"Zeros of Airy functiion and relaxation process",
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,
$\X(t)=(X_1(t), \dots, X_N(t)),
t \in [0,\infty), 2 \leq N < \infty$,
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 $\beta=2$
starting from the first $N$ zeros of $\Ai(z)$,
$0 > a_1 > \cdots > a_N$, $N \geq 2$.
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
$\Y(t)=(Y_1(t), \dots, Y_N(t)), t \in [0, \infty)$, 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 $\mu_{\Ai}$.
The stationary state $\mu_{\Ai}$ 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.
