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Universality properties of Gelfand-Tsetlin patterns

A standard Gelfand-Tsetlin pattern of depth $n$ is a configuration of particles in $\{1,...,n\} \times \R$. For each $r \in \{1,...,n\}$, $\{r\} \times \R$ is referred to as the $r^\text{th}$ level of the pattern. A standard Gelfand-Tsetlin pattern has exactly $r$ particles on each level $r$, and particles on adjacent levels satisfy an interlacing constraint. Probability distributions on the set of Gelfand-Tsetlin patterns of depth $n$ arise naturally as distributions of eigenvalue minor processes of random Hermitian matrices of size $n$. We consider such probability spaces when the distribution of the matrix is unitarily invariant, prove a determinantal structure for a broad subclass, and calculate the correlation kernel. In particular we consider the case where the eigenvalues of the random matrix are fixed. This corresponds to choosing uniformly from the set of Gelfand-Tsetlin patterns whose $n^\text{th}$ level is fixed at the eigenvalues of the matrix. Fixing $q_n \in \{1,...,n\}$, and letting $n \to \infty$ under the assumption that $\frac{q_n}n \to \a \in (0,1)$ and the empirical distribution of the particles on the $n^\text{th}$ level converges weakly, the asymptotic behaviour of particles on level $q_n$ is relevant to free probability theory. Saddle point analysis is used to identify the set in which these particles behave asymptotically like a determinantal random point field with the Sine kernel.