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

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.

Full text

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arXiv:1106.4360v1 [math.PR] 22 Jun 2011
Markov property of determinantal processes
with extended sine, Airy, and Bessel kernels
Makoto Katori∗and Hideki Tanemura†
22 June 2011
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 ξ =?
kernel, which is specified by an entire function Φ having zeros in supp ξ. Using such
entire functions Φ, we define new topologies called the Φ-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 Φ-moderate topologies, so that the probability
distributions are continuous with respect to initial configurations ξ with ξ(R) = ∞. 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.
j∈Λδxj, in the sense that any
multitime correlation function is given by a determinant associated with the correlation
Keywords Determinantal processes · Correlation kernels · Random matrix theory ·
Infinite particle systems · Markov property · Entire function and topology
Mathematics Subject Classification (2010) 15B52, 30C15, 47D07, 60G55, 82C22
∗Department of Physics, Faculty of Science and Engineering, Chuo University, Kasuga, Bunkyo-ku, Tokyo
112-8551, Japan; e-mail: katori@phys.chuo-u.ac.jp
†Department of Mathematics and Informatics, Faculty of Science, Chiba University, 1-33 Yayoi-cho,
Inage-ku, Chiba 263-8522, Japan; e-mail: tanemura@math.s.chiba-u.ac.jp
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1Introduction
Let M be the space of nonnegative integer-valued Radon measures on R, which is a Pol-
ish space with the vague topology: let C0(R) be the set of all continuous real-valued functions
with compact supports, and we say ξn,n ∈ N converges to ξ vaguely, if limn→∞
?
ξ(K) = ♯{xj : xj ∈ K} < ∞ for any compact subset K ⊂ R. We call an element ξ of
M an unlabeled configuration, and a sequence of points x a labeled configuration. A prob-
ability measure on a configuration space is called a determinantal point process or Fermion
point process, if its correlation functions are generally represented by determinants [32, 33].
In the present paper we say that an M-valued process Ξ(t) is determinantal, if the multi-
time correlation functions for any chosen series of times are represented by determinants.
In other words, a determinantal process is an M-valued process such that, for any integer
M ∈ N = {1,2,...}, f = (f1,f2,...,fM) ∈ C0(R)M, a sequence of times t = (t1,t2,...,tM)
with 0 < t1 < ··· < tM < ∞, if we set χtm(x) = efm(x)− 1,1 ≤ m ≤ M, the moment
generating function of multitime distribution, Ψt[f] ≡ E
is given by a Fredholm determinant
?
?
Rϕ(x)ξn(dx) =
Rϕ(x)ξ(dx) for any ϕ ∈ C0(R).
?
Each element ξ of M can be represented as ξ(·) =
j∈Λδxj(·) with an index set Λ and a sequence of points x = (xj)j∈Λ in R satisfying
?
exp
??M
m=1
?
Rfm(x)Ξ(tm,dx)
??
,
Ψt[f] =Det
(s,t)∈{t1,t2,...,tM}2,
(x,y)∈R2
= Detδstδ(x − y) + K(s,x;t,y)χt(y)
?
,(1.1)
with a locally integrable function K called a correlation kernel [14, 16]. Finite- and infinite-
dimensional determinantal processes were introduced as multi-matrix models [7, 22], tiling
models [11], and surface growth models [30], and they have been extensively studied. In
the present paper we study the infinite-dimensional determinantal processes describing one-
dimensional infinite particle systems with long-range repulsive interactions. They are ob-
tained by taking appropriate N → ∞ limits of the N-particle systems of noncolliding diffu-
sion processes, which dynamically simulate the eigenvalue statistics of the Gaussian random
matrix ensembles studied in the random matrix theory [21, 8]. The purpose of the present
paper is to prove Markov property of the three infinite-dimensional determinantal processes
with the correlation kernels called the extended sine, Airy, and Bessel kernels, which are
reversible with respect to the probability measures obtained in the bulk scaling limit and
