Exact finite-size corrections for the spanning-tree model under different boundary conditions

TL;DR

This paper derives exact finite-size corrections for the spanning-tree model on finite square lattices with various boundary conditions, providing precise asymptotic expansions and confirming conformal field theory predictions.

Contribution

It presents exact asymptotic expansions of the spanning-tree partition function for different boundary conditions and establishes identities relating these functions, clarifying discrepancies with conformal field theory.

Findings

Exact asymptotic expansions for each boundary condition

Identities relating spanning-tree partition functions

Corner free energy matches conformal field theory predictions

Abstract

We express the partition functions of the spanning tree on finite square lattices under five different sets of boundary conditions (free, cylindrical, toroidal, M\"obius strip, and Klein bottle) in terms of a principal partition function with twisted boundary conditions. Based on these expressions, we derive the exact asymptotic expansions of the logarithm of the partition function for each case. We have also established several groups of identities relating spanning-tree partition functions for the different boundary conditions. We also explain an apparent discrepancy between logarithmic correction terms in the free energy for a two dimensional spanning tree model with periodic and free boundary conditions and conformal field theory predictions. We have obtain corner free energy for the spanning tree under free boundary conditions in full agreement with conformal field theory predictions.