Elliptic Determinantal Processes and Elliptic Dyson Models
We introduce seven families of stochastic systems of interacting particles in one-dimension corresponding to the seven families of irreducible reduced affine root systems. We prove that they are determinantal in the sense that all spatio-temporal correlation functions are given by determinants contr...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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Інститут математики НАН України
2017
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| Цитувати: | Elliptic Determinantal Processes and Elliptic Dyson Models / M. Katori // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 43 назв. — англ. |
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| author | Katori, M. |
| author_facet | Katori, M. |
| citation_txt | Elliptic Determinantal Processes and Elliptic Dyson Models / M. Katori // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 43 назв. — англ. |
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| description | We introduce seven families of stochastic systems of interacting particles in one-dimension corresponding to the seven families of irreducible reduced affine root systems. We prove that they are determinantal in the sense that all spatio-temporal correlation functions are given by determinants controlled by a single function called the spatio-temporal correlation kernel. For the four families AN₋₁, BN, CN and DN, we identify the systems of stochastic differential equations solved by these determinantal processes, which will be regarded as the elliptic extensions of the Dyson model. Here we use the notion of martingales in probability theory and the elliptic determinant evaluations of the Macdonald denominators of irreducible reduced affine root systems given by Rosengren and Schlosser.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 079, 36 pages
Elliptic Determinantal Processes
and Elliptic Dyson Models
Makoto KATORI
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
Received April 19, 2017, in final form September 29, 2017; Published online October 04, 2017
https://doi.org/10.3842/SIGMA.2017.079
Abstract. We introduce seven families of stochastic systems of interacting particles in one-
dimension corresponding to the seven families of irreducible reduced affine root systems.
We prove that they are determinantal in the sense that all spatio-temporal correlation func-
tions are given by determinants controlled by a single function called the spatio-temporal
correlation kernel. For the four families AN−1, BN , CN and DN , we identify the systems
of stochastic differential equations solved by these determinantal processes, which will be
regarded as the elliptic extensions of the Dyson model. Here we use the notion of mar-
tingales in probability theory and the elliptic determinant evaluations of the Macdonald
denominators of irreducible reduced affine root systems given by Rosengren and Schlosser.
Key words: elliptic determinantal processes; elliptic Dyson models; determinantal martin-
gales; elliptic determinant evaluations; irreducible reduced affine root systems
2010 Mathematics Subject Classification: 60J65; 60G44; 82C22; 60B20; 33E05; 17B22
1 Introduction and main results
Stochastic analysis on interacting particle systems is important to provide useful models describ-
ing equilibrium and non-equilibrium phenomena studied in statistical physics [15]. Determinan-
tal process is a stochastic system of interacting particles which is integrable in the sense that all
spatio-temporal correlation functions are given by determinants controlled by a single function
called the spatio-temporal correlation kernel [4, 20]. Since the generating functions of correlation
functions are generally given by the Laplace transforms of probability densities, the stochastic
integrability of determinantal processes is proved by showing that the Laplace transform of any
multi-time joint probability density is expressed by the spatio-temporal Fredholm determinant
associated with the correlation kernel. The purpose of this paper is to present new kinds of
determinantal processes in which the interactions between particles are described by the loga-
rithmic derivatives of Jacobi’s theta functions. A classical example of determinantal processes
is Dyson’s Brownian motion model with parameter β = 2, which is a dynamical version of the
eigenvalue statistics of random matrices in the Gaussian unitary ensemble (GUE), and we call it
simply the Dyson model [6, 15, 38]. We will extend the Dyson model to the elliptic-function-level
in this paper. We use the notion of martingales in probability theory [14, 15] and the elliptic de-
terminantal evaluations of the Macdonald denominators of seven families of irreducible reduced
affine root systems given by Rosengren and Schlosser [34] (see also [26, 42]).
Among the seven families of irreducible reduced affine root systems, R = AN−1, BN , B
∨
N , CN ,
C∨N , BCN , and DN , we reported the results only for the system R = AN−1 in the previous papers
This paper is a contribution to the Special Issue on Elliptic Hypergeometric Functions and Their Applications.
The full collection is available at https://www.emis.de/journals/SIGMA/EHF2017.html
mailto:katori@phys.chuo-u.ac.jp
https://doi.org/10.3842/SIGMA.2017.079
https://www.emis.de/journals/SIGMA/EHF2017.html
2 M. Katori
[16, 17] as follows. Assume 0 < t∗ <∞, 0 < r <∞, 0 < N <∞, and let
A2πr
N (t∗ − t, x) =
[
1
2πr
∂
∂v
log ϑ1(v; τ)
]
v=x/2πr, τ=iN (t∗−t)/2πr2
, t ∈ [0, t∗), (1.1)
where θ1(v; τ) denotes one of the Jacobi theta functions. See Appendix A for the Jacobi theta
functions and related functions. For N ∈ {2, 3, . . . }, we define the Weyl chamber
WN =
{
x = (x1, x2, . . . , xN ) ∈ RN : x1 < x2 < · · · < xN
}
.
We introduced a one-parameter (β > 0) family of systems of stochastic differential equations
(SDEs) for t ∈ [0, t∗) [17]
(AN−1) X
AN−1
j (t) = uj +Wj(t) +
β
2
∫ t
0
A2πr
N
(
t∗ − s,
N∑
`=1
X
AN−1
` (s)− κN
)
ds
+
β
2
∑
1≤k≤N,
k 6=j
∫ t
0
A2πr
N
(
t∗ − s,X
AN−1
j (s)−XAN−1
k (s)
)
ds,
j = 1, . . . , N, in R, (1.2)
for u = (u1, . . . , uN ) ∈WN , where Wj(t), t ≥ 0, j = 1, . . . , N are independent one-dimensional
standard Brownian motions, and
κN =
{
πr(N − 1), if N is even,
πr(N − 2), if N is odd.
We called this family of N -particle systems on R, XAN−1(t) =
(
X
AN−1
1 (t), . . . , X
AN−1
N (t)
)
,
t ∈ [0, t∗), the elliptic Dyson model of type A with parameter β. By (A.9) in Appendix A.2,
(1.2) gives
dX
AN−1
j (t) ∼ dWj(t) +
β
2
v
AN−1
j −XAN−1
j (t)
t∗ − t
dt, j = 1, 2, . . . , N, in t ↑ t∗,
with
v
AN−1
j =
πr
N
(2j − 1), if N is even,
2πr
N
(j − 1), if N is odd,
j = 1, 2, . . . , N,
which give equidistant-spacing configurations vAN−1 =
(
v
AN−1
1 , . . . , v
AN−1
N
)
. This implies that
the elliptic Dyson model of type A is realized as a system of interacting Brownian bridges (see, for
instance, [5, Part I, Section IV.4.22]) pinned at the configuration vAN−1 at time t∗ [17]. When the
system of SDEs is temporally homogeneous, it is well known that the corresponding Kolmogorov
(Fokker–Planck) equation for transition probability density can be transformed into a Calogero–
Moser–Sutherland quantum system (see, for instance, [8, Chapter 11]). The present system
of interacting Brownian bridges is temporally inhomogeneous, however, and the Kolmogorov
equation is mapped into a Schrödinger-type equation with time-dependent Hamiltonian and
the time-dependent ground energy [17]. The obtained quantum system is elliptic, but different
from the elliptic Calogero–Moser–Sutherland model extensively studied as a quantum integrable
system [7, 25, 31, 39, 40]. We found at the same time that the interaction among particles
vanishes when the parameter is chosen to be a special value (β = 2 in our case) as found in
Elliptic Determinantal Processes and Elliptic Dyson Models 3
the usual Calogero–Moser–Sutherland models. We applied the determinantal-martingale-me-
thod [14] and proved that the elliptic Dyson model of type A with β = 2 is an integrable
stochastic process in a sense that it is determinantal for a set of observables [16, 17].
In the present paper, we report the results for other six systems, R = BN , B
∨
N , CN , C
∨
N ,
BCN , DN . Here we first construct the six families of determinantal processes (Theorem 1.1).
Then for the three families BN , CN and DN , we clarify the systems of SDEs (with parameter
β = 2) which are solved by our new determinantal processes (Theorem 1.2).
For N ∈ {2, 3, . . . }, 0 < r <∞, define
W(0,πr)
N =
{
x = (x1, x2, . . . , xN ) ∈ RN : 0 < x1 < x2 < · · · < xN < πr
}
.
Let u = (u1, u2, . . . , uN ) ∈W(0,πr)
N , and
τR(t) =
iNR(t∗ − t)
2πr2
, (1.3)
where the numbers NR are given by
NR =
N, R = AN−1,
2N − 1, R = BN ,
2N, R = B∨N , C
∨
N ,
2(N + 1), R = CN ,
2N + 1, R = BCN ,
2(N − 1), R = DN .
(1.4)
For t ∈ [0, t∗) put
DR
u (t,x) =
cR0
(
τR(t)
)
cR0
(
τR(0)
) N∏
`=1
ϑ1
(
cR1 x`/2πr; c
R
2 τ
R(t)
)
ϑ1
(
cR1 u`/2πr; c
R
2 τ
R(0)
)
×
∏
1≤j<k≤N
ϑ1
(
(xk − xj)/2πr; τR(t)
)
ϑ1
(
(uk − uj)/2πr; τR(0)
) ϑ1((xk + xj)/2πr; τ
R(t)
)
ϑ1
(
(uk + uj)/2πr; τR(0)
)
for R = BN , B
∨
N , CN , C
∨
N , (1.5)
DBCN
u (t,x) =
cBCN0
(
τBCN (t)
)
cBCN0
(
τBCN (0)
) N∏
`=1
ϑ1
(
x`/2πr; τ
BCN (t)
)
ϑ1
(
u`/2πr; τBCN (0)
) ϑ0(x`/πr; 2τBCN (t)
)
ϑ0
(
u`/πr; 2τBCN (0)
)
×
∏
1≤j<k≤N
ϑ1
(
(xk − xj)/2πr; τBCN (t)
)
ϑ1
(
(uk − uj)/2πr; τBCN (0)
) ϑ1((xk + xj)/2πr; τ
BCN (t)
)
ϑ1
(
(uk + uj)/2πr; τBCN (0)
) , (1.6)
DDN
u (t,x) =
cDN0
(
τDN (t)
)
cDN0
(
τDN (0)
)
×
∏
1≤j<k≤N
ϑ1
(
(xk − xj)/2πr; τDN (t)
)
ϑ1
(
(uk − uj)/2πr; τDN (0)
) ϑ1((xk + xj)/2πr; τ
DN (t)
)
ϑ1
(
(uk + uj)/2πr; τDN (0)
) , (1.7)
where
cR0 (τ) = η(τ)−N(N−1) for R = BN , CN ,
c
B∨N
0 (τ) = η(τ)−(N−1)
2
η(2τ)−(N−1),
c
C∨N
0 (τ) = η(τ)−(N−1)
2
η(τ/2)−(N−1),
4 M. Katori
cBCN0 (τ) = η(τ)−N(N−1)η(2τ)−N ,
cDN0 (τ) = η(τ)−N(N−2), (1.8)
and
cBN1 = c
C∨N
1 = 1, c
B∨N
1 = cCN1 = 2,
cBN2 = cCN2 = 1, c
B∨N
2 = 2, c
C∨N
2 = 1/2. (1.9)
Here η(τ) denotes the Dedekind modular function (see, for instance, [32, Section 23.15])
η(τ) = eτπi/12
∞∏
n=1
(
1− e2nτπi
)
, =τ > 0. (1.10)
We call (1.5)–(1.7) the determinantal martingale-functions [14] (see Sections 2.2 and 2.5).
