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

Abstract

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

Figures (2)

• Figure 1: Examples of $L \in \mathcal{L}_{\tau}^{(N)}[a,b]$ for $N=1,2,3$. Each arc represents an edge, with edge lengths not exceeding $\tau$. When $b-a$ exceeds $\tau$, $\mathcal{L}_{\tau}^{(1)}[a,b]=\varnothing$.
• Figure 2: A sample lace $L$ in $\mathcal{L}^{(2)}_{\tau}$ and a compatible edge $s't'$ in $\mathcal{C}_{\tau}(L)\backslash \mathcal{C}(L)$. The black edges are shorter than $\tau$, while the blue edge is longer than $\tau+1$ in length. The right figure provides a diagrammatic representation of the left graph, where each black line denotes a self-avoiding subpath of $\omega$. In particular, $\omega_{s_1}=\omega_{t_1}=o$, $\omega_{s_2}=\omega_{t_2}=x$ and $\omega_{s'}=\omega_{t'}=y$. These subpaths exhibit mutual avoidance effects.

Theorems & Definitions (16)

• Definition 1
• Remark 1
• Theorem 1.1
• Proposition 1.2
• Proposition 1.3
• Theorem 2.1: $hs02$
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Remark 2