1998/10/03 by Alexei Borodin, Borodin, Alexei
Mathematics · Medicine · Physics and Astronomy · #15A52 #20C32 #60G55 #Advanced Neuroimaging Techniques and Applications #Condensed Matter (cond-mat) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #High Energy Physics - Theory (hep-th) #Point processes and geometric inequalities #Probability (math.PR) #Random Matrices and Applications #Representation Theory (math.RT) #cond-mat #gr-qc #hep-th #math.PR #math.RT #msc:15A52 #msc:20C32 #msc:60G55 #nlin.SI #solv-int
paper · pdf · doi:10.48550/arxiv.math/9810013
AMSTeX, 17 pages
arxiv created 1998/10/03 · openalex publication_date 1998/10/03 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study a 2-parametric family of probability measures on the space of countable point configurations on the punctured real line (the points of the random configuration are concentrated near zero). These measures (or, equivalently, point processes) have been introduced in Part II (A. Borodin, math.RT/9804087) in connection with the problem of harmonic analysis on the infinite symmetric group. The main result of the present paper is a determinantal formula for the correlation functions. The formula involves a kernel called the matrix Whittaker kernel. Each of its two diagonal blocks governs the projection of the process on one of the two half-lines; the corresponding kernel on the half-line was studied in Part III (A. Borodin and G. Olshanski, math/RT/9804088). While the diagonal blocks of the matrix Whitaker kernel are symmetric, the whole kernel turns out to be J-symmetric, i.e., symmetric with respect to a natural indefinite inner product. We also discuss a rather surprising connection of our processes with the recent work by B. Eynard and M. L. Mehta (cond-mat/9710230) on correlations of eigenvalues of coupled random matrices.