Freezing Limits for Beta-Cauchy Ensembles

Michael Voit

Symmetry, integrability and geometry: methods and applications, Tome 18 (2022)

Résumé

Bessel processes associated with the root systems $A_{N-1}$ and $B_N$ describe interacting particle systems with $N$ particles on $\mathbb R$; they form dynamic versions of the classical $\beta$-Hermite and Laguerre ensembles. In this paper we study corresponding Cauchy processes constructed via some subordination. This leads to $\beta$-Cauchy ensembles in both cases with explicit distributions. For these distributions we derive central limit theorems for fixed $N$ in the freezing regime, i.e., when the parameters tend to infinity. The results are closely related to corresponding known freezing results for $\beta$-Hermite and Laguerre ensembles and for Bessel processes.

Détail

Keywords: Cauchy processes, Bessel processes, freezing, zeros of classical orthogonal polynomials, Calogero–Moser–Sutherland particle models.
Mots-clés : $\beta$-Hermite ensembles, $\beta$-Laguerre ensembles

Bibliographie

[1] Anderson G.W., Guionnet A., Zeitouni O., An Introduction to random matrices, Cambridge Studies in Advanced Mathematics, 118, Cambridge University Press, Cambridge, 2009 | DOI | MR

[2] Andraus S., Hermann K., Voit M., “Limit theorems and soft edge of freezing random matrix models via dual orthogonal polynomials”, J. Math. Phys., 62 (2021), 083303, 26 pp., arXiv: 2009.01418 | DOI | MR

[3] Andraus S., Katori M., Miyashita S., “Interacting particles on the line and Dunkl intertwining operator of type $A$: application to the freezing regime”, J. Phys. A, 45 (2012), 395201, 26 pp., arXiv: 1202.5052 | DOI | MR

[4] Andraus S., Katori M., Miyashita S., “Two limiting regimes of interacting Bessel processes”, J. Phys. A, 47 (2014), 235201, 30 pp., arXiv: 1309.2733 | DOI | MR

[5] Andraus S., Miyashita S., “Two-step asymptotics of scaled Dunkl processes”, J. Math. Phys., 56 (2015), 103302, 23 pp., arXiv: 1412.2832 | DOI | MR

[6] Andraus S., Voit M., “Central limit theorems for multivariate Bessel processes in the freezing regime II: The covariance matrices”, J. Approx. Theory, 246 (2019), 65–84, arXiv: 1902.06840 | DOI | MR

[7] Andraus S., Voit M., “Limit theorems for multivariate Bessel processes in the freezing regime”, Stochastic Process. Appl., 129 (2019), 4771–4790, arXiv: 1804.03856 | DOI | MR

[8] Anker J.-P., “An introduction to Dunkl theory and its analytic aspects”, Analytic, Algebraic and Geometric Aspects of Differential Equations, Trends Math., Birkhäuser/Springer, Cham, 2017, 3–58, arXiv: 1611.08213 | DOI | MR

[9] Arista J., Demni N., “Explicit expressions of the Hua–Pickrell semigroup”, Theory Probab. Appl., 67 (2022), 208–228, arXiv: 2008.07195 | DOI | MR

[10] Assiotis T., “Hua–Pickrell diffusions and Feller processes on the boundary of the graph of spectra”, Ann. Inst. Henri Poincaré Probab. Stat., 56 (2020), 1251–1283, arXiv: 1703.01813 | DOI | MR

[11] Assiotis T., Bedert B., Gunes M.A., Soor A., “Moments of generalized Cauchy random matrices and continuous-Hahn polynomials”, Nonlinearity, 34 (2021), 4923–4943, arXiv: 2009.04752 | DOI | MR

[12] Berg C., Forst G., Potential theory on locally compact abelian groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, 87, Springer-Verlag, New York – Heidelberg, 1975 | DOI | MR

[13] Borodin A., Olshanski G., “Infinite random matrices and ergodic measures”, Comm. Math. Phys., 223 (2001), 87–123, arXiv: math-ph/0010015 | DOI | MR

[14] Chybiryakov O., Gallardo L., Yor M., “Dunkl processes and their radial parts relative to a root system”, Harmonic and Stochastic Analysis of Dunkl Processes, Hermann, Paris, 2008, 113–198 http://dml.mathdoc.fr/item/hal-00345627

