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 $\beta /2$ times the inverse of particle distance. When $\beta =2$, 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 $\beta =2$ 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 $\mathbb{R}$. We consider Dyson's model with $\beta =2$ starting from the first $N$ zeros of $\Ai(z)$, $0>a_1>...>a_N$, $N\ge 2$. In order to properly control the effect of such initial confinement of particles in the negative region of $\mathbb{R}$, we put the drift term to each Brownian motion, which increases in time as a parabolic function : $Y_j(t)=X_j(t)+t^2/4+\{d_1+{\sum }_{\ell =1}^N(1/a_{\ell })\}t,1\le j\le N$, 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 $\mathbb{R}$ 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 $\mathbb{R}$ and in which the Tracy-Widom distribution describes the rightmost particle position.

AMS-LaTeX, 33 pages, no figure, v4: minor corrections made for publication in J. Stat. Phys

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