In the interval [0, πr], we consider the N -particle system of one-dimensional standard Brow-
nian motions B(t) = (B1(t), . . . , BN (t)), t ≥ 0 started at u = (u1, . . . , uN ) ∈W(0,πr)
N with either
an absorbing or reflecting boundary condition at the endpoints of the interval, 0 and πr. The
transition probability density of each particle is generally denoted by p[0,πr]. The boundary con-
ditions at 0 and πr are indicated by b and b′, respectively, and the transition probability density
with the specified boundary conditions b, b′ is written as p
[0,π]
b,b′ . In this paper the absorbing
(resp. reflecting) boundary condition is abbreviated as ‘a’ (resp. ‘r’). By the reflection princi-
ple of Brownian motion (see, for instance, [5, Appendixes 1.5 and 1.6]), if both boundaries are
absorbing, the transition probability density is given by
p[0,πr]aa (t, y|x) =
∞∑
k=−∞
{
pBM(t, y + 2πrk|x)− pBM(t, y + 2πrk| − x)
}
, (1.11)
and if both are reflecting, it is given by
p[0,πr]rr (t, y|x) =
∞∑
k=−∞
{
pBM(t, y + 2πrk|x) + pBM(t, y + 2πrk| − x)
}
, (1.12)
for x, y ∈ [0, πr], t ≥ 0, where pBM(t, y|x) denotes the transition probability density of the
one-dimensional standard Brownian motion
pBM(t, y|x) =
e−(x−y)
2/2t
√
2πt
, t > 0,
δ(x− y), t = 0.
(1.13)
We write the probability law of such a system of boundary-conditioned Brownian motions in
[0, πr] as P
[0,πr]
u . In P
[0,πr]
u , put
Tcollision = inf
{
t > 0: Bj(t) = Bk(t) for any j 6= k
}
, (1.14)
i.e., the first collision-time of the N -particle system of Brownian motions in the interval [0, πr].
Let 1(ω) be the indicator function of a condition ω; 1(ω) = 1 if ω is satisfied, and 1(ω) = 0
otherwise. Then we define
PRu
∣∣
Ft = 1(Tcollision > t)DR
u (t,B(t))P
[0,πr]
u
∣∣
Ft , t ∈ [0, t∗), (1.15)
where DR
u are given by (1.5)–(1.7) and Ft denotes the filtration associated with the Brownian
motion (see Section 2.1). That is, the Radon–Nikodym derivative of PRu with respect to the
Elliptic Determinantal Processes and Elliptic Dyson Models 5
Wiener measure P
[0,πr]
u of Brownian motion is given by 1(Tcollision > t)DR
u (t,B(t)) at each time
t ∈ [0, t∗). Therefore, a stochastic process with N particles governed by PRu is well-defined as
a realization of non-intersecting paths which are absolutely continuous to the N -particle paths
of independent Brownian motions in [0, πr].
For y ∈ R, δy(·) denotes the delta measure such that δy({x}) = 1 if x = y and δy({x}) = 0
otherwise. The first theorem of this paper is the following.
Theorem 1.1. Assume that 0 < t∗ < ∞, 0 < r < ∞. For each N ∈ {2, 3, . . . }, R = BN ,
B∨N , CN , C
∨
N , BCN , DN , u = (u1, . . . , uN ) ∈ W(0,πr)
N , PRu defined by (1.15) gives a probability
measure and defines a measure-valued stochastic process
ΞR(t, ·) =
N∑
j=1
δXR
j (t)(·), t ∈ [0, t∗). (1.16)
The process
((
ΞR(t)
)
t∈[0,t∗),P
R
u
)
is determinantal with the spatio-temporal correlation kernel
KR
u (s, x; t, y) =
N∑
j=1
p[0,πr](s, x|uj)MR
u,uj (t, y)− 1(s > t)p[0,πr](s− t, x|y), (1.17)
(s, x), (t, y) ∈ [0, t∗)× [0, πr], where
MR
u,uj (t, x) =
∫ ∞
−∞
ΦR
u,uj (x+ ix̃)pBM(t, x̃|0)dx̃ (1.18)
with the sets of entire functions (the elliptic Lagrange interpolation functions)
ΦR
u,uj (z) =
ϑ1
(
cR1 z/2πr; c
R
2 τ
R(0)
)
ϑ1
(
cR1 uj/2πr; c
R
2 τ
R(0)
)
×
∏
1≤`≤N,
6̀=j
ϑ1
(
(z − u`)/2πr; τR(0)
)
ϑ1
(
(uj − u`)/2πr; τR(0)
) ϑ1((z + u`)/2πr; τ
R(0)
)
ϑ1
(
(uj + u`)/2πr; τR(0)
)
for R = BN , B
∨
N , CN , C
∨
N , (1.19)
ΦBCN
u,uj (z) =
ϑ1
(
z/2πr; τBCN (0)
)
ϑ1
(
uj/2πr; τBCN (0)
) ϑ0(z/πr; 2τBCN (0)
)
ϑ0
(
uj/πr; 2τBCN (0)
)
×
∏
1≤`≤N,
` 6=j
ϑ1
(
(z − u`)/2πr; τBCN (0)
)
ϑ1
(
(uj − u`)/2πr; τBCN (0)
) ϑ1((z + u`)/2πr; τ
BCN (0)
)
ϑ1
(
(uj + u`)/2πr; τBCN (0)
) , (1.20)
ΦDN
u,uj (z) =
∏
1≤`≤N,
6̀=j
ϑ1
(
(z − u`)/2πr; τDN (0)
)
ϑ1
(
(uj − u`)/2πr; τDN (0)
) ϑ1((z + u`)/2πr; τ
DN (0)
)
ϑ1
(
(uj + u`)/2πr; τDN (0)
) , (1.21)
j = 1, 2, . . . , N . Here τR(t) are given by (1.3) with (1.4), and cR1 , cR2 for R = BN , B
∨
N , CN , C
∨
N
are given by (1.9).
The second theorem of this paper is the following.
Theorem 1.2. Let N ∈ {2, 3, . . . }, 0 < t∗ < ∞, 0 < r < ∞. Assume u = (u1, . . . , uN ) ∈
W(0,πr)
N . In the interval [0, πr], for
((
ΞBN (t)
)
t∈[0,t∗),P
BN
u
)
we put an absorbing boundary con-
dition at 0 and a reflecting boundary condition at πr, for
((
ΞCN (t)
)
t∈[0,t∗),P
CN
u
)
we put an
6 M. Katori
absorbing boundary condition both at 0 and πr, and for
((
ΞDN (t)
)
t∈[0,t∗),P
DN
u
)
we put a re-
flecting boundary condition both at 0 and πr, respectively. If we set ΞR(t) =
N∑
j=1
δXR
j (t), then
XR(t) =
(
XR
1 (t), . . . , XR
N (t)
)
, R = BN , CN , DN , solve the following systems of SDEs
(BN ) XBN
j (t) = uj +Wj(t) +
∫ t
0
A2πr
2N−1
(
t∗ − s,XBN
j (s)
)
ds
+
∑
1≤k≤N,
k 6=j
∫ t
0
{
A2πr
2N−1
(
t∗ − s,XBN
j (s)−XBN
k (s)
)
+A2πr
2N−1
(
t∗ − s,XBN
j (s) +XBN
k (s)
)}
ds, j = 1, 2, . . . , N,
in [0, πr] with a reflecting boundary condition at πr, (1.22)
(CN ) XCN
j (t) = uj +Wj(t) + 2
∫ t
0
A2πr
2(N+1)
(
t∗ − s, 2XCN
j (s)
)
ds
+
∑
1≤k≤N,
k 6=j
∫ t
0
{
A2πr
2(N+1)
(
t∗ − s,XCN
j (s)−XCN
k (s)
)
+A2πr
2(N+1)
(
t∗ − s,XCN
j (s) +XCN
k (s)
)}
ds, j = 1, 2, . . . , N,
in [0, πr], (1.23)
(DN ) XDN
j (t) = uj +Wj(t) +
∑
1≤k≤N,
k 6=j
∫ t
0
{
A2πr
2(N−1)
(
t∗ − s,XDN
j (s)−XDN
k (s)
)
+A2πr
2(N−1)
(
t∗ − s,XDN
j (s) +XDN
k (s)
)}
ds, j = 1, 2, . . . , N,
in [0, πr] with a reflecting boundary condition both at 0 and πr, (1.24)
where Wj(t), t ≥ 0, j = 1, 2, . . . , N are independent one-dimensional standard Brownian mo-
tions.
We call the systems (1.22)–(1.24) the elliptic Dyson models of types B, C, D, respectively.
By (A.9) in Appendix A.2, we see that in t ↑ t∗
dXR
j (t) ∼ dWj(t) +
vRj −XR
j (t)
t∗ − t
dt, j = 1, 2, . . . , N, R = BN , CN , DN
with
vBNj =
2j − 1
2N − 1
πr, vCNj =
j
N + 1
πr, vDNj =
j − 1
N − 1
πr, j = 1, 2, . . . , N.