[15] de Boor C., Saff E.B., “Finite sequences of orthogonal polynomials connected by a Jacobi matrix”, Linear Algebra Appl., 75 (1986), 43–55 | DOI | MR

[16] Deift P.A., Orthogonal polynomials and random matrices: a Riemann–Hilbert approach, Courant Lecture Notes in Mathematics, 3, New York University, Courant Institute of Mathematical Sciences, New York; Amer. Math. Soc., Providence, RI, 1999 | MR

[17] Demni N., “Generalized Bessel function of type $D$”, SIGMA, 4 (2008), 075, 7 pp., arXiv: 0811.0507 | DOI | MR

[18] Dumitriu I., Edelman A., “Matrix models for beta ensembles”, J. Math. Phys., 43 (2002), 5830–5847, arXiv: math-ph/0206043 | DOI | MR

[19] Dumitriu I., Edelman A., “Eigenvalues of Hermite and Laguerre ensembles: large beta asymptotics”, Ann. Inst. H. Poincaré Probab. Statist., 41 (2005), 1083–1099, arXiv: math-ph/0403029 | DOI | MR

[20] Forrester P.J., Log-gases and random matrices, London Mathematical Society Monographs Series, 34, Princeton University Press, Princeton, NJ, 2010 | DOI | MR

[21] Gorin V., Kleptsyn V., “Universal objects of the infinite beta random matrix theory”, J. European Math. Soc. (to appear) , arXiv: 2009.02006

[22] Hua L.K., Harmonic analysis of functions of several complex variables in the classical domains, Translations of Mathematical Monographs, 6, Amer. Math. Soc., Providence, R.I., 1979 | MR

[23] Ismail M.E.H., Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, 98, Cambridge University Press, Cambridge, 2005 | DOI | MR

[24] Neretin Yu.A., “Matrix beta-integrals: an overview”, Geometric Methods in Physics, Trends Math., Birkhäuser/Springer, Cham, 2015, 257–272, arXiv: 1411.2110 | DOI | MR

[25] Pickrell D., “Measures on infinite-dimensional Grassmann manifolds”, J. Funct. Anal., 70 (1987), 323–356 | DOI | MR

[26] Rösler M., “Generalized Hermite polynomials and the heat equation for Dunkl operators”, Comm. Math. Phys., 192 (1998), 519–542, arXiv: q-alg/9703006 | DOI | MR

[27] Rösler M., “Dunkl operators: theory and applications”, Orthogonal Polynomials and Special Functions (Leuven, 2002), Lecture Notes in Math., 1817, Springer, Berlin, 2003, 93–135, arXiv: math.CA/0210366 | DOI | MR

[28] Rösler M., Voit M., “Markov processes related with Dunkl operators”, Adv. in Appl. Math., 21 (1998), 575–643 | DOI | MR

[29] Rösler M., Voit M., “Dunkl theory, convolution algebras, and related Markov processes”, Harmonic and Stochastic Analysis of Dunkl Processes, Hermann, Paris, 2008, 1–112

[30] Rösler M., Voit M., “Elementary symmetric polynomials and martingales for Heckman–Opdam processes”, Hypergeometry, Integrability and Lie Theory, Contemp. Math., 780, Amer. Math. Soc., Providence, RI, 2022, 243–262, arXiv: 2108.03228 | DOI | MR

[31] Sato K., Lévy processes and infinitely divisible distributions, Cambridge Studies in Advanced Mathematics, 68, Cambridge University Press, Cambridge, 2013 | MR

[32] Szegö G., Orthogonal polynomials, American Mathematical Society Colloquium Publications, 23, Amer. Math. Soc., New York, 1939 | DOI | MR

[33] van Diejen J.F., Vinet L. (Editors), Calogero–Moser–Sutherland models, CRM Series in Mathematical Physics, Springer-Verlag, New York, 2000 | DOI | MR

[34] Vinet L., Zhedanov A., “A characterization of classical and semiclassical orthogonal polynomials from their dual polynomials”, J. Comput. Appl. Math., 172 (2004), 41–48 | DOI | MR

[35] Voit M., “Central limit theorems for multivariate Bessel processes in the freezing regime”, J. Approx. Theory, 239 (2019), 210–231, arXiv: 1805.08585 | DOI | MR

[36] Voit M., Woerner J.H.C., “Functional central limit theorems for multivariate Bessel processes in the freezing regime”, Stoch. Anal. Appl., 39 (2021), 136–156, arXiv: 1901.08390 | DOI | MR