Determinantal structures in the $q$-Whittaker measure

Abstract

The $q$-Whittaker measure is a probability measure on the set of partitions. We show a Fredholm determinant formula for an expectation value with respect to this measure, which is the $q$-Laplace transform of the marginal distribution on the last element of the partition. Contrary to the typical approaches in integrable probability, our method does not focus on the $q$-moment generating function but explains the origin of its determinantal structure using Ramanujan's summation formula and the Frobenius determinant. This method can be applied to analyze on the fluctuations in the stationary situations in $q$-TASEP, higher spin exclusion process, and the directed polymer in the beta-distributed random environment.