The above implies that the elliptic Dyson models of types B, C, and D are realized as the
systems of interacting Brownian bridges pinned at the configurations vR =
(
vR1 , . . . , v
R
N
)
, R =
BN , CN , DN , at time t∗. Note that vBN1 > 0, vBNN = πr; vCN1 > 0, vCNN < πr; vDN1 = 0,
vDNN = πr, corresponding to the situation such that the absorbing boundary condition is imposed
at 0 for BN , and at 0 and πr for CN , while the reflecting boundary condition is imposed at
other endpoints of the interval [0, πr].
We see
lim
t∗→∞
A2πr
N (t∗ − t, x) =
1
2r
cot
( x
2r
)
,
Elliptic Determinantal Processes and Elliptic Dyson Models 7
by (1.1) and (A.4). Hence in the limit t∗ → ∞, the systems of SDEs (1.2) with β = 2, and of
(1.22)–(1.24) become the following temporally homogeneous systems of SDEs for t ∈ [0,∞):
(AN−1) X
AN−1
j (t) = uj +Wj(t)−
1
2r
∫ t
0
tan
(
1
2r
N∑
`=1
X
AN−1
` (s)
)
ds
+
1
2r
∑
1≤k≤N,
k 6=j
∫ t
0
cot
(
X
AN−1
j (s)−XAN−1
k (s)
2r
)
ds,
j = 1, 2, . . . , N, in R, (1.25)
(BN ) XBN
j (t) = uj +Wj(t) +
1
2r
∫ t
0
cot
(
XBN
j (s)
2r
)
ds
+
1
2r
∑
1≤k≤N,
k 6=j
∫ t
0
{
cot
(
XBN
j (s)−XBN
k (s)
2r
)
+ cot
(
XBN
j (s) +XBN
k (s)
2r
)}
ds, j = 1, 2, . . . , N,
in [0, πr] with a reflecting boundary condition at πr, (1.26)
(CN ) XCN
j (t) = uj +Wj(t) +
1
r
∫ t
0
cot
(
XCN
j (s)
r
)
ds
+
1
2r
∑
1≤k≤N,
k 6=j
∫ t
0
{
cot
(
XCN
j (s)−XCN
k (s)
2r
)
+ cot
(
XCN
j (s) +XCN
k (s)
2r
)}
ds, j = 1, 2, . . . , N, in [0, πr], (1.27)
(DN ) XDN
j (t) = uj +Wj(t) +
1
2r
∑
1≤k≤N,
k 6=j
∫ t
0
{
cot
(
XDN
j (s)−XDN
k (s)
2r
)
+ cot
(
XDN
j (s) +XDN
k (s)
2r
)}
ds, j = 1, 2, . . . , N,
in [0, πr] with a reflecting boundary condition both at 0 and πr, (1.28)
respectively. We will call the above systems (1.25)–(1.28) the trigonometric Dyson models of
types A, B, C and D, respectively.
Moreover, for
lim
r→∞
1
2r
cot
( x
2r
)
=
1
x
, lim
r→∞
1
2r
tan
( x
2r
)
= 0,
the r →∞ limit of the systems (1.25)–(1.28) are given as follows, for t ∈ [0,∞)
(AN−1) X
AN−1
j (t) = uj +Wj(t) +
∑
1≤k≤N,
k 6=j
∫ t
0
1
X
AN−1
j (s)−XAN−1
k (s)
ds,
j = 1, 2, . . . , N, in R, (1.29)
8 M. Katori
(CN ) XCN
j (t) = uj +Wj(t) +
∫ t
0
1
XCN
j (s)
ds
+
∑
1≤k≤N,
k 6=j
∫ t
0
{
1
XCN
j (s)−XCN
k (s)
+
1
XCN
j (s) +XCN
k (s)
}
ds,
j = 1, 2, . . . , N, in [0,∞), (1.30)
(DN ) XDN
j (t) = uj +Wj(t)
+
∑
1≤k≤N,
k 6=j
∫ t
0
{
1
XDN
j (s)−XDN
k (s)
+
1
XDN
j (s) +XDN
k (s)
}
ds, (1.31)
j = 1, 2, . . . , N, in [0,∞) with a reflecting boundary condition at 0.
That is, the system (1.25) is reduced to (1.29), (1.26) and (1.27) are degenerated to (1.30),
and (1.28) to (1.31). The system (1.29) is the original Dyson model (Dyson’s Brownian motion
model with parameter β = 2) [6, 15]. The SDEs (1.30) and (1.31) are known as the system
of noncolliding absorbing Brownian motions (or the Dyson model of type C) and the system
of noncolliding reflecting Brownian motions (or the Dyson model of type D), respectively. See
[2, 18, 19, 23, 24, 28, 41].
The correlation kernel for a determinantal process is in general a function of two points on the
spatio-temporal plane, say, (s, x) and (t, y). In the present formula (1.17) in Theorem 1.1, the
dependence of the correlation kernel KR
u on one spatio-temporal point (s, x) is explicitly given by
the transition probability density p[0,πr] of a single Brownian motion in an interval [0, πr] with
given boundary conditions. The dependence of KR
u on another spatio-temporal point (t, y) is
described by the following two procedures. The information of interaction among particles and
initial configuration is expressed by a set of entire functions
{
ΦR
u,uj (z)
}N
j=1
, (1.19)–(1.21), which
are static functions without time-variable t. Then evolution in time t is given by the integral
transformation (1.18) specified by the transition probability density pBM with time duration t
of a single Brownian motion. One of the benefits of such separation of static information
and dynamics in the formula is that we can trace a relaxation process to equilibrium, which is
a typical non-equilibrium phenomenon, for any initial configuration in W(0,πr)
N [14]. In order
to demonstrate this fact, in Section 5 we will take the temporally homogeneous limit t∗ → ∞
to make the systems have equilibrium processes and study the relaxation to equilibria for the
trigonometric Dysom models of types C and D. Another possible benefit of the present formula
for spatio-temporal correlation kernels is that the infinite particle systems will be studied if we
can control the set of entire functions
{
ΦR
u,uj (z)
}N
j=1
in the infinite-particle limit N → ∞. For
the original Dyson model [22] and other related systems [21, 23], we have applied the Hadamard
theorem on the Weierstrass canonical product-formulas of entire functions [27] to analyze the
infinite particle systems. In the present case with (1.19)–(1.21), we have to treat the rations of
infinite products of the Jacobi theta functions in this limit N →∞. Asymptotic analysis with
N →∞ will be an interesting future problem.
The paper is organized as follows. In Section 2 we explain the relationship between determi-
nantal martingales and determinantal processes used in this paper, and determinantal equalities
(Lemma 2.4) obtained from the elliptic determinant evaluations of the Macdonald denomina-
tors given by Rosengren and Schlosser [34]. There we explain how to derive the determinan-
tal martingale-functions (1.5)–(1.7), which define the interacting systems of Brownian motions
by (1.15). Proofs of Theorems 1.1 and 1.2 are given in Sections 3 and 4, respectively. We study
the temporally homogeneous limit t∗ → ∞ in Section 5, and relaxation processes to equilibria
are clarified for the trigonometric Dyson models of types C and D. Concluding remarks and
Elliptic Determinantal Processes and Elliptic Dyson Models 9
open problems are given in Section 6. Notations and formulas of the Jacobi theta functions and
related functions are listed in Appendix A.
2 Determinantal martingales and determinantal equalities
2.1 Notion of martingale
Martingales are the stochastic processes preserving their mean values and thus they represent
fluctuations. A typical example of martingale is the one-dimensional standard Brownian motion
as explained below. Let B(t), t ≥ 0 denote the position of the standard Brownian motion in R
starting from the origin 0 at time t = 0. The transition probability density from the position
x ∈ R to y ∈ R in time duration t ≥ 0 is given by (1.13). Let B(R) be the Borel set on R. Then
for an arbitrary time sequence 0 ≡ t0 < t1 < · · · < tM < ∞, M ∈ N ≡ {1, 2, . . . }, and for any
Am ∈ B(R), m = 1, 2, . . . ,M ,
P
[
B(tm) ∈ Am, m = 1, 2, . . . ,M
]
=
∫
A1
dx(1) · · ·
∫
AM
dx(M)
M∏
m=1
pBM
(
tm − tm−1, x(m)|x(m−1)
)
with x(0) ≡ 0. The collection of all paths is denoted by Ω and there is a subset Ω̃ ⊂ Ω such
that P[Ω̃] = 1 and for any realization of path ω ∈ Ω̃, B(t) = B(t, ω), t ≥ 0 is a real continuous
function of t. In other words, B(t), t ≥ 0 has a continuous path almost surely (a.s. for short).
For each t ∈ [0,∞), we write the smallest σ-field (completely additive class of events) generated
by the Brownian motion up to time t as Ft = σ(B(s) : 0 ≤ s ≤ t). We have a nondecreasing
family {Ft : t ≥ 0} such that Fs ⊂ Ft for 0 ≤ s < t < ∞, which we call a filtration, and
put F =
⋃
t≥0Ft. The triplet (Ω,F ,P) is called the probability space for the one-dimensional
standard Brownian motion, and P is especially called the Wiener measure. The expectation with
respect to the probability law P is written as E. When we see pBM(t, y|x) as a function of y, it
is nothing but the probability density of the normal distribution with mean x and variance t,
and hence it is easy to verify that
E[B(t)|Fs] =
∫ ∞
−∞
xpBM(t− s, x|B(s))dx = B(s) a.s. 0 ≤ s < t <∞,
which means that B(t), t ≥ 0 is a martingale. We see, however, B(t)n, t ≥ 0, n ∈ {2, 3, . . . } are
not martingales, since the generating function of E[B(t)n|Fs], n ∈ N0 ≡ {0, 1, 2, . . . }, 0 < s ≤
t <∞, with parameter α ∈ C is calculated as
∞∑
n=0
αn
n!
E[B(t)n|Fs] = E[eαB(t)|Fs] =
∫ ∞
−∞
eαx
e−(x−B(s))2/2(t−s)√
2π(t− s)
dx
= eαB(s)+(t−s)α2/2 6= eαB(s) a.s.
