Finite-size corrections for universal boundary entropy in bond percolation

TL;DR

This paper derives exact and asymptotic formulas for the boundary entropy in bond percolation on a square lattice, revealing universal corrections and a crossover phenomenon related to boundary loop weights.

Contribution

It provides the first rigorous finite-size corrections for boundary entropy in bond percolation, including universal logarithmic terms and a crossover at a specific boundary weight.

Findings

Exact expressions for boundary entropy on strip and cylinder geometries.

Asymptotic analysis of finite-size corrections to arbitrary order.

Observation of a crossover at a special boundary loop weight.

Abstract

We compute the boundary entropy for bond percolation on the square lattice in the presence of a boundary loop weight, and prove explicit and exact expressions on a strip and on a cylinder of size $L$. For the cylinder we provide a rigorous asymptotic analysis which allows for the computation of finite-size corrections to arbitrary order. For the strip we provide exact expressions that have been verified using high-precision numerical analysis. Our rigorous and exact results corroborate an argument based on conformal field theory, in particular concerning universal logarithmic corrections for the case of the strip due to the presence of corners in the geometry. We furthermore observe a crossover at a special value of the boundary loop weight.