Zeros of Airy Function and Relaxation Process
arXiv:0906.3666 · doi:10.1007/s10955-009-9829-7
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 times the inverse of particle distance. When , 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 and 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 . We consider Dyson's model with starting from the first zeros of $\Ai(z)$, , . In order to properly control the effect of such initial confinement of particles in the negative region of , we put the drift term to each Brownian motion, which increases in time as a parabolic function : , where $d_1=\Ai'(0)/\Ai(0)$. We show that, as the 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 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 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