Abstract:
In a series of works published in the 1990s, Kerov put forth various
applications of the circle of ideas centered at the Markov moment problem
to the limiting shape of random continual diagrams arising in representation
theory and spectral theory. We demonstrate on several examples that his
approach is also adequate to study the fluctuations about the limiting shape.
In the random matrix setting, we compare two continual diagrams: one
is constructed from the eigenvalues of the matrix and the critical points of
its characteristic polynomial, whereas the second one is constructed from
the eigenvalues of the matrix and those of its principal submatrix. The fluctuations
of the latter diagram were recently studied by Erdős and Schröder;
we discuss the fluctuations of the former, and compare the two limiting
processes.
For Plancherel random partitions, the Markov correspondence establishes
the equivalence between Kerov's central limit theorem for the Young diagram
and the Ivanov–Olshanski central limit theorem for the transition measure.
We outline a combinatorial proof of the latter, and compare the limiting
process with the ones of random matrices.
Key words and phrases:interlacing sequences, Markov moment problem, continual diagrams, random matrices, central limit theorem.
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