From Gumbel to Tracy-Widom II, via integer partitions

by Dan Betea (University of Bonn)

Location: 200B.02.18, Leuven
Time: Wednesday May 8, 2019 at 10:30

There are two natural well-studied measures on integer partitions: Plancherel and uniform. In the scaling limit, their first parts behave differently and on a different scale: Plancherel shows random matrix-type Tracy-Widom statistics -the Baik-Deift-Johansson theorem, while uniform shows Gumbel (a maximum of independent Gaussians)-the Erdos-Lehner theorem. In this talk, based on joint work with Jérémie Bouttier, we study the ‘finite temperature/cylindric’ extension of the Plancherel measure, coming from counting standard Young tableaux of skew shape and interpolating between Plancherel and uniform. Upon passing to the grand canonical ensemble, this measure becomes determinantal with kernel given by the finite temperature Bessel kernel, which we introduce. In the critical regime, edge fluctuations are governed by the finite temperature Tracy-Widom distribution of Johansson, observed in random matrix models and interpolating between Tracy-Widom and Gumbel statistics. The critical finite-temperature regime can thus be viewed as a balancing between thermal (Gumbel) and quantum (Tracy-Widom) fluctuations in a way we make precise. A time-extension of the result yields, upon scaling, the recently discovered finite temperature/periodic extended Airy process of Le Doussal-Majumdar-Schehr.