Now we assume that B̃(t), t ≥ 0 is a one-dimensional standard Brownian motion which is
independent of B(t), t ≥ 0, and its probability space is denoted by
(
Ω̃, F̃ , P̃
)
. Then we introduce
a complex-valued martingale called the complex Brownian motion
Z(t) = B(t) + iB̃(t), t ≥ 0,
with i =
√
−1. The probability space of Z(t), t ≥ 0 is given by the product space (Ω,F ,P) ⊗
(Ω̃, F̃ , P̃) and we write the expectation as E = E ⊗ Ẽ. For the complex Brownian motion, by
10 M. Katori
the independence of its real and imaginary parts, we see that
∞∑
n=0
αn
n!
E
[
Z(t)n|Fs ⊗ F̃s
]
= E
[
eαZ(t)|Fs ⊗ F̃s
]
= E
[
eαB(t)|Fs
]
× Ẽ
[
eiαB̃(t)|F̃s
]
= eαB(s)+(t−s)α2/2 × eiαB̃(s)−(t−s)α2/2 = eαZ(s) =
∞∑
n=0
αn
n!
Z(s)n
a.s. 0 ≤ s < t <∞, α ∈ C,
which implies that for any n ∈ N0, Z(t)n, t ≥ 0 is a martingale. This observation will be
generalized as the following stronger statement; if F is an entire and non-constant function,
then F (Z(t)), t ≥ 0 is a time change of a complex Brownian motion (see, for instance, [33,
Theorem 2.5 in Section V.2]). Hence F (Z(t)), t ≥ 0 is a martingale:
E[F (Z(t))|Fs ⊗ F̃s] = F (Z(s)) a.s. 0 ≤ s < t <∞.
If we take the expectation Ẽ with respect to =Z(·) = B̃(·) of the both sides of the above equality,
we have
E
[
Ẽ[F (Z(t))]
∣∣Fs] = Ẽ[F (Z(s))] a.s. 0 ≤ s < t <∞.
In this way we can obtain a martingale F̂ (t, B(t)) ≡ Ẽ[F (Z(t))], t ≥ 0 with respect to the
filtration Ft of the one-dimensional Brownian motion. The present argument implies that if
we have proper entire functions, then we will obtain useful martingales describing intrinsic
fluctuations involved in interacting particle systems.
2.2 Basic equalities and determinantal martingales
Let fj , j ∈ I, be an infinite series of linearly independent entire functions, where the index set
I is Z = {. . . ,−1, 0, 1, 2, . . . } or N. For N ∈ {2, 3 . . . }, we assume
u = (u1, u2, . . . , uN ) ∈WN . (2.1)
We then define a set of N distinct entire and non-constant functions of z ∈ C by
Φu,uj (z) =
det
1≤`,m≤N
[f`(um + (z − um)δmj)]
det
1≤`,m≤N
[f`(um)]
, j = 1, 2, . . . , N. (2.2)
By definition, it is easy to verify that
Φu,uj (uk) = δjk, j, k ∈ {1, 2, . . . , N}. (2.3)
It should be noted that given (2.1), Φu,uj (z), j ∈ {1, 2, . . . , N} satisfying (2.3) can be regarded
as the Lagrange interpolation functions [10, 11, 12]. We can prove the following lemmas.
Lemma 2.1. Let N ∈ {2, 3, . . . }. The functions (2.2) have the following expansions
Φu,uj (z) =
N∑
k=1
φu,uj (k)fk(z), j ∈ {1, 2, . . . , N}, (2.4)
and the coefficients φu,uj (k), j, k ∈ {1, 2, . . . , N} satisfy the relations
N∑
`=1
fj(u`)φu,u`(k) =
det
1≤m,n≤N
[fm+(j−m)δmk(un)]
det
1≤m,n≤N
[fm(un)]
, j ∈ Z, k ∈ {1, 2, . . . , N}. (2.5)
Elliptic Determinantal Processes and Elliptic Dyson Models 11
In particular, if j is an element of {1, 2, . . . , N},
N∑
`=1
fj(u`)φu,u`(k) = δjk, j, k ∈ {1, 2, . . . , N}. (2.6)
Proof. Consider an N ×N matrix
fu = (f`(um))1≤`,m≤N ,
and write its determinant as |fu| 6= 0. The minor determinant |fu(k, j)| is defined as the deter-
minant of the (N − 1)× (N − 1) matrix, fu(k, j), which is obtained from fu by deleting the k-th
row and the j-th column, k, j ∈ {1, 2, . . . , N}. Then the determinant in the numerator of (2.2)
is expanded along the j-th column and we obtain (2.4) with
φu,uj (k) = (−1)j+k
|fu(k, j)|
|fu|
. (2.7)
Hence
N∑
`=1
fj(u`)φu,u`(k) =
1
|fu|
N∑
`=1
(−1)`+k|fu(k, `)|fj(u`),
which proves (2.5), for the summation in the r.h.s. is the expansion of det
1≤m,n≤N
[fm+(j−m)δmk(un)]
along the k-th row. It is immediate to conclude (2.6) from (2.5), if both of j and k are elements
of {1, 2, . . . , N}. �
Then we obtain the following equalities.
Lemma 2.2.
(i) For z ∈ C,
N∑
`=1
fj(u`)Φu,u`(z) = fj(z), j ∈ {1, 2, . . . , N}. (2.8)
(ii) For z = (z1, z2, . . . , zN ) ∈ CN , the equality
det
1≤j,k≤N
[Φu,uj (zk)] =
det
1≤j,k≤N
[fj(zk)]
det
1≤j,k≤N
[fj(uk)]
, z = (z1, z2, . . . , zN ) ∈ CN (2.9)
holds.
Proof. (i) Multiply the both sides of (2.6) by fk(z) and take summation over k ∈ {1, 2, . . . , N}.
Then use (2.4) to obtain (2.8). (ii) Consider N equations by putting z = zj , j ∈ {1, 2, . . . , N}
in (2.8). Then calculate the determinant |fz| to prove the determinantal equality (2.9). �
We consider N pairs of independent copies (Bk(t), B̃k(t)), k = 1, 2, . . . , N , of (B(t), B̃(t)),
t ≥ 0, and define N independent complex Brownian motions
Zk(t) = uk +Bk(t) + iB̃k(t), t ≥ 0, k = 1, 2, . . . , N, (2.10)
each of which starts from uk ∈ R. The probability law and expectation of them are given
by Pu ⊗ P̃ and Eu ⊗ Ẽ, respectively. Then for each complex Brownian motion Zk(t), t ≥ 0,
12 M. Katori
k = 1, 2, . . . , N , we have N distinct time-changes of complex Brownian motions, Φu,uj (Zk(t)),
t ≥ 0, j = 1, 2, . . . , N , started from the real values, 0 or 1; Φu,uj (Zk(0)) = Φu,uj (uk) = δjk,
j, k = 1, 2, . . . , N . Therefore, we can conclude that, if we take expectation Ẽ, we will obtain N
distinct martingales
Mu,uj (t, Bk(t)) ≡ Ẽ
[
Φu,uj (Bk(t) + iB̃k(t))
]
=
det
1≤`,m≤N
[
f̂`(tδmj , um + (Bk(t)− um)δmj)
]
det
1≤`,m≤N
[
f̂`(0, um)
] , t ≥ 0, j = 1, 2, . . . , N,
for each one-dimensional Brownian motion Bk(t), t ≥ 0, k = 1, 2, . . . , N , where
f̂`(t, x) ≡ Ẽ
[
f`(x+ iB̃(t))
]
=
∫ ∞
−∞
f`(x+ ix̃)pBM(t, x̃|0)dx̃, ` ∈ I.
By this definition, f̂`(0, x) = f`(x), ` ∈ I. Then we define a multivariate function of t ≥ 0 and
x = (x1, x2, . . . , xN ) ∈ RN by
Du(t,x) = det
1≤j,k≤N
[Mu,uj (t, xk)], (2.11)
which gives a martingale as a functional of t ≥ 0 and B(t) = (B1(t), B2(t), . . . , BN (t)), t ≥ 0,
Eu[Du(t,B(t))|Fs] = Du(s,B(s)), 0 ≤ s < t <∞.
By the multi-linearity of determinant and (2.9), we see
Du(t,x) = Ẽ
[
det
1≤j,k≤N
[
Φu,uj (xk + iB̃k(t))
]]
=
det
1≤j,k≤N
[
f̂j(t, xk)
]
det
1≤j,k≤N
[
f̂j(0, uk)
] . (2.12)
We call Du(t,B(t)), t ≥ 0, the determinantal martingale [14].
2.3 Auxiliary measure and spatio-temporal Fredholm determinant
Now we introduce an auxiliary measure P̂u, which is complex-valued in general, but is absolutely
continuous to the Wiener measure Pu as
P̂u
∣∣
Ft = Du(t,B(t))Pu
∣∣
Ft , t ≥ 0. (2.13)
(Note that this is generally different from PRu given by (1.15), since the condition 1(Tcollision > t)
is omitted, and hence it is not the measure representing any noncolliding particles.) Here we
consider the corresponding auxiliary system of N particles on R governed by P̂u
X̂(t) =
(
X̂1(t), . . . , X̂N (t)
)
, t ≥ 0,
starting from (2.1) and each particle of which has a continuous path a.s. We consider the
unlabeled configuration of X̂(t) as
Ξ̂(t, ·) =
N∑
j=1
δ
X̂j(t)
(·), t ≥ 0. (2.14)
Consider an arbitrary number M ∈ N and an arbitrary set of strictly increasing times t =
{t1, t2, . . . , tM}, 0 ≡ t0 < t1 < · · · < tM <∞. Let Cc(R) be the set of all continuous real-valued
Elliptic Determinantal Processes and Elliptic Dyson Models 13
functions with compact supports on R. For g = (gt1 , gt1 , . . . , gtM ) ∈ Cc(R)M , we consider the
following functional of g
L̂u[t,g] = Êu
[
exp
{
M∑
m=1
∫
R
gtm(x)Ξ̂(tm, dx)
}]
, (2.15)
which is the Laplace transform of the multi-time distribution function P̂u on a set of times t
with the functions g. If we put
χt = egt − 1,
then (2.15) can be written as
L̂u[t,g] = Êu
M∏
m=1
N∏
j=1
{
1 + χtm
(
X̂j(tm)
)} . (2.16)
Explicit expression of (2.16) is given by the following multiple integrals
L̂u[t,g] =
∫
RN
dx(1) · · ·
∫
RN
dx(M)Du
(
tM ,x
(M)
)
×
M∏
m=1
N∏
j=1
[
pBM
(
tm − tm−1, x(m)
j |x
(m−1)
j
){
1 + χtm
(
x
(m)
j
)}]
,
where x
(0)
j = uj , j = 1, 2, . . . , N , x(m) =
(
x
(m)
1 , . . . , x
(m)
N
)
, and dx(m) =
N∏
j=1
dx
(m)
j , m =
1, 2, . . . ,M .
