A noncolliding diffusion process is a conditional process of $N$ independent one-dimensional diffusion processes such that the particles never collide with each other. This process realizes an interacting particle system with long-ranged strong repulsive forces acting between any pair of particles. When the individual diffusion process is a one-dimensional Brownian motion, the noncolliding process is equivalent in distribution with the eigenvalue process of an $N \times N$ Hermitian-matrix-valued process, which we call Dyson's model. For any deterministic initial configuration of $N$ particles, distribution of particle positions of the noncolliding Brownian motion on the real line at any fixed time $t >0$ is a determinantal point process. We can prove that the process is determinantal in the sense that the multi-time correlation function for any chosen series of times, which determines joint distributions at these times, is also represented by a determinant. We study the asymptotic behavior of the system, when the number of Brownian motions $N$ in the system tends to infinity. This problem is concerned with the random matrix theory on the asymptotics of eigenvalue distributions, when the matrix size becomes infinity. In the present paper, we introduce a variety of noncolliding diffusion processes by generalizing the noncolliding Brownian motion, some of which are temporally inhomogeneous. We report the results of our research project to construct and study finite and infinite particle systems with long-ranged strong interactions realized by noncolliding processes.
Deep Dive into Noncolliding processes, matrix-valued processes and determinantal processes.
A noncolliding diffusion process is a conditional process of $N$ independent one-dimensional diffusion processes such that the particles never collide with each other. This process realizes an interacting particle system with long-ranged strong repulsive forces acting between any pair of particles. When the individual diffusion process is a one-dimensional Brownian motion, the noncolliding process is equivalent in distribution with the eigenvalue process of an $N \times N$ Hermitian-matrix-valued process, which we call Dyson’s model. For any deterministic initial configuration of $N$ particles, distribution of particle positions of the noncolliding Brownian motion on the real line at any fixed time $t >0$ is a determinantal point process. We can prove that the process is determinantal in the sense that the multi-time correlation function for any chosen series of times, which determines joint distributions at these times, is also represented by a determinant. We study the asymptotic beh
In a system of N independent one-dimensional diffusion processes, if we impose a condition such that the particles never collide with each other, then we obtain an interacting particle system with long-ranged strong repulsive forces acting between any pair of particles. We call such a system a noncolliding diffusion process. In 1962 Dyson [14] showed that, when the individual diffusion process is a onedimensional Brownian motion, the obtained noncolliding process, the noncolliding Brownian motion, is related to a matrix-valued process. He introduced a Hermitian-matrix-valued process having Brownian motions as its diagonal elements, and complex Brownian motions as off-diagonal elements. The size of the matrix is supposed to be N × N. By virtue of the Hermitian property, all eigenvalues of the matrix are real, and Dyson derived a system of N-simultaneous stochastic differential equations for the process of N eigenvalues on the real line R. In the present paper we call this stochastic process of eigenvalues Dyson's model. (Strictly speaking, it is a special case of Dyson's Brownian motion models with the parameter β = 2 as explained below.) If we regard each eigenvalue as a particle position in one dimension, Dyson's model is considered to be a one-dimensional system of interacting Brownian motions. Dyson showed that this system is nothing but the noncolliding Brownian motion [4,23].
A probability distribution on the space of particle configurations is called a determinantal point process or a Fermion point process, if its correlation functions are generally represented by determinants [65,66,27]. The noncolliding Brownian motion provides us examples of determinantal point processes: for any deterministic initial configuration of N particles, distribution of particle positions on R at any fixed time t > 0 is a determinantal point process [45]. Moreover, by using the method developed by Eynard and Mehta for multi-layer random matrix models [16,50], we can show that the multi-time correlation functions for any chosen series of times, which determine joint distributions at these times, are also represented by determinants [37,43,44,45]. In the present paper we call such a stochastic process that any multi-time correlation function is given by a determinant a determinantal process [43].
We study the asymptotic behavior of the system, when the number of Brownian motions N in the system tends to infinity. Since, as explained above, the noncolliding Brownian motion can be realized by the eigenvalue process and the correlation functions are expressed by determinants of matrices, this problem is concerned with the asymptotics of eigenvalue distributions, when the matrix size becomes infinity. The latter problem is one of the main topics of the random matrix theory [50]. In other words, our research project reported in this paper is to construct infinite particle systems with long-ranged strong interactions by applying the results of recent development of the random matrix theory [37,40,42,43,44,45].
In the present paper, we introduce a variety of noncolliding diffusion processes by generalizing the noncolliding Brownian motion. In Section 2 first we explain basic properties of diffusion processes treated in this paper, such as Brownian motions, Brownian bridges, absorbing Brownian motions, Bessel processes, Bessel bridges, and generalized meanders. The transition probability density of a noncolliding diffusion process is expressed by a determinant of a matrix, each element of which is the transition probability density of the individual diffusion process in one dimension (the Karlin-McGregor formula). This formula provides a useful tool for us to analyze noncolliding diffusion processes. In Section 3 we state the Karlin-McGregor formula and present basic properties of noncolliding diffusion processes. When such a Hermitian-matrix-valued process is given that its elements are one-dimensional diffusion processes, it will be a fundamental and interesting problem to determine a system of stochastic differential equations for eigenvalue process of the given matrixvalued process. Bru’s theorem [8,9] and its generalization [39,40] give answers to this problem. In Section 4 we give a generalized version of Bru’s theorem and show its applications. The determinantal structures of multi-time correlation functions of noncolliding processes are explained in Section 5. There asymptotics in N → ∞ are also discussed [37,42,43,44,45].
When we impose the noncolliding condition on a finite time-interval (0, T ), T ∈ (0, ∞) instead of an infinite time-interval (0, ∞), the noncolliding diffusion processes become temporally inhomogeneous, even if individual one-dimensional diffusion processes are temporally homogeneous. In Section 6 we discuss these temporally inhomogeneous noncolliding processes. These processes are not determinantal any more, and make a new family of processes, which we call Pfaffian processes [37,42]. In the last sec
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