The Ramdom Cluster Model and New Summation and Integration Identities

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 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}.