The following was proved as Theorem 1.3 in [14].
Proposition 2.3. Put
K̂u(s, x; t, y) =
N∑
j=1
pBM(s, x|uj)Mu,uj (t, y)− 1(s > t)pBM(s− t, x|y),
s, t > 0, x, y ∈ R. Then the following equality holds for an arbitrary number M ∈ N, an
arbitrary set of strictly increasing times t = {t1, t2, . . . , tM}, 0 ≡ t0 < t1 < · · · < tM < ∞, and
g = (gt1 , gt1 , . . . , gtM ) ∈ Cc(R)M ,
L̂u[t,g] = Det
s,t∈t,
x,y∈R
[
δstδ(x− y) + K̂u(s, x; t, y)χt(y)
]
,
where the r.h.s. denotes the spatio-temporal Fredholm determinant with the kernel K̂u defined by
Det
s,t∈t,
x,y∈R
[
δstδ(x− y) + K̂u(s, x; t, y)χt(y)
]
(2.17)
≡
∑
0≤Nm≤N,
1≤m≤M
∫
M∏
m=1
WNm
M∏
m=1
dx
(m)
Nm
Nm∏
j=1
χtm
(
x
(m)
j
)
det
1≤j≤Nm, 1≤k≤Nn,
1≤m,n≤M
[
K̂u
(
tm, x
(m)
j ; tn, x
(n)
k
)]
,
where dx
(m)
Nm
=
Nm∏
j=1
dx
(m)
j , m = 1, 2, . . . ,M , and the term with Nm = 0, 1 ≤ ∀m ≤ M in the
r.h.s. should be interpreted as 1.
14 M. Katori
This proposition is general and it proves that the auxiliary system (2.14) is determinantal.
The measure (2.13) which governs this particle system is, however, complex-valued in general,
and hence the system is unphysical. The problem is to clarify the proper conditions which
should be added to (2.13) to construct a non-negative-definite real measure, i.e., the probability
measure, which defines a physical system of interacting particles. As a matter of course, this
problem depends on the choice of an infinite set of linearly independent entire functions fj ,
j ∈ I.
2.4 Elliptic determinant evaluations of the Macdonald denominators
Here we report the results when we choose the entire functions fj , j ∈ Z as follows
f
AN−1
j (z; τ) = eiJ
AN−1 (j)z/rϑ1
(
JAN−1(j)τ +
NAN−1z
2πr
+
1− (−1)N
4
;NAN−1τ
)
,
fRj (z; τ) = eiJ
R(j)z/rϑ1
(
JR(j)τ +
NRz
2πr
;NRτ
)
− e−iJR(j)z/rϑ1
(
JR(j)τ − N
Rz
2πr
;NRτ
)
for R = BN , B
∨
N ,
fRj (z; τ) = eiJ
R(j)z/rϑ1
(
JR(j)τ +
NRz
2πr
+
1
2
;NRτ
)
− e−iJR(j)z/rϑ1
(
JR(j)τ − N
Rz
2πr
+
1
2
;NRτ
)
for R = CN , C
∨
N , BCN ,
fDNj (z; τ) = eiJ
DN (j)z/rϑ1
(
JDN (j)τ +
NDN z
2πr
+
1
2
;NDN τ
)
+ e−iJ
DN (j)z/rϑ1
(
JDN (j)τ − N
DN z
2πr
+
1
2
;NDN τ
)
, (2.18)
z ∈ C, j ∈ Z, with N ∈ N, 0 < r <∞, and τ ∈ C with 0 < =τ <∞, where
JR(j) =
j − 1, R = AN−1, BN , B
∨
N , DN ,
j, R = CN , BCN ,
j − 1/2, R = C∨N ,
(2.19)
and NR are given by (1.4). These functions (2.18) were used to express the determinant evalua-
tions by Rosengren and Schlosser [34] for the Macdonald denominators WR(x) for seven families
of irreducible reduced affine root systems R = AN−1, BN , B
∨
N , CN , C
∨
N , BCN , DN . Note that, if
x ∈ R, τ ∈ iR, and 0 < =τ <∞, then fR(x; τ) ∈ R for R = BN , B
∨
N , DN and fR(x; τ) ∈ iR for
R = CN , C
∨
N , BCN .
Assume that, for u = (u1, u2, . . . , uN ) ∈WN , det
1≤j,k≤N
[fj(uk)] is factorized in the form
det
1≤j,k≤N
[fj(uk)] = k0ksym(u)
N∏
`=1
k1(u`)
∏
1≤j<k≤N
k2(uk, uj), (2.20)
where k0 is a constant, k1 is a single variable function, k2 is an antisymmetric function of two vari-
ables, k2(u, v) = −k2(v, u), and ksym(u) is a symmetric function of u which cannot be factorized
as
N∏
`=1
k1(u`). For u = (u1, . . . , uj−1, uj , uj+1, . . . , uN ) ∈WN , we replace the j-th component uj
Elliptic Determinantal Processes and Elliptic Dyson Models 15
by a variable z and write the obtained vector as u(j)(z) = (u1, . . . , uj−1, z, uj+1, . . . , uN ). Under
the assumption (2.20), Φu,uj (z) defined by (2.2) is also factorized as
Φu,uj (z) =
ksym(u(j)(z))
ksym(u)
k1(z)
k1(uj)
∏
1≤`≤N,
`6=j
k2(z, u`)
k2(uj , u`)
, j = 1, 2, . . . , N. (2.21)
Then the determinantal equality (2.9) becomes
det
1≤j,k≤N
ksym(u(j)(zk))
ksym(u)
k1(zk)
k1(uj)
∏
1≤`≤N,
` 6=j
k2(zk, u`)
k2(uj , u`)
=
ksym(z)
ksym(u)
N∏
`=1
k1(z`)
k1(u`)
∏
1≤j<k≤N
k2(zk, zj)
k2(uk, uj)
, z ∈ CN , u ∈WN . (2.22)
In [34] Rosengren and Schlosser gave the elliptic determinant evaluations of the Macdonald
denominators of seven families of irreducible reduced affine root systems, AN−1, BN , B∨N , CN ,
C∨N , BCN and DN . From their determinantal equalities [34, Proposition 6.1], it is easy to
verify that our seven choices of fj , j ∈ Z given by (2.18) allow the factorization (2.20) for their
determinants. As functions of τ , we write
q(τ) = eτπi, q0(τ) =
∞∏
n=1
(
1− q(τ)2n
)
. (2.23)
Lemma 2.4. For the seven choices of fRj (·; τ), j ∈ Z, =τ > 0, given by (2.18), the equality
det
1≤j,k≤N
[
fRj (uk; τ)
]
= kR0 (τ)kRsym(u; τ)
N∏
`=1
kR1 (u`; τ)
∏
1≤j<k≤N
kR2 (uk, uj ; τ), (2.24)
holds with the following factors,
k
AN−1
0 (τ) = i−(N−1)(3N+1−(−1)N )/2q(τ)−(N−1)(3N−2)/8q0(τ)−(N−1)(N−2)/2,
kBN0 (τ) = 2q(τ)−N(N−1)/4q0(τ)−N(N−1),
k
B∨N
0 (τ) = 2q(τ)−N(N−1)/4q0(τ)−(N−1)
2
q0(2τ)−(N−1),
kCN0 (τ) = i−Nq(τ)−N
2/4q0(τ)−N(N−1),
k
C∨N
0 (τ) = i−Nq(τ)−N(2N−1)/8q0(τ)−(N−1)
2
q0(τ/2)−(N−1),
kBCN0 (τ) = i−Nq(τ)−N(N+1)/4q0(τ)−N(N−1)q0(2τ)−N ,
kDN0 (τ) = 4q(τ)−N(N−1)/4q0(τ)−N(N−2), (2.25)
k
AN−1
sym (u; τ) = ϑ1
N∑
j=1
uj − κN
2πr
; τ
,
kRsym(u; τ) = 1 for R = BN , B
∨
N , CN , C
∨
N , BCN , DN ,
k
AN−1
1 (u; τ) = kDN1 (u; τ) = 1,
kBN1 (u; τ) = ϑ1
( u
2πr
; τ
)
, k
B∨N
1 (u; τ) = ϑ1
( u
πr
; 2τ
)
,
16 M. Katori
kCN1 (u; τ) = ϑ1
( u
πr
; τ
)
, k
C∨N
1 (u; τ) = ϑ1
( u
2πr
;
τ
2
)
,
kBCN1 (u; τ) = ϑ1
( u
2πr
; τ
)
ϑ0
( u
πr
; 2τ
)
,
k
AN−1
2 (u, v; τ) = ϑ1
(
u− v
2πr
; τ
)
,
kR2 (u, v; τ) = ϑ1
(
u− v
2πr
; τ
)
ϑ1
(
u+ v
2πr
; τ
)
for R = BN , B
∨
N , CN , C
∨
N , BCN , DN . (2.26)
2.5 Determinantal martingale-functions
Since fRj (z; τ), j ∈ Z given by (2.18) are entire and non-constant functions of z ∈ C,
f̂Rj (t, x; τ) = Ẽ[fRj (x+ iB̃(t); τ)], t ∈ [0,∞), j ∈ Z, (2.27)
give the single-variable martingale-functions such that, if f̂Rj (t, B(t); τ), t ≥ 0, j ∈ Z are finite,
then they give martingales with respect to the filtration Ft of one-dimensional Brownian mo-
tion B(t), t ≥ 0,
E[f̂Rj (t, B(t); τ)|Fs] = f̂Rj (s,B(s); τ), 0 ≤ s < t <∞ a.s. j ∈ Z.
We have found that f̂Rj (t, x; τ)’s are expressed using fRj (t, x; ·) by shifting
τ → τ − iNRt
2πr2
with (1.4) and multiplying time-dependent factors. With 0 < t∗ <∞, we put (1.3).