the soft-edge scaling limit of the eigenvalue distribution in the Gaussian unitary ensemble
(GUE), and in the hard-edge scaling limit of that in the chiral Gaussian unitary ensemble
(chGUE), respectively. See, for instance, [19] and the references therein.
Dyson [6] introduced a stochastic model of particles in R with a log-potential, which
obeys the stochastic differential equations (SDEs):
dXj(t) = dBj(t) +
?
1≤k≤N,k?=j
dt
Xj(t) − Xk(t),1 ≤ j ≤ N,t ∈ [0,∞),(1.2)
where Bj(t)’s are independent one-dimensional standard Brownian motions. This is a spe-
cial case with the parameter β = 2 of Dyson’s Brownian motion models [6, 21] but we
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will call it simply the Dyson model in this paper. It is equivalent to a system of one-
dimensional Brownian motions conditioned never to collide with each other [19]. For the
solution Xj(t), j = 1,2,...,N of (1.2) with initial values Xj(0) = xj, j = 1,2,...,N, we
put Ξ(t,·) =?N
fixed unlabeled configuration ξN, the process ({Ξ(t)}t∈[0,∞),PξN) is determinantal with the
correlation kernel KξNspecified by an entire function Π0, which is explicitly given by (2.1)
below in the present paper. Then for configurations ξ ∈ M with ξ(R) = ∞, a sufficient con-
dition was given so that the sequence of the processes ({Ξ(t)}t∈[0,∞),Pξ∩[−L,L]) converges to
an M-valued determinantal process as L → ∞ in the sense of finite dimensional distributions
in the vague topology, where ξ ∩ [−L,L] denotes the restricted measure of ξ on an interval
[−L,L]. Note that ξ ∈ M implies ξ ∩ [−L,L](R) < ∞ for 0 < L < ∞. The limit process
is the Dyson model with an infinite number of particles and denoted by ({Ξ(t)}t∈[0,∞),Pξ).
Recently, in [18] the tightness of the sequence of processes was proved. The class of config-
urations satisfying our condition, denoted by Y, is large enough to carry the Poisson point
processes, Gibbs states with regular conditions, as well as the determinantal (Fermion) point
process µsinwith the sine kernel:
?
where i =√−1.
From the uniqueness of solutions of (1.2), the Dyson model with a finite number of
particles is a diffusion process (i.e. it is a strong Markov process having a continuous path
almost surely). Although the process ({Ξ(t)}t∈[0,∞),Pξ) is given as the limit of a sequence
of diffusion processes ({Ξ(t)}t∈[0,∞),Pξ∩[−L,L]), L ∈ N, the Markov property could be lost in
it. In the Dyson model, because of the long-range interaction, if the number of particles is
infinite, the probability distribution Pξis not vaguely continuous with respect to the initial
configuration ξ. Therefore the Markovianity of the process is not readily concluded in the
infinite-particle limit of a sequence of Markov processes in the vague topology.
For a probability measure µ on M we put Pµ(·) =?
to find a subset?Y of M with a topology T , which is stronger than the vague topology, and
1. Pξ(·) is continuous with respect to ξ in the topology T ,
2. Pµ(Ξ(t) ∈?Y) = 1, for any t ∈ [0,∞),
j=1δXj(t)(·) and ξN(·) =?N
j=1δxj(·). The (unlabeled) Dyson model starting
from ξNis denoted by ({Ξ(t)}t∈[0,∞),PξN). In the previous paper [16] we showed that for any
Ksin(x,y) ≡
1
2π
|k|≤π
dk eik(y−x)=sin{π(y − x)}
π(y − x)
,x,y ∈ R,(1.3)
Mµ(dξ)Pξ(·). Suppose that ξ(R) = ∞,
µ-almost surely. For proving Markov property of the process ({Ξ(t)}t∈[0,∞),Pµ), it is sufficient
a sequence of probability measures {µN}N∈Nsuch that
3. µN({η ∈?Y : η(R) = N}) = 1, N ∈ N,
4. ({Ξ(t)}t∈[0,∞),PµN) converges ({Ξ(t)}t∈[0,∞),Pµ) as N → ∞ in the sense of finite di-
mensional distributions in the topology T .
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In this paper first we will show that these conditions are satisfied in the case that µ = µsin,
if we use Y as?Y with the topology T called the Φ0-moderate topology defined by (2.2) given
process ({Ξ(t)}t∈[0,∞),Psin) associated with the extended sine kernel Ksinwith density 1:
1
2π
|k|≤π


