Painlevé Functions and difference equations

But perhaps the most dramatic discoveries were published from 1973 to 1981 on the spin correlation functions of the Ising model and most particularly the results that in the scaling limit where $T\rightarrow T_c$ and the separation of the spins $N \rightarrow \infty$ such that |T-T_c|N=r is fixed that the correlation function [14] $<\sigma_{0,0}\sigma_{0,N}\gt$divided by |T-T_c|^1/4 is  

$$G_{\pm}(2r)=(1\mp\eta)\eta^{-1/2}{\rm exp}\int_r^{\infty}dx{1\over 4}x\eta^{-2}[(1-\eta^2)^2-\eta'^2]$$ (2)

where $\eta(x)$ satisfies the third equation of Painlevé [15]  

$${d^2\eta\over dx^2}={1\over \eta}({d\eta\over dx})^2-{1\over... ...+{1\over x}(\alpha\eta^2+\beta)+\gamma\eta^3+{\delta\over \eta}$$ (3)

with $\alpha=\beta=0$ and $\gamma=-\delta=1$ and

$$\eta(x)\rightarrow 1-{2\over \pi}K_0(2x)~~~{\rm as}~~x\rightarrow \infty$$ (4)

where K₀(2x) is the modified Bessel function of order zero. Furthermore on the lattice all the correlation functions satisfy quadratic difference equations [16].

This discovery of Painlevé equations in the Ising model was the beginning of a host of developments in mathematical physics which continues in an ever expanding volume to this day. It lead Sato, Miwa, and Jimbo [17] to their celebrated series of work on isomonodromic deformation and to the solution of the distribution of eigenvalue of the GUE random matrix problem [18] in terms of a Painlevé V function. This has subsequently been extended by many people, including one of our original collaborators, Craig Tracy [19], to so many branches of physics and mathematics including random matrix theory matrix models in quantum gravity and random permutations that entire semester long work shops are now devoted to the subject. Indeed a recent book on special functions [20] characterized Painlevé functions and ''the special functions of the 21st century.'' Rarely has the solution to one small problem in physics had so many ramifications in so many different fields.