Explicit description of Christoffel deformations and Palm measures of the Plancherel measure, the $z$-measures and the Gamma process

Abstract

Christoffel deformation of a measure on the real line consists of multipying this measure by a squared polynomial having its roots in $\R$. We introduce Christoffel deformations of discrete orthogonal polynomial ensembles by considering the Christoffel deformations of the underlying measure, and prove that this construction extends to more general point processes describing distribution on partitions: the poissonized Plancherel measure and the $z$-measures. These deformations contain the theory of Palm measures, and for example, explicit formulas for the Palm measures of the poissonized Plancherel measure provide a description of the TASEP with initial wedge condition with frozen particles. We also obtain new formulas for Palm measures of the $z$-measures. The extension to the Plancherel measure is obtained via a limit transition from the Charlier ensemble, while the extension to the $z$-measures follows from an analytic continuation argument. A limit procedure starting from the non-degenerate $z$-measures leads to a deformation of the Gamma process introduced by Borodin and Olshanski.