The O(n) model on the annulus

TL;DR

This paper derives an explicit scaling limit formula for the critical O(n) model on an annulus using Coulomb gas methods, capturing boundary effects and providing new results for percolation and self-avoiding loops.

Contribution

It introduces a Coulomb gas-based explicit formula for the O(n) model's partition function on an annulus, accounting for magnetic charge asymmetry and null state decoupling.

Findings

Matches previous conjectures and known results for special n values

Provides new formulas for percolation probabilities and self-avoiding loop partition functions

Offers explicit examples in logarithmic conformal field theory for n→0

Abstract

We use Coulomb gas methods to propose an explicit form for the scaling limit of the partition function of the critical O(n) model on an annulus, with free boundary conditions, as a function of its modulus. This correctly takes into account the magnetic charge asymmetry and the decoupling of the null states. It agrees with an earlier conjecture based on Bethe ansatz and quantum group symmetry, and with all known results for special values of n. It gives new formulae for percolation (the probability that a cluster connects the two opposite boundaries) and for self-avoiding loops (the partition function for a single loop wrapping non-trivially around the annulus.) The limit n->0 also gives explicit examples of partition functions in logarithmic conformal field theory.