Optimal Cooperation and Submodularity for Computing Potts' Partition Functions with a Large Number of State

TL;DR

This paper introduces an efficient polynomial-time combinatorial algorithm for computing the dominant contribution to the Potts model's partition function in the large-q limit, leveraging submodular function minimization.

Contribution

It presents the optimal cooperation algorithm, a novel method for minimizing a specific submodular function related to the Potts model, applicable to any lattice.

Findings

Algorithm operates in polynomial time

Effective for large-q Potts models

Practical implementation details provided

Abstract

The partition function of the q-state Potts model with random ferromagnetic couplings in the large-q limit is generally dominated by the contribution of a single diagram of the high temperature expansion. Computing this dominant diagram amounts to minimizing a particular submodular function. We provide a combinatorial optimization algorithm, the optimal cooperation algorithm, which works in polynomial time for any lattice. Practical implementation and the speed of the method is also discussed.