Alex Moll: Random Partitions and the Correspondence Principle

The classical inviscid Hopf-Burgers equation v_t + v v_x = 0 with periodic boundary conditions is a completely integrable system for v: T -> R on the unit circle T.  Its hierarchy of commuting conservation laws can be quantized, and the resulting quantum Hamiltonians are simultaneously diagonalized on Schur polynomials.  The decomposition of a Glauber coherent state around a classical configuration v into eigenfunctions defines a Schur measure on partitions with specializations determined by v.  In the semi-classical limit, we prove a concentration of profiles of Young diagrams around the push-forward along v: T-> R of the uniform measure on T, recovering Okounkov (2003) and in particular Vershik-Kerov (1977) for v(x) = 2 cos x.  Moreover, the global fluctuations converge to a Gaussian process, the push-forward along v: T -> R of H^{1/2} noise on T.  Although no large deviation principle is currently known for arbitrary Schur measures, our results are as predicted by the quantum-classical correspondence principle.  Our proofs exploit the integrability of the model via spectral theory of Toeplitz operators with symbol v described by Nazarov-Sklyanin (2013) and extend to Jack measures and the periodic Benjamin-Ono system.  The talk will begin with the asymptotics of a single Poisson random variable at high intensity and the correspondence principle for a single harmonic oscillator.