A tree-decomposed transfer matrix for computing exact Potts model partition functions for arbitrary graphs, with applications to planar graph colourings

TL;DR

This paper introduces a novel tree-decomposed transfer matrix algorithm for efficiently computing exact Potts model partition functions and chromatic polynomials on arbitrary graphs, especially planar ones, with significant speed improvements.

Contribution

The authors develop a general, efficient algorithm combining tree decomposition and transfer matrix methods for exact partition function calculations on arbitrary graphs, notably improving performance on planar graphs.

Findings

Achieved sub-exponential average running time of ~ exp(1.516 sqrt(N)) for large planar graphs.

Compared chromatic roots statistics of random planar graphs with regular lattice data.

Demonstrated the algorithm's effectiveness on graphs with up to 100 vertices.

Abstract

Combining tree decomposition and transfer matrix techniques provides a very general algorithm for computing exact partition functions of statistical models defined on arbitrary graphs. The algorithm is particularly efficient in the case of planar graphs. We illustrate it by computing the Potts model partition functions and chromatic polynomials (the number of proper vertex colourings using Q colours) for large samples of random planar graphs with up to N=100 vertices. In the latter case, our algorithm yields a sub-exponential average running time of ~ exp(1.516 sqrt(N)), a substantial improvement over the exponential running time ~ exp(0.245 N) provided by the hitherto best known algorithm. We study the statistics of chromatic roots of random planar graphs in some detail, comparing the findings with results for finite pieces of a regular lattice.