Point processes and the infinite symmetric group. Part III: Fermion point processes

In Part I (G.Olshanski, math.RT/9804086) and Part II (A.Borodin, math.RT/9804087) we developed an approach to certain probability distributions on the Thoma simplex. The latter has infinite dimension and is a kind of dual object for the infinite symmetric group. Our approach is based on studying the correlation functions of certain related point stochastic processes. In the present paper we consider the so-called tail point processes which describe the limit behavior of the Thoma parameters (coordinates on the Thoma simplex) with large numbers. The tail processes turn out to be stationary processes on the real line. Their correlation functions have determinantal form with a kernel which generalizes the well-known sine kernel arising in random matrix theory. Our second result is a law of large numbers for the Thoma parameters. We also produce Sturm-Liouville operators commuting with the Whittaker kernel introduced in Part II and with the generalized sine kernel.

[1]  Daryl J. Daley,et al.  An Introduction to the Theory of Point Processes , 2013 .

[2]  A. Borodin.  Point Processes and the Infinite Symmetric Group. Part II: Higher Correlation Functions , 1998, math/9804087.

[3]  G. Olshanski.  Point processes and the infinite symmetric group. Part I: The general formalism and the density function , 1998, math/9804086.

[4]  S. Kerov.  Anisotropic young diagrams and jack symmetric functions , 1997, math/9712267.

[5]  S. Kerov,et al.  The boundary of the Young graph with Jack edge multiplicities , 1997, q-alg/9703037.

[6]  A. Okounkov.  Thoma's theorem and representations of the infinite bisymmetric group , 1994 .

[7]  M. Wadati,et al.  Eigenvalue Distribution of Random Matrices at the Spectrum Edge , 1993 .

[8]  Y. Chen,et al.  Transitions in spectral statistics , 1993, cond-mat/9311001.

[9]  P. Forrester.  The spectrum edge of random matrix ensembles , 1993 .

[10]  Yang Chen,et al.  New family of unitary random matrices. , 1993, Physical review letters.

[11]  M. Wadati,et al.  Correlation Functions of Random Matrix Ensembles Related to Classical Orthogonal Polynomials. III , 1992 .

[12]  S. Kerov.  Combinatorial examples in the theory of AF-algebras , 1992 .

[13]  Yu. A. Brychkov,et al.  Integrals and series , 1992 .

[14]  Robert C. Griffiths,et al.  On the distribution of points in a poisson dirichlet process , 1988, Journal of Applied Probability.

[15]  J. Stembridge.  A characterization of supersymmetric polynomials , 1985 .

[16]  Robert C. Griffiths,et al.  On the distribution of allele frequencies in a diffusion model , 1979 .

[17]  J. Kingman.  Random partitions in population genetics , 1978, Proceedings of the Royal Society of London. Series B. Biological Sciences.

[18]  A. Vershik,et al.  Limit Measures Arising in the Asympyotic Theory of Symmetric Groups. I. , 1977 .

[19]  W. R. Buckland.  Multiple Hypergeometric Functions and Applications , 1977 .

[20]  O. Macchi.  The coincidence approach to stochastic point processes , 1975, Advances in Applied Probability.

[21]  O. Macchi,et al.  Detection and ``emission'' processes of quantum particles in a ``chaotic state'' , 1973 .

[22]  W. Ewens.  The sampling theory of selectively neutral alleles. , 1972, Theoretical population biology.

[23]  B. V. Bronk,et al.  Exponential Ensemble for Random Matrices , 1965 .

[24]  E. Thoma.  Die unzerlegbaren, positiv-definiten Klassenfunktionen der abzählbar unendlichen, symmetrischen Gruppe , 1964 .

[25]  A. Erdélyi,et al.  Tables of integral transforms , 1955 .

[26]  Grigori Olshanski,et al.  POINT PROCESSES AND THE INFINITE SYMMETRIC GROUP , 1998 .

[27]  C. Tracy,et al.  SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS FOR A CLASS OF OPERATOR DETERMINANTS , 1995 .

[28]  A. Vershik,et al.  Harmonic analysis on the infinite symmetric group. A deformation of the regular representation , 1993 .

[29]  Craig A. Tracy,et al.  Mathematical Physics © Springer-Verlag 1994 Level Spacing Distributions and the Bessel Kernel , 1993 .

[30]  Mathematical Physics © Springer-Verlag 1994 Level-Spacing Distributions and the Airy Kernel , 1992 .

[31]  D. Aldous.  Exchangeability and related topics , 1985 .

[32]  Anatoly M. Vershik,et al.  Asymptotic theory of characters of the symmetric group , 1981 .

[33]  I. G. MacDonald,et al.  Symmetric functions and Hall polynomials , 1979 .

[34]  O. Macchi.  The Fermion Process — A Model of Stochastic Point Process with Repulsive Points , 1977 .

[35]  Anatoly Vershik,et al.  The Erwin Schrr Odinger International Institute for Mathematical Physics a New Approach to Representation Theory of Symmetric Groups , 2022 .

[36]  Craig A. Tracy,et al.  Mathematical Physics © Springer-Verlag 1994 Fredholm Determinants, Differential Equations and Matrix Models , 2022 .