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Infinite random matrices and ergodic measures

Alexei Borodin·Grigori Olshanski
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Abstract

We introduce and study a 2-parameter family of unitarily invariant probability measures on the space of infinite Hermitian matrices. We show that the decomposition of a measure from this family on ergodic components is described by a determinantal point process on the real line. The correlation kernel for this process is explicitly computed. At certain values of parameters the kernel turns into the well-known sine kernel which describes the local correlation in Circular and Gaussian Unitary Ensembles. Thus, the random point configuration of the sine process is interpreted as the random set of ``eigenvalues'' of infinite Hermitian matrices distributed according to the corresponding measure.

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Alexei Borodin, Grigori Olshanski. 2000-10-11. Infinite random matrices and ergodic measures. https://doi.org/10.1007/s002200100529

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