$q$-deformation of random partitions, determinantal structure, and Riemann-Hilbert problem

We study $q$-deformation of probability measures on partitions, i.e., $q$-deformed random partitions. We in particular consider the $q$-Plancherel measure and show a determinantal formula for the correlation function using a $q$-deformation of the discrete Bessel kernel. We also investigate Riemann-Hilbert problems associated with the corresponding orthogonal polynomials and obtain $q$-Painlevé equations from the $q$-difference Lax formalism.