Lemma 2.5. For fRj , j ∈ Z given by (2.18), we obtain the following infinite series of linearly
independent martingale-functions
f̂Rj
(
t, x; τR(0)
)
= eJ
R(j)2t/2r2fRj
(
x; τR(t)
)
, t ∈ [0, t∗), j ∈ Z, (2.28)
where JR(j), j ∈ Z are given by (2.19).
Proof. By (2.18), it is enough to calculate
Ẽ
[
e±iJ
R(j)(x+iB̃(t))/rϑ1
(
JR(j)τ ± N
R(x+ iB̃(t))
2πr
+ α;NRt
)]
(2.29)
with a constant α. By the definition (A.1) of ϑ1, this is equal to
e±iJ
R(j)xi
∑
n∈Z
(−1)neN
Rτπi(n−1/2)2+(JR(j)τ±NRx/2πr+α)πi(2n−1)
× Ẽ
[
e∓{J
R(j)+NR(2n−1)/2}B̃(t)/r
]
.
Since
Ẽ
[
e∓{J
R(j)+NR(2n−1)/2}B̃(t)/r
]
= e{J
R(j)+NR(2n−1)/2}2t/2r2
= eJ
R(j)2t/2r2eN
R(−iNRt/2πr2)πi(n−1/2)2+JR(j)(−iNRt/2πr2)πi(2n−1),
we can see that (2.29) is equal to
eJ
R(j)2t/2r2e±iJ
R(j)x/rϑ1
(
JR(j)
(
τ − iNRt
2πr2
)
± N
Rx
2πr
+ α;NR
(
τ − iNRt
2πr2
))
.
Therefore, if we set τ = iNRt∗/2πr
2, (2.28) are obtained. �
Elliptic Determinantal Processes and Elliptic Dyson Models 17
We call the multivariate function Du(t,x) of x ∈ RN defined by (2.11) the determinantal
martingale-function, since if it is finite, it gives the determinantal martingale when we put the
N -dimensional Brownian motion B(t), t ≥ 0 into x. The following is proved.
Proposition 2.6. For fRj , j ∈ Z given by (2.18), the determinantal martingale-functions are
factorized as follows
DR
u (t,x) =
cR0
(
τR(t)
)
cR0
(
τR(0)
) kRsym(x; τR(t)
)
kRsym
(
u; τR(0)
) N∏
`=1
kR1
(
x`; τ
R(t)
)
kR1
(
u`; τR(0)
) ∏
1≤j<k≤N
kR2
(
xk, xj ; τ
R(t)
)
kR2
(
uk, uj ; τR(0)
) , (2.30)
where
c
AN−1
0 (τ) = η(τ)−(N−1)(N−2)/2 (2.31)
and for R = BN , B
∨
N , CN , C
∨
N , BCN , DN , cR0 (τ) are given by (1.8). For R = BN , B
∨
N , CN , C
∨
N ,
BCN , DN , (2.30) are written as (1.5)–(1.7) with (1.8) and (1.9).
Proof. By the second equality of (2.12) and Lemma 2.5,
DR
u (t,x) =
det
1≤j,k≤N
[
eJ
R(j)2t/2r2fRj (xk; τ
R(t))
]
det
1≤j,k≤N
[
fRj (uk; τR(0))
] = e
(t/2r2)
N∑
j=1
JR(j)2 det
1≤j,k≤N
[
fRj (xk; τ
R(t))
]
det
1≤j,k≤N
[
fRj (uk; τR(0))
] ,
where the multi-linearity of determinant was used in the second equality. Then applying
Lemma 2.4, we obtain (2.30) with
cR0 (τR(t))
cR0 (τR(0))
= e
(t/2r2)
N∑
j=1
JR(j)2 kR0 (τR(t))
kR0 (τR(0))
.
By (2.25) and the summation formulas
N∑
j=1
j = N(N + 1)/2,
N∑
j=1
j2 = N(N + 1)(2N + 1)/6,
we obtain the expressions (1.8) using the Dedekind modular function (1.10), since η(τ) =
eτπi/12q0(τ) with (2.23). Thus we can derive readily the expressions (1.5)–(1.7) with (1.8)
and (1.9) from (2.30). �
3 Proof of Theorem 1.1
Assume t ∈ [0, t∗). By (A.3), we see that if 0 ≤ x ≤ 2πr, 0 ≤ kR1
(
x; τR(t)
)
< ∞ for R =
BN , C
∨
N , BCN ; if 0 ≤ x ≤ πr, 0 ≤ kR1
(
x; τR(t)
)
< ∞ for R = B∨N , CN ; and kDN1 ≡ 1. We also
see that if
|x1| < x2 < · · · < xN−1 < xN ∧ (2πr − xN ), (3.1)
then 0 <
∏
1≤j<k≤N
kR2
(
xk, xj ; τ
R(t)
)
< ∞. The inequality |x1| < x2 means that x2 is greater
than both of x1 and its reflection at 0, and xN−1 < xN ∧ (2πr−xN ) means that xN−1 is smaller
than both of xN and its reflection at πr. The above observation implies that if u ∈W(0,πr)
N , then
0 ≤
N∏
`=1
kR1
(
B`(t); τ
R(t)
) ∏
1≤j<k≤N
kR2
(
Bk(t), Bj(t); τ
R(t)
)
<∞,
for 0 ≤ t ≤ Tcollision, a.s. in P
[0,πr]
u .
18 M. Katori
Due to the Karlin–McGregor–Lindström–Gessel–Viennot (KMLGV) formula [9, 13, 29] for
non-intersecting paths, the transition probability density of the N particle system in [0, πr]
governed by the probability law PRu defined by (1.15) is given by
pR(t,y|s,x) =
cR0
(
τR(t)
)
cR0
(
τR(s)
) N∏
`=1
kR1
(
y`; τ
R(t)
)
kR1
(
x`; τR(s)
) ∏
1≤j<k≤N
kR2
(
yk, yj ; τ
R(t)
)
kR2
(
xk, xj ; τR(s)
)
× det
1≤n,n′≤N
[
p[0,πr](t− s, yn|xn′)
]
, 0 ≤ s < t < t∗, x,y ∈W(0,πr)
N . (3.2)
Since we consider the measure-valued stochastic process (1.16), the configuration is unlabeled
and hence all the observables at each time should be symmetric functions of particle positions.
For an arbitrary number M ∈ N and an arbitrary strictly increasing series of times 0 ≡ t0 < t1 <
· · · < tM < t∗, let gtm(x), m = 1, 2, . . . ,M , be symmetric functions of x ∈W(0,πr)
N . Then (1.15)
and (3.2) give
ERu
[
M∏
m=1
gtm(X(tm))
]
= E
[0,πr]
u
[
M∏
m=1
gtm(B(tm))1(Tcollision > tM )DR
u (t,B(tM ))
]
=
∫
W(0,πr)
N
dx(1) · · ·
∫
W(0,πr)
N
dx(M)
M∏
m=1
det
1≤n,n′≤N
[
p[0,πr]
(
tm − tm−1, x(m)
n |x
(m−1)
n′
)]
gtm
(
x(m)
)
×
cR0
(
τR(tM )
)
cR0
(
τR(0)
) N∏
`=1
kR1
(
x
(M)
` ; τR(tM )
)
kR1
(
u`; τR(0)
) ∏
1≤j<k≤N
kR2
(
x
(M)
k , x
(M)
j ; τR(tM )
)
kR2
(
uk, uj ; τR(0)
) . (3.3)
By definition of determinant, the above is equal to∑
σ1∈SN
· · ·
∑
σM∈SN
∫
W(0,πr)
N
dx(1) · · ·
∫
W(0,πr)
N
dx(M)
×
M∏
m=1
{
sgn(σm)
N∏
n=1
p[0,πr]
(
tm − tm−1, x(m)
σ(m)(n)
|x(m−1)
σ(m−1)(n)
)
gtm
(
x(m)
)
×
cR0
(
τR(tm)
)
cR0
(
τR(tm−1)
) N∏
`=1
kR1
(
x
(m)
` ; τR(tm)
)
kR1
(
x
(m−1)
` ; τR(tm−1)
)
×
∣∣∣∣∣∣
∏
1≤j<k≤N
kR2
(
x
(m)
σ(m)(k)
, x
(m)
σ(m)(j)
; τR(tm)
)
kR2
(
x
(m−1)
σ(m−1)(k)
, x
(m−1)
σ(m−1)(j)
; τR(tm−1)
)
∣∣∣∣∣∣
,
where SN denotes the collection of all permutations of N indices {1, 2, . . . , N}, σ(m) ≡ σ1 ◦ σ2 ◦
· · · ◦ σm, m ≥ 1, σ(0) = id, and x
(0)
j = uj , j = 1, . . . , N . Here we used the fact that gtm(x)’s and
N∏
`=1
kR1 (x`; ·) are symmetric functions of x. By the definition of W(0,πr)
N and the fact that
∏
1≤j<k≤N
kR2 (xσ(k), xσ(j); ·) = sgn(σ)
∏
1≤j<k≤N
kR2 (xk, xj ; ·), σ ∈ SN ,
this is equal to∫
[0,πr]N
dx(1) · · ·
∫
[0,πr]N
dx(M)
M∏
m=1
{
N∏
n=1
p[0,πr]
(
tm − tm−1, x(m)
n |x(m−1)n
)
gtm
(
x(m)
)}
Elliptic Determinantal Processes and Elliptic Dyson Models 19
×
cR0
(
τR(tM )
)
cR0
(
τR(0)
) N∏
`=1
kR1
(
x
(M)
` ; τR(tM )
)
kR1
(
u`; τR(0)
) ∏
1≤j<k≤N
kR2
(
x
(M)
k , x
(M)
j ; τR(tM )
)
kR2
(
uk, uj ; τR(0)
) .
Therefore, we obtain the equality
ERu
[
M∏
m=1
gtm(X(tm))
]
= E
[0,πr]
u
[
M∏
m=1
gtm(B(tm))DR
u (tM ,B(tM ))
]
, (3.4)
that is, the noncolliding condition, Tcollision > tM , can be omitted in (3.3). The expression (3.4)
is interpreted as the expectation of the product of symmetric functions
M∏
m=1
gtm(·) with respect
to the signed measure
P̂Ru
∣∣
Ft = DR
u (t,B(t))P
[0,πr]
u
∣∣
Ft , t ∈ [0, t∗), (3.5)
which is a modification of (2.13) obtained by replacing Pu by P
[0,πr]
u and Du by DR
u . Then
we can apply Proposition 2.3, where we replace pBM by p[0,πr] and use the proper martingale
functions given by (1.18) with (1.19)–(1.21). Thus the proof is completed.
