The emptiness formation probability of the XY chain, in the double-scaling limit near its critical lines, is governed by a Painleve V tau function; the result is exact at the Ising point and numerically supported elsewhere.
A representation of joint moments of CUE characteristic polynomials in terms of Painleve functions
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abstract
We establish a representation of the joint moments of the characteristic polynomial of a CUE random matrix and its derivative in terms of a solution of the sigma-Painleve V equation. The derivation involves the analysis of a formula for the joint moments in terms of a determinant of generalised Laguerre polynomials using the Riemann-Hilbert method. We use this connection with the sigma-Painleve V equation to derive explicit formulae for the joint moments and to show that in the large-matrix limit the joint moments are related to a solution of the sigma-Painleve III equation. Using the conformal block expansion of the tau-functions associated with the sigma-Painleve V and the sigma-Painleve III equations leads to general conjectures for the joint moments.
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Emptiness formation probability and Painlev\'e V equation in the XY spin chain
The emptiness formation probability of the XY chain, in the double-scaling limit near its critical lines, is governed by a Painleve V tau function; the result is exact at the Ising point and numerically supported elsewhere.