Gap Probabilities for the Generalized Bessel Process: A Riemann-Hilbert Approach

We consider the gap probability for the Generalized Bessel process in the single-time and multi-time case, a determinantal process which arises as critical limiting kernel in the study of self-avoiding squared Bessel paths. We prove that such gap probabilities, i.e. the scalar and matrix Fredholm determinant of the process respectively, can be expressed in terms of determinants of suitable Its-Izergin-Korepin-Slavnov integrable kernels and therefore they can be related in a canonical way to Riemann-Hilbert problems. Moreover, such Fredholm determinants can be interpreted as isomonodromic τ-functions in the sense of Jimbo, Miwa and Ueno; in particular, in the single-time case we are able to construct an explicit Lax pair. On the other hand, in the multi-time case we explicitly define a completely new multi-time kernel and we proceed with the study of gap probabilities as in the single-time case.

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

[2]  Michio Jimbo,et al.  Monodromy preserving deformation of linear ordinary differential equations with rational coefficients: I. General theory and τ-function , 1981 .

[3]  M. L. Mehta,et al.  Matrices coupled in a chain: I. Eigenvalue correlations , 1998 .

[4]  M. Girotti.  Riemann-Hilbert approach to gap probabilities for the Bessel process , 2013, 1306.5663.

[5]  M. Bertola.  The Dependence on the Monodromy Data of the Isomonodromic Tau Function , 2009, 1601.04790.

[6]  Michio Jimbo,et al.  Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. III , 1981 .

[7]  M. Jimbo,et al.  Monodromy perserving deformation of linear ordinary differential equations with rational coefficients. II , 1981 .

[8]  K. Johansson.  Course 1 - Random Matrices and Determinantal Processes , 2006 .

[9]  Vladimir E. Korepin,et al.  Differential Equations for Quantum Correlation Functions , 1990 .

[10]  A. Kuijlaars,et al.  Non-Intersecting Squared Bessel Paths and Multiple Orthogonal Polynomials for Modified Bessel Weights , 2007, 0712.1333.

[11]  A. Kuijlaars,et al.  Non-Intersecting Squared Bessel Paths: Critical Time and Double Scaling Limit , 2010, 1011.1278.

[12]  Александр Борисович Сошников,et al.  Детерминантные случайные точечные поля@@@Determinantal random point fields , 2000 .

[13]  M. Cafasso,et al.  The Transition between the Gap Probabilities from the Pearcey to the Airy Process—a Riemann–Hilbert Approach , 2010, 1005.4083.

[14]  Samuel Karlin,et al.  COINCIDENT PROPERTIES OF BIRTH AND DEATH PROCESSES , 1959 .

[15]  K. Johansson.  Random matrices and determinantal processes , 2005, math-ph/0510038.

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

[17]  W. Wasow.  Asymptotic expansions for ordinary differential equations , 1965 .

[18]  B. Simon.  Trace ideals and their applications , 1979 .

[19]  Riemann–Hilbert approach to multi-time processes: The Airy and the Pearcey cases , 2011, 1104.4940.

[20]  Integrable Fredholm Operators and Dual Isomonodromic Deformations , 1997, solv-int/9706002.

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

[22]  Athanassios S. Fokas,et al.  Painleve Transcendents: The Riemann-hilbert Approach , 2006 .

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