The fixed irreducible bridge ensemble for self-avoiding walks

TL;DR

This paper introduces the fixed irreducible bridge ensemble for self-avoiding walks, conjectures its relation to SLE in the unit strip, and provides numerical evidence supporting the conjecture and boundary scaling estimates.

Contribution

It defines a new ensemble for self-avoiding walks, conjectures its connection to SLE, and offers numerical support and boundary exponent estimates.

Findings

Numerical evidence supports the conjectured relationship between the ensemble and SLE.

The paper estimates the boundary scaling exponent for the ensemble.

Conjectures the convergence of scaled bridge heights to a stable distribution.

Abstract

We define a new ensemble for self-avoiding walks in the upper half-plane, the fixed irredicible bridge ensemble, by considering self-avoiding walks in the upper half-plane up to their $n$-th bridge height, $Y_n$, and scaling the walk by $1/Y_n$ to obtain a curve in the unit strip, and then taking $n\to\infty$. We then conjecture a relationship between this ensemble to $\SLE$ in the unit strip from $0$ to a fixed point along the upper boundary of the strip, integrated over the conjectured exit density of self-avoiding walk spanning a strip in the scaling limit. We conjecture that there exists a positive constant $\sigma$ such that $n^{-\sigma}Y_n$ converges in distribution to that of a stable random variable as $n\to\infty$. Then the conjectured relationship between the fixed irreducible bridge scaling limit and $\SLE$ can be described as follows: If one takes a SAW considered up to $Y_n$ and scales by $1/Y_n$ and then weights the walk by $Y_n$ to an appropriate power, then in the limit $n\to\infty$, one should obtain a curve from the scaling limit of the self-avoiding walk spanning the unit strip. In addition to a heuristic derivation, we provide numerical evidence to support the conjecture and give estimates for the boundary scaling exponent.