Bulk, surface and corner free energy series for the chromatic polynomial on the square and triangular lattices

TL;DR

This paper introduces an efficient algorithm to compute the chromatic polynomial's free energy series for square and triangular lattices, extending known series and verifying analytical formulas with high-order terms.

Contribution

It presents a new sparse matrix transfer matrix algorithm for large-q series expansion of the chromatic polynomial on 2D lattices, extending previous results and conjecturing exact formulas.

Findings

Extended the series for the square lattice by 32 terms, up to order q^{-79}.

Verified Baxter's formula for the triangular lattice bulk free energy up to q^{-40}.

Conjectured exact product formulas for surface and corner free energies.

Abstract

We present an efficient algorithm for computing the partition function of the q-colouring problem (chromatic polynomial) on regular two-dimensional lattice strips. Our construction involves writing the transfer matrix as a product of sparse matrices, each of dimension ~ 3^m, where m is the number of lattice spacings across the strip. As a specific application, we obtain the large-q series of the bulk, surface and corner free energies of the chromatic polynomial. This extends the existing series for the square lattice by 32 terms, to order q^{-79}. On the triangular lattice, we verify Baxter's analytical expression for the bulk free energy (to order q^{-40}), and we are able to conjecture exact product formulae for the surface and corner free energies.