Bulk, surface and corner free energy series for the chromatic polynomial on the square and triangular lattices
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.
Keywords
Cite
@article{arxiv.1005.3609,
title = {Bulk, surface and corner free energy series for the chromatic polynomial on the square and triangular lattices},
author = {Jesper Lykke Jacobsen},
journal= {arXiv preprint arXiv:1005.3609},
year = {2015}
}
Comments
17 pages. Version 2: added 4 further term to the series