The Ising correlation C(M, N) for ν = −k

Abstract

Abstract We present Painlevé VI sigma form equations for the general Ising low and high temperature two-point correlation functions C(M, N) with M ⩽ N in the special case ν = −k where ν = sinh 2E h /k B T/sinh 2E v /k B T. More specifically four different non-linear ODEs depending explicitly on the two integers M and N emerge: these four non-linear ODEs correspond to distinguish respectively low and high temperature, together with M + N even or odd. These four different non-linear ODEs are also valid for M ⩾ N when ν = −1/k. For the low-temperature row correlation functions C(0, N) with N odd, we exhibit again for this selected ν = −k condition, a remarkable phenomenon of a Painlevé VI sigma function being the sum of four Painlevé VI sigma functions having the same Okamoto parameters. We show in this ν = −k case for T < T c and also T > T c, that C(M, N) with M ⩽ N is given as an N × N Toeplitz determinant.