Two-step asymptotics of scaled Dunkl processes

Dunkl processes are generalizations of Brownian motion obtained by using the differential-difference operators known as Dunkl operators as a replacement of spatial partial derivatives in the heat equation. Special cases of these processes include Dyson's Brownian motion model and the Wishart-Laguerre eigenvalue processes, which are well-known in random matrix theory. It is known that the dynamics of Dunkl processes is obtained by transforming the heat kernel using Dunkl's intertwining operator. It is also known that, under an appropriate scaling, their distribution function converges to a steady-state distribution which depends only on the coupling parameter $\beta$ as the process time $t$ tends to infinity. We study scaled Dunkl processes starting from an arbitrary initial distribution, and we derive expressions for the intertwining operator in order to calculate the asymptotics of the distribution function in two limiting situations. In the first one, $\beta$ is fixed and $t$ tends to infinity (approach to the steady state), and in the second one, $t$ is fixed and $\beta$ tends to infinity (strong-coupling limit). We obtain the deviations from the limiting distributions in both of the above situations, and we find that they are caused by the two different mechanisms which drive the process, namely, the drift and exchange mechanisms. We find that the deviation due to the drift mechanism decays as $t^{-1}$, while the deviation due to the exchange mechanism decays as $t^{-1/2}$.

[1]  Charles F. Dunkl,et al.  Differential-difference operators associated to reflection groups , 1989 .

[2]  Quantum Calogero-Moser models: integrability for all root systems , 2000, hep-th/0005277.

[3]  E. Youssfi,et al.  The Dunkl intertwining operator , 2009 .

[4]  Meinhard E. Mayer,et al.  Group theory and physics , 1994 .

[5]  W. Fulton.  Young Tableaux: With Applications to Representation Theory and Geometry , 1996 .

[6]  Polynuclear growth model with external source and random matrix model with deterministic source. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[7]  Michael Voit,et al.  Markov Processes Related with Dunkl Operators , 1998 .

[8]  Pandey,et al.  Higher-order correlations in spectra of complex systems. , 1985, Physical review letters.

[9]  P. G. de Gennes,et al.  Soluble Model for Fibrous Structures with Steric Constraints , 1968 .

[10]  M. Katori,et al.  Noncolliding Brownian Motion and Determinantal Processes , 2007, 0705.2460.

[11]  Verbaarschot,et al.  Spectral density of the QCD Dirac operator near zero virtuality. , 1993, Physical Review Letters.

[12]  P. Forrester,et al.  Polynomial Eigenfunctions of the Calogero—Sutherland—Moser Models with Exchange Terms , 2000 .

[13]  Makoto Katori,et al.  Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems , 2004 .

[14]  K. Takeuchi,et al.  Universal fluctuations of growing interfaces: evidence in turbulent liquid crystals. , 2010, Physical review letters.

[15]  Anthony J. Guttmann,et al.  Vicious walkers, friendly walkers and Young tableaux: II. With a wall , 2000 .

[16]  Yuan Xu,et al.  Orthogonal Polynomials of Several Variables: Subject index , 2001 .

[17]  P. Forrester,et al.  The Calogero-Sutherland Model and Generalized Classical Polynomials , 1996, solv-int/9608004.

[18]  Anthony J. Guttmann,et al.  Vicious walkers and Young tableaux I: without walls , 1998 .

[19]  Harmonic polynomials and peak sets of reflection groups , 1989 .

[20]  M. Yor,et al.  Some new examples of Markov processes which enjoy the time-inversion property , 2005 .

[21]  Seiji Miyashita,et al.  Two limiting regimes of interacting Bessel processes , 2013, 1309.2733.

[22]  J. Baik,et al.  A model for the bus system in Cuernavaca (Mexico) , 2005, math/0510414.

[23]  Scaling limit of vicious walks and two-matrix model. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  Gregory Schehr,et al.  Extremes of N Vicious Walkers for Large N: Application to the Directed Polymer and KPZ Interfaces , 2012, 1203.1658.

[25]  Seiji Miyashita,et al.  Interacting particles on the line and Dunkl intertwining operator of type A: application to the freezing regime , 2012, 1202.5052.

[26]  Alan Edelman,et al.  Eigenvalues of Hermite and Laguerre ensembles: large beta asymptotics , 2005 .

[27]  M. Rösler.  Generalized Hermite Polynomials and the Heat Equation for Dunkl Operators , 1998 .

[28]  Michael E. Fisher,et al.  Walks, walls, wetting, and melting , 1984 .

[29]  M. Rösler.  POSITIVITY OF DUNKL'S INTERTWINING OPERATOR , 1997 .

[30]  Essam,et al.  Vicious walkers and directed polymer networks in general dimensions. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[31]  L. Vinet,et al.  The Dunkl oscillator in three dimensions , 2013, 1312.3877.

[32]  F. Dyson.  A Brownian‐Motion Model for the Eigenvalues of a Random Matrix , 1962 .

[33]  P. Deift.  Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach , 2000 .

[34]  M. Katori,et al.  Noncolliding Squared Bessel Processes , 2010, 1008.0144.

[35]  Spohn,et al.  Universal distributions for growth processes in 1+1 dimensions and random matrices , 2000, Physical review letters.

[36]  Charles F. Dunkl,et al.  Integral Kernels with Reflection Group Invariance , 1991, Canadian Journal of Mathematics.

[37]  Neil O'Connell,et al.  Eigenvalues of the Laguerre Process as Non-Colliding Squared Bessel Processes , 2001 .

[38]  P. Graczyk,et al.  Multidimensional Yamada-Watanabe theorem and its applications to particle systems , 2011, 1108.1459.

[39]  N. Demni,et al.  Dunkl kernel associated with dihedral group , 2015, 1504.07220.

[40]  P. Forrester.  Log-Gases and Random Matrices (LMS-34) , 2010 .

[41]  L. Vinet,et al.  Supersymmetric quantum mechanics with reflections , 2011, 1107.5844.