4 Proof of Theorem 1.2
Theorem 1.2 is concluded from the following key lemma.
Lemma 4.1. The transition probability densities pBN , pCN and pDN given by (3.2) for R =
BN , CN , and DN solve the following backward Kolmogorov equations
−∂p
BN (t,y|s,x)
∂s
=
1
2
N∑
j=1
∂2pBN (t,y|s,x)
∂x2j
+
N∑
j=1
A2πr
2N−1(t∗ − s, xj)
∂pBN (t,y|s,x)
∂xj
+
∑
1≤j,k≤N,
j 6=k
(
A2πr
2N−1(t∗ − s, xj − xk) +A2πr
2N−1(t∗ − s, xj + xk)
)∂pBN (t,y|s,x)
∂xj
, (4.1)
−∂p
CN (t,y|s,x)
∂s
=
1
2
N∑
j=1
∂2pCN (t,y|s,x)
∂x2j
+ 2
N∑
j=1
A2πr
2(N+1)(t∗ − s, 2xj)
∂pCN (t,y|s,x)
∂xj
+
∑
1≤j,k≤N,
j 6=k
(
A2πr
2(N+1)(t∗ − s, xj − xk) +A2πr
2(N+1)(t∗ − s, xj + xk)
)∂pCN (t,y|s,x)
∂xj
, (4.2)
−∂p
DN (t,y|s,x)
∂s
=
1
2
N∑
j=1
∂2pDN (t,y|s,x)
∂x2j
+
∑
1≤j,k≤N,
j 6=k
(
A2πr
2(N−1)(t∗ − s, xj − xk) +A2πr
2(N−1)(t∗ − s, xj + xk)
)∂pDN (t,y|s,x)
∂xj
, (4.3)
for x,y ∈W(0,πr)
N , 0 ≤ s < t < t∗, under the conditions
lim
s↑t
pBN (t,y|s,x) = lim
s↑t
pCN (t,y|s,x) = lim
s↑t
pDN (t,y|s,x) =
N∏
j=1
δ(xj − yj). (4.4)
20 M. Katori
Proof. For given t ∈ (0, t∗) and y ∈W(0,πr)
N , put
u(s,x) = w(s,x)q[0,πr](t− s,y|x), 0 < s < t, x ∈W(0,πr)
N (4.5)
with the KMLGV determinant
q[0,πr](t− s,y|x) = det
1≤j,k≤N
[
p[0,πr](t− s, yj |xk)
]
, (4.6)
where w(s,x) is a C1,2-function which will be specified later, and p[0,πr] = p
[0,πr]
aa given by (1.11)
for R = CN , p[0,πr] = p
[0,πr]
rr given by (1.12) for R = DN . Since p[0,πr] is the transition probability
density of a boundary-conditioned Brownian motion, it solves the diffusion equation
−∂p
[0,πr](t− s, y|x)
∂s
=
1
2
∂2p[0,πr](t− s, y|x)
∂x2
with lim
s↑t
p[0,πr](t − s, y|x) = δ(x − y). The KMLGV determinant (4.6) is the summation of
products of p[0,πr]’s, and hence
−∂q(t− s,y|x)
∂s
=
1
2
N∑
j=1
∂2q(t− s,y|x)
∂x2j
is satisfied. It is easy to verify that
lim
s↑t
q(t− s,y|x) =
N∏
j=1
δ(xj − yj). (4.7)
Therefore, the following equation holds ∂
∂s
+
1
2
N∑
j=1
∂2
∂x2j
u(s,x)
= q[0,πr](t− s,y|x)
∂
∂s
+
1
2
N∑
j=1
∂2
∂x2j
w(s,x) +
N∑
j=1
∂w(s,x)
∂xj
∂q[0,πr](t− s,y|x)
∂xj
.
Since
N∑
j=1
∂w(s,x)
∂xj
∂u(s,x)
∂xj
= q[0,πr](t− s,y|x)
N∑
j=1
(
∂w(s,x)
∂xj
)2
+ w(s,x)
N∑
j=1
∂w(s,x)
∂xj
∂q[0,πr](t− s,y|x)
∂xj
,
the above equation is written as ∂
∂s
+
1
2
N∑
j=1
∂2
∂x2j
− 1
w(s,x)
N∑
j=1
∂w(s,x)
∂xj
∂
∂xj
u(s,x)
= q[0,πr](t− s,y|x)
∂
∂s
+
1
2
N∑
j=1
∂2
∂x2j
w(s,x)− 1
w(s,x)
N∑
j=1
(
∂w(s,x)
∂xj
)2
. (4.8)
Elliptic Determinantal Processes and Elliptic Dyson Models 21
Now we put
w(s,x) = wR(s,x) (4.9)
≡
gR(s)
N∏
`=1
ϑ1
(
cR1 x`
2πr
; τR(s)
) ∏
1≤j<k≤N
ϑ1
(
xk − xj
2πr
; τR(s)
)
ϑ1
(
xk + xj
2πr
; τR(s)
)
−1
for R = BN , CN , and
w(s,x) = wDN (s,x)
≡
gDN (s)
∏
1≤j<k≤N
ϑ1
(
xk − xj
2πr
; τDN (s)
)
ϑ1
(
xk + xj
2πr
; τDN (s)
)
−1
(4.10)
for R = DN , where gR(s), R = BN , CN , DN , are C1-functions of s, which will be determined
later, and cBN1 = 1, cCN1 = 2. We define αR = 1 for R = BN and CN , and αR = 0 for R = DN .
We put cDN1 = 1 for convention. By (1.1) with (A.7), we see that
− 1
wR(s,x)
∂wR(s,x)
∂xj
= αRcR1 A
2πr
NR
(
t∗ − s, cR1 xj
)
+
∑
1≤k≤N,
k 6=j
(
A2πr
NR(t∗ − s, xj − xk) +A2πr
NR(t∗ − s, xj + xk)
)
, (4.11)
j = 1, 2, . . . , N , and if we multiply (4.11) by ∂pR(t,y|s,x)/∂xj and take summation over j =
1, 2, . . . , N for R = BN , CN , DN , they give the drift terms in (4.1)–(4.3). Therefore, if we can
choose gR(s) so that the equation ∂
∂s
+
1
2
N∑
j=1
∂2
∂x2j
wR(s,x)− 1
wR(s,x)
N∑
j=1
(
∂wR(s,x)
∂xj
)2
= 0 (4.12)
holds, then the r.h.s. of (4.8) vanishes and we can conclude that u(s,x) given in the form (4.5)
with (4.9) and (4.10) solve the Kolmogorov equations (4.1)–(4.3).
From (4.9) and (4.10) with (1.3), we find that
1
wR(s,x)
∂wR(s,x)
∂s
= − 1
gR(s)
dgR(s)
ds
+ αR
iNR
2πr2
N∑
j=1
ϑ̇1
(
cR1 xj/2πr; τ
R(s)
)
ϑ1
(
cR1 xj/2πr; τ
R(s)
)
+
1
2
iNR
2πr2
∑
1≤j,k≤N,
j 6=k
{
ϑ̇1
(
(xj − xk)/2πr; τR(s)
)
ϑ1
(
(xj − xk)/2πr; τR(s)
) +
ϑ̇1
(
(xj + xk)/2πr; τ
R(s)
)
ϑ1
(
(xj + xk)/2πr; τR(s)
)} ,
where ϑ̇1(v; τ) = ∂ϑ1(v; τ)/∂τ . If we use the equation (A.2), then the above is written as
1
wR(s,x)
∂wR(s,x)
∂s
= − 1
gR(s)
dgR(s)
ds
+ αR
NR
8π2r2
N∑
j=1
ϑ′′1
(
cR1 xj/2πr; τ
R(s)
)
ϑ1
(
cR1 xj/2πr; τ
R(s)
)
+
1
2
NR
8π2r2
∑
1≤j,k≤N,
j 6=k
{
ϑ′′1
(
(xj − xk)/2πr; τR(s)
)
ϑ1
(
(xj − xk)/2πr; τR(s)
) +
ϑ′′1
(
(xj + xk)/2πr; τ
R(s)
)
ϑ1
(
(xj + xk)/2πr; τR(s)
)} ,
22 M. Katori
where ϑ′′1(v; τ) = ∂2ϑ1(v; τ)/∂v2. From (4.11), we find that
∂2wR(s,x)
∂x2j
= −wR(s,x)
∂
∂xj
{
αRcR1 A
2πr
NR
(
t∗ − s, cR1 xj
)
+
∑
1≤k≤N,
k 6=j
(
A2πr
NR(t∗ − s, xj − xk) +A2πr
NR(t∗ − s, xj + xk)
)}
+
1
wR(s,x)
(
∂wR(s,x)
∂xj
)2
,
and hence
1
2
N∑
j=1
∂2wR(s,x)
∂x2j
− 1
wR(s,x)
N∑
j=1
(
∂wR(s,x)
∂xj
)2
= −1
2
wR(s,x)S
with
S =
N∑
j=1
[
∂
∂xj
{
αRcR1 A
2πr
NR
(
t∗ − s, cR1 xj
)
+
∑
1≤k≤N,
k 6=j
(
A2πr
NR(t∗ − s, xj − xk) +A2πr
NR(t∗ − s, xj + xk)
)}
+
{
αRcR1 A
2πr
NR
(
t∗ − s, cR1 xj
)
+
∑
1≤k≤N,
k 6=j
(
A2πr
NR(t∗ − s, xj − xk) +A2πr
NR(t∗ − s, xj + xk)
)}2
]
.
