The random cluster model and new summation and integration identities

Details The random cluster model and new summation and integration identities The random cluster model and new summation and integration identities

2005-01-11

arxiv.org/abs/cond-mat/0501228v2

J. Phys. A: Math. Gen. 38, 6271-6276 (2005).

Physics

Condensed Matter

Statistical Mechanics

15 pages

Scientific paper

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}.

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