Nonlinear Waves Seminar: Igor Rumanov
Quantum Painleve II (QPII) and classical Painleve II (PII): beta ensembles for beta = 6.
Igor Rumanov
Applied Mathematics,Ìý
Date and time:Ìý
Tuesday, September 23, 2014 - 4:30pm
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ECOT 226
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Quantum Painleve equations are Fokker-Planck/non-stationary Schroedinger equations satisfied by certain eigenvalue probabilities of Dyson beta ensembles of random matrices which can be interpreted as certain correlation functions (or rather conformal blocks) of conformal field theory. QPII describes the soft edge limit of beta ensembles.
Using explicit Lax pairs with QPII solutions as eigenvector components we found for even integer values of beta, the case beta=6 is further studied here. This is the smallest even beta, when the corresponding Lax pair and its relation to classical PII have not been known before, unlike cases beta = 2 and 4. It turns out that again everything can be expressed in terms of the Hastings-McLeod solution of PII. A second order nonlinear ODE for the log-derivative of Tracy-Widom distribution for beta = 6, involving the PII function in the coefficients, is found, which allows one to compute asymptotics for the distribution function. The ODE is a consequence of a linear system of three ODEs for which the local singularity analysis yields series solutions with exponents in the set 4/3, 1/3 and -2/3.