(Two processes having the same state space are said to be equivalent if they have the same
finite-dimensional distributions, and we also say that each one is a version of other or they
are versions of the same process. See Section 1.1 of [31].) Hence, we conclude that the
determinantal process with the extended sine kernel is a Markov process which is reversible
with respect to the determinantal point process µsin(Theorem 2.6).
The above strategy will be also used to show Markovianity of the infinite-dimensional
determinantal process ({Ξ(t)}t∈[0,∞),PJν) associated with the extended Bessel kernel KJν:




x,y ∈ R+≡ {x ∈ R : x ≥ 0}, where Jν(·) is the Bessel function with index ν > −1 defined
by
∞
?
with the gamma function Γ(x) =?∞


with J′
below. We also show that the process ({Ξ(t)}t∈[0,∞),Pµsin) is a version of the determinantal
Ksin(s,x;t,y) ≡
?
?1
Ksin(x,y)
?∞
dkek2(t−s)/2+ik(y−x)− 1(s > t)p(s − t,x|y)
=









0
dueπ2u2(t−s)/2cos{πu(y − x)} if t > s
if t = s
−
1
dueπ2u2(t−s)/2cos{πu(y − x)} if t < s.
(1.4)
KJν(s,x;t,y) =



















?1
Jν(2√x)√yJ′
0
due−2u(s−t)Jν(2√ux)Jν(2√uy)ifs < t
ν(2√y) −√xJ′
x − y
due−2u(s−t)Jν(2√ux)Jν(2√uy)
ν(2√x)Jν(2√y)
ift = s
−
?∞
1
if s > t,
(1.5)
Jν(z) =
n=0
(−1)n
Γ(n + 1)Γ(n + 1 + ν)
?z
2
?2n+ν
,z ∈ C
0e−uux−1du (Theorem 2.7). This process is reversible
with respect to the determinantal point process µJνwith the Bessel kernel
Jν(2√x)√yJ′
KJν(x,y) =

ν(2√y) −√xJ′
x − y
ν(2√x)Jν(2√y)
ifx ?= y,
x = y,
(Jν(2√x)2− Jν+1(2√x)Jν−1(2√x) if
(1.6)
ν(x) = dJν(x)/dx.
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Finally we will study the infinite-dimensional determinantal process ({Ξ(t)}t∈[0,∞),PAi)
associated with the extended Airy kernel KAi:


x,y ∈ R, where Ai(·) is the Airy function defined by
Ai(z) =
2π
R
KAi(s,x;t,y) ≡





?∞
−
0
?0
due−u(t−s)/2Ai(u + x)Ai(u + y)if t ≥ s
−∞
due−u(t−s)/2Ai(u + x)Ai(u + y) if t < s,
(1.7)
1
?
dkei(zk+k3/3),z ∈ C,
which is reversible with respect to the determinantal point process µAiwith the Airy kernel


and shows the same asymptotic behavior with the function ? ρ(x) ≡ (√−x/π)1(x < 0) as
and 1(ω) = 0 otherwise (see (4.12) in Lemma 4.3). Therefore the repulsive force from
infinite number of particles in the negative region x < 0 causes a positive drift with infinite
strength. To compensate it in order to obtain a stationary system, another negative drift with
infinite strength should be included in the process. For this reason we have to modify our
strategy and we introduce not only a sequence of probability measures but also a sequence of
approximate processes with finite numbers of particles ({Ξ? ρN(t)}t∈[0,∞),PξN),N ∈ N, having
negative drifts such that their strength diverges as N goes to infinity. To determine the N-
dependence of negative drifts , we use the estimate (4.13) in Lemma 4.3. Then we will show
that the limiting process is a version of ({Ξ(t)}t∈[0,∞),PAi) and it is Markovian (Theorem
2.8).
KAi(x,y) =

Ai(x)Ai′(y) − Ai′(x)Ai(y)
x − y
(Ai′(x))2− x(Ai(x))2
if x ?= y
if x = y,
(1.8)
where Ai′(x) = dAi(x)/dx. In this process the density ρAi(x) of particles is not bounded
x → −∞, where 1(ω) is the indicator function of a condition ω; 1(ω) = 1 if ω is satisfied,
Remark 1.
their stationary measures have been constructed and Markovianity of systems was proved in
Borodin and Olshanski [3, 4, 5], Olshanski [24], and Borodin and Gorin [2]. They determined
the state spaces and transition functions associated with Feller semi-groups and concluded
that these infinite-dimensional determinantal processes are strong Feller processes. In the
present paper, we first prove Markovianity of the determinantal processes with the extended
sine kernel Ksin, the extended Airy kernel KAi, and the extended Bessel kernel KJν, which
have been well-studied in the random matrix theory and its related fields. To these processes,
the argument by [3, 4, 5, 24, 2] can not be applied, and the strong Feller property has not
yet proved. (See also Remark 2 given at the end of Section 2.)
Other processes having infinite-dimensional determinantal point processes as
The paper is organized as follows. In Section 2 preliminaries and main results are given.
In Section 3 the basic properties of the correlation functions are summarized. Section 4 is
devoted to proofs of results by using Lemma 4.3. In Section 5 Lemma 4.3 is proved by using
the estimate of Hermite polynomials given by Plancherel and Rotach [29].
5