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Quantum Painleve II (QPII) and Painleve representation of Tracy-Widom distribution for \(\beta = 6\) Rumanov, Igor


Quantum Painleve equations (QP) are Fokker-Planck (or non-stationary Schroedinger) equations in two independent variables (``time" and ``space")with diffusion-drift operators being quantized Painleve Hamiltonians. They are satisfied by certain eigenvalue probabilities of beta ensembles. E.g. QPII describes the soft edge limit of beta ensembles while QPIII does so for the hard edge. QP are relatively simple instances of (confluent) Belavin-Polyakov-Zamolodchikov (BPZ) equations of conformal field theory (CFT). The general multidimensional linear BPZ PDEs also naturally arise from beta-ensemble integrals. While CFT is known as quantum integrable theory, we show on the example of QPII how classical integrable structure can be extended to all values of beta. Using explicit Lax pair for even integer beta with QPII solution as eigenvector component (known before only for beta = 2 and 4), the case beta=6 is further studied. It turns out that again everything depends on the Hastings-McLeod solution of Painleve II (PII). The main result is a second order nonlinear ODE for the log-derivative of Tracy-Widom distribution for beta = 6, involving the PII function in the coefficients. Beyond even integers, the derived nonlinear integrable system associated with QPII possesses identically nice analytic properties for all beta, e.g. the Painleve property. Its general solutions are related by a Cole-Hopf transform with two linearly independent solutions of QPII.

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