The random cluster model and new summation and integration identities

TL;DR

This paper evaluates the free energy of the random cluster model at criticality for 0 < q < 4, deriving new summation identities and a closed-form integral evaluation for q=2, revealing dependence on rationality conditions.

Contribution

It provides explicit evaluations of the free energy at criticality and introduces new summation identities and integral formulas related to the random cluster model.

Findings

Explicit free energy expressions depending on rationality of a specific angle

New summation identities derived from the analysis

Closed-form integral evaluation for q=2 case

Abstract

We explicitly evaluate the free energy of the random cluster model at its critical point for 0 < q < 4 using an exact result due to Baxter, Temperley and Ashley. It is found that the resulting expression assumes a form which depends on whether $\pi/2\cos^{-1}[\sqrt(q)/2]$ is a rational number, and if it is a rational number whether the denominator is an odd integer. Our consideration leads to new summation identities and, for q = 2, a closed-form evaluation of the integral [1/(4\pi^2)] \int_0^{2\pi}dx \int_0^{2\pi}dy ln[A + B + C - A cos x - B cos y - C cos(x + y)] = -\ln(2S) + (2/\pi)[Ti_2(AS) + Ti_2(BS) + Ti_2(CS)], where A, B, C >=0 and S = 1/\sqrt{AB+BC+CA}.