Gelfand-Tsetlin模式的普适性
Universality properties of Gelfand-Tsetlin patterns
摘要：
深度$n$的标准Gelfand-Tsetlin图案是$\{1,...,n\} \times \R$中的颗粒配置。对于每个$r \in \{1,...,n\}$，$\{r\} \times \R$被称为模式的$r^\text{th}$级别。一个标准的Gelfand-Tsetlin模式在每个级别$r$上都有精确的$r$粒子，并且相邻级别上的粒子满足交错约束。在深度为$n$的Gelfand-Tsetlin模式集上的概率分布自然地作为大小为$n$的随机厄米特矩阵的特征值小过程的分布而出现。我们认为这样的概率空间时，矩阵的分布是酉不变的，证明了一个广泛的子类的行列式结构，并计算相关核。特别是，我们考虑的情况下，随机矩阵的特征值是固定的。这对应于从其$n^\text{th}$水平固定在矩阵的特征值处的Gelfand-Tsetlin模式的集合中均匀地选择。固定$q_n \in \{1,...,n\}$，在假设$\frac{q_n}n \to \a \in (0,1)$和粒子在$n^\text{th}$水平上的经验分布弱收敛的条件下，使$n \to \infty$，粒子在$q_n$水平上的渐近行为与自由概率论有关.鞍点分析被用来确定这些粒子的行为渐近像一个决定性的随机点场与正弦核的集合。
英文摘要：
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.
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