Article

# Markov property of determinantal processes with extended sine, Airy, and Bessel kernels

06/2011;
Source: arXiv

ABSTRACT When the number of particles is finite, the noncolliding Brownian motion (the
Dyson model) and the noncolliding squared Bessel process are determinantal
diffusion processes for any deterministic initial configuration $\xi=\sum_{j \in \Lambda} \delta_{x_j}$, in the sense that any multitime correlation
function is given by a determinant associated with the correlation kernel,
which is specified by an entire function $\Phi$ having zeros in $\supp \xi$.
Using such entire functions $\Phi$, we define new topologies called the
$\Phi$-moderate topologies. Then we construct three infinite-dimensional
determinantal processes, as the limits of sequences of determinantal diffusion
processes with finite numbers of particles in the sense of finite dimensional
distributions in the $\Phi$-moderate topologies, so that the probability
distributions are continuous with respect to initial configurations $\xi$ with
$\xi(\R)=\infty$. We show that our three infinite particle systems are versions
of the determinantal processes with the extended sine, Bessel, and Airy
kernels, respectively, which are reversible with respect to the determinantal
point processes obtained in the bulk scaling limit and the soft-edge scaling
limit of the eigenvalue distributions of the Gaussian unitary ensemble, and the
hard-edge scaling limit of that of the chiral Gaussian unitary ensemble studied
in the random matrix theory. Then Markovianity is proved for the three
infinite-dimensional determinantal processes.

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### Keywords

$\Phi$-moderate topologies

Bessel

bulk scaling limit

chiral Gaussian unitary ensemble

determinantal processes

diffusion processes

Dyson model

eigenvalue distributions

entire function $\Phi$

entire functions $\Phi$

finite numbers

Gaussian unitary ensemble

hard-edge scaling limit

infinite-dimensional determinantal processes

initial configurations $\xi$

noncolliding Brownian motion

noncolliding squared Bessel process

particles

random matrix theory

three infinite particle systems