Stochastic Analysis Seminar – Samuel Johnston (King’s College London)

Two finite sets S and T of real numbers are said to interlace if for every s < s’ in S, there exists t in T such that s < t < s’. We are interested in Gelfand-Tsetlin patterns, which are triangular arrays of real numbers whose rows interlace. Gelfand-Tsetlin patterns arise naturally when considering the eigenvalues of successive minors of Hermitian matrices. Recent calculations in free probability by Shlyakhtenko and Tao have suggested that large random Gelfand-Tsetlin patterns have their behaviour governed by a variational principle. With this prediction as a motivation, we study the asymptotic behaviour of interlacing patterns from the perspective of statistical physics, ultimately showing that interlacing patterns have a notion of “free entropy” or “surface tension” that matches the Lagrangian in Shlyakhtenko and Tao’s variational principle. Along the way, we discuss ideas from integrable probability, statistical physics, Kasteleyn theory and domino tilings, large deviations, and free probability.

 This talk is based in part on joint work with Joscha Prochno (Passau).