## Electronic Communications in Probability

### Eigenvalues of the Laguerre Process as Non-Colliding Squared Bessel Processes

#### Abstract

Let $A(t)$ be an $n\times p$ matrix with independent standard complex Brownian entries and set $M(t)=A(t)^*A(t)$. This is a process version of the Laguerre ensemble and as such we shall refer to it as the Laguerre process;. The purpose of this note is to remark that, assuming $n > p$, the eigenvalues of $M(t)$ evolve like $p$ independent squared Bessel processes of dimension $2(n-p+1)$, conditioned (in the sense of Doob) never to collide. More precisely, the function $h(x)=\prod_{i < j}(x_i-x_j)$ is harmonic with respect to $p$ independent squared Bessel processes of dimension $2(n-p+1)$, and the eigenvalue process has the same law as the corresponding Doob $h$-transform. In the case where the entries of $A(t)$ are real Brownian motions, $(M(t))_{t > 0}$ is the Wishart process considered by Bru (1991). There it is shown that the eigenvalues of $M(t)$ evolve according to a certain diffusion process, the generator of which is given explicitly. An interpretation in terms of non-colliding processes does not seem to be possible in this case. We also identify a class of processes (including Brownian motion, squared Bessel processes and generalised Ornstein-Uhlenbeck processes) which are all amenable to the same $h$-transform, and compute the corresponding transition densities and upper tail asymptotics for the first collision time.

#### Article information

Source
Electron. Commun. Probab., Volume 6 (2001), paper no. 11, 107-114.

Dates
Accepted: 31 August 2001
First available in Project Euclid: 19 April 2016

https://projecteuclid.org/euclid.ecp/1461097556

Digital Object Identifier
doi:10.1214/ECP.v6-1040

Mathematical Reviews number (MathSciNet)
MR1871699

Zentralblatt MATH identifier
1011.15012

Subjects
Primary: 15A52
Secondary: 60J65: Brownian motion [See also 58J65] 62E10: Characterization and structure theory

Rights

#### Citation

König, Wolfgang; O'Connell, Neil. Eigenvalues of the Laguerre Process as Non-Colliding Squared Bessel Processes. Electron. Commun. Probab. 6 (2001), paper no. 11, 107--114. doi:10.1214/ECP.v6-1040. https://projecteuclid.org/euclid.ecp/1461097556

#### References

• T. Akuzawa and M. Wadati (1997), Laguerre ensemble and integrable systems. Chaos, Solitons and Fractals 8, no. 1, 99-107.
• A.N. Borodin and P. Salminen (1996), Handbook of Brownian Motion: Facts and Formulae. Birkhäuser, Berlin.
• M.-F. Bru (1991), Wishart processes. J. Theoret. Probab. 3, no. 4, 725-751.
• P. Carmona, F. Petit and Marc Yor (2001), Exponential functionals of Lévy processes. to appear in a Birkhäuser volume on Lévy processes, edited by T. Mikosch.
• F.J Dyson (1962), A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys. 3, 1191-1198.
• D. Grabiner (1999), Brownian motion in a Weyl chamber, non-colliding particles, and random matrices. Ann. Inst. H. Poincaré Probab. Statist. 35, no. 2, 177-204.
• D. Hobson and W. Werner (1996), Non-colliding Brownian motion on the circle. Bull. Math. Soc. 28, 643-650.
• A.T. James (1964), Distributions of matrix variates and latent roots derived from normal samples. Ann. Math. Statist. 35, 475-501.
• S.P. Karlin and G. MacGregor (1959), Coincidence probabilities. Pacif. J. Math. 9, 1141–1164.
• W.S. Kendall (1990). The diffusion of Euclidean shape. In: Disorder in Physical Systems, eds. G. Grimmett and D. Welsh, Oxford University Press, 203-217.
• I.G. Macdonald (1979), Symmetric Functions and Hall Polynomials. Oxford University Press.
• M.L. Mehta (1991), Random Matrices. Second Edition. Academic Press.
• J.R. Norris, L.C.G. Rogers and David Williams (1986), Brownian motions of ellipsoids. Trans. Amer. Math. Soc. 294, 757-765.
• E.J. Pauwels and L.C.G. Rogers (1988). Skew-product decompositions of Brownian motions. Contemporary Mathematics 73, 237-262.
• D. Revuz and Marc Yor (1991), Continuous Martingales and Brownian Motion. Springer, Berlin.