[Paper Review] The near-critical two-point function for weakly self-avoiding walk in high dimensions

Gordon Slade|arXiv (Cornell University)|Jul 31, 2020

Stochastic processes and statistical mechanics5 citations

TL;DR

This paper uses the lace expansion to analyze the two-point function of weakly self-avoiding walk on $ℤ^d$ for $d>4$ near the critical point, establishing an upper bound of $|x|^{-(d-2)}\exp[-c|x|/\xi]$, where the correlation length $\xi$ diverges as a square root at criticality. It further proves a plateau in the two-point function on a discrete torus, offering a new elementary proof for simple random walk in $d>2$.

ABSTRACT

We use the lace expansion to study the long-distance decay of the two-point function of weakly self-avoiding walk on the integer lattice $\mathbb{Z}^d$ in dimensions $d>4$, in the vicinity of the critical point, and prove an upper bound $|x|^{-(d-2)}\exp[-c|x|/\xi]$, where the correlation length $\xi$ has a square root divergence at the critical point. As one application, we prove that the two-point function for weakly self-avoiding walk on a discrete torus in dimensions $d>4$ has a ``plateau.'' A byproduct of the latter is an elementary proof of a similar plateau for simple random walk on a torus in dimensions $d>2$.

Motivation & Objective

• To understand the long-distance decay of the two-point function for weakly self-avoiding walk near the critical point in high dimensions.
• To establish rigorous upper bounds on the two-point function in the vicinity of the critical point.
• To analyze the behavior of the two-point function on a discrete torus, particularly the emergence of a plateau.
• To provide an elementary proof of the plateau phenomenon for simple random walk on a torus in dimensions $d>2$.

Proposed method

• Application of the lace expansion to control the long-range behavior of the two-point function in weakly self-avoiding walk.
• Derivation of an upper bound involving $|x|^{-(d-2)}\exp[-c|x|/\xi]$, with $\xi$ representing the correlation length.
• Analysis of the divergence of $\xi$ as $\xi \sim (p_c - p)^{-1/2}$ near the critical point $p_c$.
• Use of the lace expansion framework to handle the self-avoidance constraint in high dimensions ($d>4$).
• Adaptation of the method to study the two-point function on a discrete torus, exploiting periodic boundary conditions.
• Derivation of a plateau in the two-point function on the torus by leveraging the correlation length behavior and lattice geometry.

Experimental results

Research questions

• RQ1 How does the two-point function of weakly self-avoiding walk decay at long distances near the critical point in high dimensions?
• RQ2 What is the precise dependence of the correlation length $\xi$ on the distance to the critical point in $d>4$?
• RQ3 Does the two-point function on a discrete torus exhibit a plateau in the high-dimensional regime?
• RQ4 Can the plateau behavior for simple random walk on a torus be proven using elementary methods derived from the weakly self-avoiding walk analysis?
• RQ5 What role does the lace expansion play in establishing decay bounds and plateau phenomena in these models?

Key findings

• An upper bound of the form $|x|^{-(d-2)}\exp[-c|x|/\xi]$ is established for the two-point function of weakly self-avoiding walk in $d>4$ near the critical point.
• The correlation length $\xi$ diverges as a square root at the critical point, i.e., $\xi \sim (p_c - p)^{-1/2}$.
• On a discrete torus in $d>4$, the two-point function exhibits a plateau, indicating a regime of slow decay before exponential decay sets in.
• An elementary proof is provided for the plateau in the two-point function of simple random walk on a torus in dimensions $d>2$.
• The lace expansion method successfully captures both the critical decay behavior and the finite-size effects on the torus.
• The results demonstrate the robustness of the plateau phenomenon across different types of lattice walks in high dimensions.