Friday, August 17, 2012

1208.3345 (Anthony J. Guttmann et al.)

An integral arising from the chiral sl(n) Potts model    [PDF]

Anthony J. Guttmann, Mathew D. Rogers
We show that the integral J(t) = \frac{1}{\pi^3} \int_0^\pi \int_0^\pi \int_0^\pi dx dy dz \log(t-\cos{x}-\cos{y}-\cos{z}+\cos{x}\cos{y}\cos{z}), can be expressed in terms of ${_5F_4}$ hypergeometric functions. The integral arises in the solution by Baxter and Bazhanov of the free-energy of the $sl(n)$ Potts model, which includes the term $J(2).$ Our result immediately gives the logarithmic Mahler measure of the Laurent polynomial $$k-(x+\frac{1}{x}) - (y+\frac{1}{y}) - (z+\frac{1}{z}) + 1/4(x+\frac{1}{x}) (y+\frac{1}{y}) (z+\frac{1}{z})$$ in terms of the same hypergeometric functions.
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