In a series of works published in the 1990-s, Kerov put forth variousapplications of the circle of ideas centred at the Markov moment problem to thelimiting shape of random continual diagrams arising in representation theoryand spectral theory. We demonstrate on several examples that his approach isalso adequate to study the fluctuations about the limiting shape. In the random matrix setting, we compare two continual diagrams: one isconstructed from the eigenvalues of the matrix and the critical points of itscharacteristic polynomial, whereas the second one is constructed from theeigenvalues of the matrix and those of its principal submatrix. Thefluctuations of the latter diagram were recently studied by Erd\H{o}s andSchr\"oder; we discuss the fluctuations of the former, and compare the twolimiting processes. For Plancherel random partitions, the Markov correspondence establishes theequivalence between Kerov's central limit theorem for the Young diagram and theIvanov--Olshanski central limit theorem for the transition measure. We outlinea combinatorial proof of the latter, and compare the limiting process with theones of random matrices.
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机译：在1990年代出版的一系列著作中，克洛夫提出了以马尔可夫矩问题为中心的思想界对在表示论和谱论中产生的随机连续图的限制形状的各种应用。我们通过几个例子证明，他的方法也足以研究极限形状的波动。在随机矩阵设置中，我们比较了两个连续图：一个是从矩阵的特征值及其特征多项式的临界点构造的，而第二个是从矩阵及其主要子矩阵的特征值构造的。 Erd \ H {o} s和Schr \“ oder最近研究了后者图的涨落；我们讨论了前者的涨落，并比较了两个限制过程。对于Plancherel随机分区，马尔可夫对应关系建立了Kerov中心极限定理的等价性。对于杨氏图和过渡测量的伊万诺夫-奥尔山斯基中心极限定理，我们概述了后者的组合证明，并将极限过程与随机矩阵的比较。
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