High-temperature series expansions for the $q$-state Potts model on a hypercubic lattice and critical properties of percolation

TL;DR

This paper develops high-temperature series expansions for the $q$-state Potts model on hypercubic lattices, enabling estimation of percolation thresholds and critical exponents across various dimensions.

Contribution

It provides extended series expansions for the Potts model and percolation properties, including new estimates of critical parameters for different dimensions.

Findings

Series expansions up to order 20 for D≤3

Estimates of percolation threshold $p_c$ and exponent $3$ for bond percolation

Extended 1/D expansion of critical coupling up to order D^{-9}

Abstract

We present results for the high-temperature series expansions of the susceptibility and free energy of the $q$-state Potts model on a $D$-dimensional hypercubic lattice $\mathbb{Z}^D$ for arbitrary values of $q$. The series are up to order 20 for dimension $D\leq3$, order 19 for $D\leq 5$ and up to order 17 for arbitrary $D$. Using the $q\to 1$ limit of these series, we estimate the percolation threshold $p_c$ and critical exponent $\gamma$ for bond percolation in different dimensions. We also extend the 1/D expansion of the critical coupling for arbitrary values of $q$ up to order $D^{-9}$.