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

TL;DR

This paper studies the behavior of self-avoiding walks interacting with spatial random permutations, revealing conditions under which long cycles are unlikely and the walk tends to a straight line, with new methods for analyzing such systems.

Contribution

It introduces the concepts of spatial strong Markov property and iterative sampling for spatial random permutations, advancing understanding of their scaling limits and correlation decay.

Findings

Long cycles are exponentially unlikely in certain regimes.

Self-avoiding walks tend to a straight line at large parameters.

Develops new methods for analyzing spatial random permutations.

Abstract

We consider nearest neighbour spatial random permutations on $\mathbb{Z}^d$. In this case, the energy of the system is proportional 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.