Form factor expansions in the 2D Ising model and Painleve VI
Vladimir V Mangazeev, Anthony J Guttmann
NUCLEAR PHYSICS B | ELSEVIER SCIENCE BV | Published : 2010
We derive a Toda-type recurrence relation, in both high- and low-temperature regimes, for the λ-extended diagonal correlation functions C(N,N;λ) of the two-dimensional Ising model, using an earlier connection between diagonal form factor expansions and tau-functions within Painlevé VI (PVI) theory, originally discovered by Jimbo and Miwa. This greatly simplifies the calculation of the diagonal correlation functions, particularly their λ-extended counterparts.We also conjecture a closed form expression for the simplest off-diagonal case C±(0,1;λ) where a connection to PVI theory is not known. Combined with the results for diagonal correlations these give all the initial conditions required fo..View full abstract
The authors thank V.V. Bazhanov, M.T. Batchelor, J. de Gier and S. Sergeev for valuable comments and interesting discussions while this work was being conducted, and P.J. Forrester, J.-M. Maillard, B.M. McCoy, B.G. Nickel and W.P. Orrick for valuable comments which significantly improved the manuscript. This work is supported by the Australian Research Council.