  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.
