Journal
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
Volume 46, Issue 33, Pages -Publisher
IOP PUBLISHING LTD
DOI: 10.1088/1751-8113/46/33/335203
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Funding
- ERC [279738-NEDFOQ]
- Ukrainian SFFR project [F53.2/028, F54.1/019]
- Program of Fundamental Research of the Physics and Astronomy Division of NASU
- IRSES project 'Random and Integrable Models in Mathematical Physics'
- Joint Program of Bilateral Seminar of CNRS
- Joint Program of Bilateral Seminar of NASU
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We elaborate on a recently conjectured relation of Painleve transcendents and 2D conformal field theory. General solutions of Painleve VI, V and III are expressed in terms of c = 1 conformal blocks and their irregular limits, AGT related to instanton partition functions in N = 2 supersymmetric gauge theories with N-f = 0, 1, 2, 3, 4. The resulting combinatorial series representations of Painleve functions provide an efficient tool for their numerical computation at finite values of the argument. The series involves sums over bipartitions which, in the simplest cases, coincide with Gessel expansions of certain Toeplitz determinants. Considered applications include Fredholm determinants of classical integrable kernels, scaled gap probability in the bulk of the Gaussian Unitary Ensemble, and all-order conformal perturbation theory expansions of correlation functions in the sine-Gordon field theory at the free-fermion point.
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