Scaling limit of ballistic self-avoiding walk interacting with spatial random permutations

Volker Betz,Lorenzo Taggi

doi:10.1214/19-ejp328

Abstract

We consider nearest neighbour spatial random permutations on $\mathbb Z ^{d}$. In this case, the energy of the system is proportional to the sum of all cycle lengths, and the system can be interpreted as an ensemble of edge-weighted, mutually self-avoiding loops. The constant of proportionality, $\alpha $, is the order parameter of the model. Our first result is that in a parameter regime of edge weights where it is known that a single self-avoiding loop is weakly space filling, long cycles of spatial random permutations are still exponentially unlikely. For our second result, we embed a self-avoiding walk into a background of spatial random permutations, and condition it to cover a macroscopic distance. For large values of $\alpha $ (where long cycles are very unlikely) we show that this walk collapses to a straight line in the scaling limit, and give bounds on the fluctuations that are almost sufficient for diffusive scaling. For proving our results, we develop the concepts of spatial strong Markov property and iterative sampling for spatial random permutations, which may be of independent interest. Among other things, we use them to show exponential decay of correlations for large values of $\alpha $ in great generality.

Highlights

Self-avoiding random walks are a classical topic of modern probability theory, many questions still remain to be answered; we refer to the classic book [34] and the more recent survey [38]
We restrict to nearest neighbour spatial random permutations, as they are most closely related to self-avoiding walks
We give a somewhat negative result: we show that if there is a regime of space filling cycles, it must start at lower α than for the case of the self-avoiding polygon

Summary

Introduction

Self-avoiding random walks are a classical topic of modern probability theory, many questions still remain to be answered; we refer to the classic book [34] and the more recent survey [38]. The reason is that with probability one (in the limit n → ∞) a short cycle from the background will cross the vertical hyperplane containing the regeneration point, and introduce correlations We solve both problems by developing a method for estimating the decay of correlations for spatial random permutations. The loop O(n) model has been conject√ured to undergo a Kosterlitz-Thouless phase transition at the critical threshold log 2 + 2 − n when n ≤ 2 [33] This is compatible with our general finding that on every vertex-transitive graph the critical threshold of spatial random permutations, which corresponds more or less to the n = 2 case, is strictly less than the critical threshold for the self-avoiding walk, corresponding to the n = 0 case.

Definitions and main results

Cycle length and partition function

Iterative sampling and the spatial Markov property

The number of regeneration points

The Markov chain of regeneration pairs