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

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

2010-05-20

arxiv.org/abs/1005.3609v2

Physics

Condensed Matter

Statistical Mechanics

17 pages. Version 2: added 4 further term to the series

Scientific paper

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.

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