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PDE for spectral gap probabilities of random single and coupled matrices and Toda lattice.

Mathematical Physics & Probability

Speaker: Igor Rumanov, UC Davis
Location: 1147 MSB
Start time: Wed, Jan 27 2010, 4:10PM

Direct connection is exposed between the equations in random matrix (RM) theory, derived by different - Tracy-Widom and Adler-Shiota-van Moerbeke - methods. Simple relations are obtained between the eigenvalue spacing probabilities considered as ratios of 1-dim. Toda lattice $\tau$-functions on one side, and auxiliary variables appearing in the approach to the probabilities using resolvent kernels of Fredholm operators on the other side. A unified description of unitary invariant RM ensembles (UE) is found in terms of universal, i.e. independent of the specific probability measure, PDE for gap probabilities. At the core of my considerations were the three-term recurrence relations for orthogonal polynomials (OP) and their relation with 1-dim. Toda lattice (or Toda-AKNS) integrable hierarchy. Toda-AKNS system provides a common structure of PDE for UE, which appears in different guises: one arises from OP-Toda lattice relations, while the other comes from Schlesinger equations for isomonodromic deformations and their tie with TW equations. Similar connections for coupled Hermitian Gaussian matrices, related to 2-dim. Toda lattice hierarchy, are found, they helped the more difficult analysis of probabilities for coupled case, with potential applications to random growth processes and other nonstationary nonlinear phenomena.