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

摘要

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.