Infinite random matrices and ergodic measures

Details Infinite random matrices and ergodic measures Infinite random matrices and ergodic measures

2000-10-11

arxiv.org/abs/math-ph/0010015v1

Comm. Math. Phys. 223 (2001), no. 1, 87--123

Physics

Mathematical Physics

36 pages

Scientific paper

10.1007/s002200100529

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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