Zeros of Airy Function and Relaxation Process

Abstract

One-dimensional system of Brownian motions called Dyson's model is the particle system with long-range repulsive forces acting between any pair of particles, where the strength of force is $β/2$ times the inverse of particle distance. When $β=2$, it is realized as the Brownian motions in one dimension conditioned never to collide with each other. For any initial configuration, it is proved that Dyson's model with $β=2$ and $N$ particles, $\X(t)=(X_1(t), ..., X_N(t)), t \in [0,\infty), 2 \leq N < \infty$, is determinantal in the sense that any multitime correlation function is given by a determinant with a continuous kernel. The Airy function $\Ai(z)$ is an entire function with zeros all located on the negative part of the real axis $\R$. We consider Dyson's model with $β=2$ starting from the first $N$ zeros of $\Ai(z)$, $0 > a_1 > ... > a_N$, $N \geq 2$. In order to properly control the effect of such initial confinement of particles in the negative region of $\R$, we put the drift term to each Brownian motion, which increases in time as a parabolic function : $Y_j(t) = X_j(t) + t^2/4 + \{d_1 + \sum_{\ell=1}^N (1/a_{\ell})\}t, 1 \leq j \leq N$, where $d_1=\Ai'(0)/\Ai(0)$. We show that, as the $N \to \infty$ limit of $\Y(t)=(Y_1(t), ..., Y_N(t)), t \in [0, \infty)$, we obtain an infinite particle system, which is the relaxation process from the configuration, in which every zero of $\Ai(z)$ on the negative $\R$ is occupied by one particle, to the stationary state $μ_{\Ai}$. The stationary state $μ_{\Ai}$ is the determinantal point process with the Airy kernel, which is spatially inhomogeneous on $\R$ and in which the Tracy-Widom distribution describes the rightmost particle position.