dimensions $d>4$. The critical point is defined as the radius of convergence of the generating function for each model. It is known that the critical point of the memory-$\tau$ walk is non-decreasing in $\tau$ and converges to that of the self-avoiding walk as $\tau$ tends to infinity. In this paper, we study the rate at which the critical point of the memory-$\tau$ walk converges to that of the self-avoiding walk and show that the order is $\tau^{-(d-2)/2}$. The proof relies on the lace expansion, introduced by Brydges and Spencer.