The random cluster model and new summation and integration identities

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 π/2 cos−1( √ q/2) is a rational number, and if it is a rational number whether the denominator is an odd integer. Our consideration lealds to new summation identities and, for q = 2, a closed-form evaluation of the integral 1 4π2 ∫ 2π 0 dθ ∫ 2π 0 dφ ln [A + B + C −A cos θ −B cosφ− C cos(θ + φ)] = − ln(2S) + (2/π)[Ti2(AS) + Ti2(BS) + Ti2(CS)], where A,B,C ≥ 0 and S = 1/ √ AB + BC + CA.