By definition (1.1),
∂A2πr
NR
(
t∗ − s, cR1 x
)
∂x
=
cR1
(2πr)2
ϑ′′1
(
cR1 x/2πr; τ
R(s)
)
ϑ1
(
cR1 x/2πr; τ
R(s)
) − cR1 (A2πr
NR
(
t∗ − s, cR1 x
))2
. (4.13)
Hence
S = αR
(
cR1
)2
(2πr)2
N∑
j=1
ϑ′′1
(
cR1 xj/2πr; τ
R(s)
)
ϑ1
(
cR1 xj/2πr; τ
R(s)
)
+
1
(2πr)2
∑
1≤j,k≤N,
j 6=k
{
ϑ′′1
(
(xj − xk)/2πr; τR(s)
)
ϑ1
(
(xj − xk)/2πr; τR(s)
) +
ϑ′′1
(
(xj + xk)/2πr; τ
R(s)
)
ϑ1
(
(xj + xk)/2πr; τR(s)
)}
+ 2αRcR1
∑
1≤j,k≤N,
j 6=k
A2πr
NR
(
t∗ − s, cR1 xj
)(
A2πr
NR(t∗ − s, xj − xk) +A2πr
NR(t∗ − s, xj + xk)
)
+
∑
1≤j,k,`≤N,
j 6=k 6=` 6=j
(
A2πr
NR(t∗ − s, xj − xk) +A2πr
NR(t∗ − s, xj + xk)
)
×
(
A2πr
NR(t∗ − s, xj − x`) +A2πr
NR(t∗ − s, xj + x`)
)
,
where we have used the fact∑
1≤j,k≤N,
j 6=k
A2πr
NR(t∗ − s, xj − xk)A2πr
NR(t∗ − s, xj + xk) = 0
Elliptic Determinantal Processes and Elliptic Dyson Models 23
concluded from (A.7). Then (4.12) holds, if
− 1
gR(s)
dgR(s)
ds
+ αR
NR −
(
cR1
)2
8π2r2
N∑
j=1
ϑ′′1
(
cR1 xj/2πr; τ
R(s)
)
ϑ1
(
cR1 xj/2πr; τ
R(s)
)
+
NR − 2
16π2r2
∑
1≤j,k≤N,
j 6=k
{
ϑ′′1
(
(xj − xk)/2πr; τR(s)
)
ϑ1
(
(xj − xk)/2πr; τR(s)
) +
ϑ′′1
(
(xj + xk)/2πr; τ
R(s)
)
ϑ1
(
(xj + xk)/2πr; τR(s)
)}
− αRcR1
∑
1≤j,k≤N,
j 6=k
A2πr
NR
(
t∗ − s, cR1 xj
)(
A2πr
NR(t∗ − s, xj − xk) +A2πr
NR(t∗ − s, xj + xk)
)
− 1
2
∑
1≤j,k,`≤N,
j 6=k 6= 6̀=j
(
A2πr
NR(t∗ − s, xj − xk) +A2πr
NR(t∗ − s, xj + xk)
)
×
(
A2πr
NR(t∗ − s, xj − x`) +A2πr
NR(t∗ − s, xj + x`)
)
= 0. (4.14)
Now we use the following expressions for A2πr
NR and its spatial derivative [16, 17]
A2πr
NR
(
t∗ − s, cR1 x
)
= ζNR
(
t∗ − s, cR1 x
)
− cR1
η1NR(t∗ − s)x
πr
,
∂A2πr
NR
(
t∗ − s, cR1 x
)
∂x
= −cR1 ℘NR
(
t∗ − s, cR1 x
)
− cR1
η1NR(t∗ − s)
πr
. (4.15)
Here η1NR(t∗ − s) is given by (A.10) and we put
ζNR(t∗ − s, x) = ζ(x|2ω1, 2ω3)
∣∣
ω1=πr,ω3=iNR(t∗−s)/2r,
℘NR(t∗ − s, x) = ℘(x|2ω1, 2ω3)
∣∣
ω1=πr,ω3=iNR(t∗−s)/2r, (4.16)
where the Weierstrass ℘ function and zeta function ζ are defined by (A.14) in Appendix A.5.
Applying (4.15) to (4.13) gives
1
(2πr)2
ϑ′′1
(
cR1 x/2πr; τ
R(s)
)
ϑ1
(
cR1 x/2πr; τ
R(s)
) =
(
ζNR
(
t∗ − s, cR1 x
)
− cR1
η1NR(t∗ − s)x
πr
)2
− ℘NR
(
t∗ − s, cR1 x
)
−
η1NR(t∗ − s)
πr
. (4.17)
Using (4.15) and (4.17), the l.h.s. of (4.14) can be written using ζNR , ℘NR and η1NR . Moreover,
from the functional equation (A.15) given in Appendix A.5, we can derive the following equalities∑
1≤j,k,`≤N,
j 6=k 6= 6̀=j
(ζNR(t∗ − s, xj − xk) + ζNR(t∗ − s, xj + xk))
× (ζNR(t∗ − s, xj − x`) + ζNR(t∗ − s, xj + x`))
= (N − 2)
∑
1≤j,k≤N,
j 6=k
(
ζNR(t∗ − s, xj − xk)2 + ζNR(t∗ − s, xj + xk)
2
)
− (N − 2)
∑
1≤j,k≤N,
j 6=k
(℘NR(t∗ − s, xj − xk) + ℘NR(t∗ − s, xj + xk)), (4.18)
∑
1≤j,k≤N,
j 6=k
ζNR(t∗ − s, xj)(ζNR(t∗ − s, xj − xk) + ζNR(t∗ − s, xj + xk))
24 M. Katori
=
1
4
∑
1≤j,k≤N,
j 6=k
(
ζNR(t∗ − s, xj − xk)2 + ζNR(t∗ − s, xj + xk)
2
)
− 1
4
∑
1≤j,k≤N,
j 6=k
(℘NR(t∗ − s, xj − xk) + ℘NR(t∗ − s, xj + xk))
+ (N − 1)
N∑
j=1
ζNR(t∗ − s, xj)2 − (N − 1)
N∑
j=1
℘NR(t∗ − s, xj), (4.19)
and ∑
1≤j,k≤N,
j 6=k
ζNR(t∗ − s, 2xj)(ζNR(t∗ − s, xj − xk) + ζNR(t∗ − s, xj + xk))
=
1
2
∑
1≤j,k≤N,
j 6=k
(
ζNR(t∗ − s, xj − xk)2 + ζNR(t∗ − s, xj + xk)
2
)
− 1
2
∑
1≤j,k≤N,
j 6=k
(℘NR(t∗ − s, xj − xk) + ℘NR(t∗ − s, xj + xk))
+
N − 1
2
N∑
j=1
ζNR(t∗ − s, 2xj)2 −
N − 1
2
N∑
j=1
℘NR(t∗ − s, 2xj). (4.20)
Since ηNR(t∗ − s,−x) = −ηNR(t∗ − s, x), we have∑
1≤j,k,`≤N,
j 6=k 6= 6̀=j
(ζNR(t∗ − s, xj − xk) + ζNR(t∗ − s, xj + xk)){(xj − xk) + (xj + x`)}
= (N − 2)
∑
1≤j,k≤N,
j 6=k
{ζNR(t∗ − s, xj − xk)(xj − xk) + ζNR(t∗ − s, xj + xk)(xj + xk)}, (4.21)
and ∑
1≤j,k,`≤N,
j 6=k 6= 6̀=j
{(xj − xk)(xj − x`) + (xj + xk)(xj + x`)}
= (N − 2)
∑
1≤j,k≤N,
j 6=k
{
(xj − xk)2 + (xj + xk)
2
}
= 4(N − 1)(N − 2)
N∑
j=1
x2j . (4.22)
Using the above formulas and the values of NR given by (1.4), we can show that (4.14) is
reduced to the following simple equations
d log gBN (s)
ds
= −N(N − 1)(2N − 1)
η12N−1(t∗ − s)
2πr
,
d log gCN (s)
ds
= −N(N2 − 1)
η12(N+1)(t∗ − s)
πr
,
d log gDN (s)
ds
= −N(N − 1)(N − 2)
η12(N−1)(t∗ − s)
πr
. (4.23)
Elliptic Determinantal Processes and Elliptic Dyson Models 25
They give the conditions for gR(s) so that (4.12) holds. Since (A.11) gives for R = BN , CN , DN
d log η
(
τBN (s)
)
ds
= (2N − 1)
η12N−1(t∗ − s)
2πr
,
d log η
(
τCN (s)
)
ds
= (N + 1)
η12(N+1)(t∗ − s)
πr
,
d log η
(
τDN (s)
)
ds
= (N − 1)
η12(N−1)(t∗ − s)
πr
,
we find that
gBN (s) = cBN η
(
τBN (s)
)−N(N−1)
,
gCN (s) = cCN η
(
τCN (s)
)−N(N−1)
,
gDN (s) = cDN η
(
τDN (s)
)−N(N−2)
,
with constants cR, R = BN , CN , DN . By (4.7), the conditions (4.4) determine the constants.
In conclusion, under the condition (4.4),
pBN (t,y|s,x) =
(
η
(
τBN (t)
)
η
(
τBN (s)
))−N(N−1) N∏
`=1
ϑ1
(
y`/2πr; τ
BN (t)
)
ϑ1
(
x`/2πr; τBN (s)
)
×
∏
1≤j<k≤N
ϑ1
(
(yk − yj)/2πr; τBN (t)
)
ϑ1
(
(xk − xj)/2πr; τBN (s)
) ϑ1((yk + yj)/2πr; τ
BN (t)
)
ϑ1
(
(xk + xj)/2πr; τBN (s)
)q[0,πr]ar (t− s,y|x)
solves (4.1),
pCN (t,y|s,x) =
(
η
(
τCN (t)
)
η
(
τCN (s)
))−N(N−1) N∏
`=1
ϑ1
(
y`/πr; τ
CN (t)
)
ϑ1
(
x`/πr; τCN (s)
)
×
∏
1≤j<k≤N
ϑ1
(
(yk − yj)/2πr; τCN (t)
)
ϑ1
(
(xk − xj)/2πr; τCN (s)
) ϑ1((yk + yj)/2πr; τ
CN (t)
)
ϑ1
(
(xk + xj)/2πr; τCN (s)
)q[0,πr]aa (t− s,y|x)
solves (4.2), and
pDN (t,y|s,x) =
(
η
(
τDN (t)
)
η
(
τDN (s)
))−N(N−2)
×
∏
1≤j<k≤N
ϑ1
(
(yk − yj)/2πr; τDN (t)
)
ϑ1
(
(xk − xj)/2πr; τDN (s)
) ϑ1((yk + yj)/2πr; τ
DN (t)
)
ϑ1
(
(xk + xj)/2πr; τDN (s)
)q[0,πr]rr (t− s,y|x)
solves (4.3). The proof is thus completed. �
Now we prove Theorem 1.2.
Proof. It is obvious that the backward Kolmogorov equations (4.1)–(4.3) correspond to the sys-
tems of SDEs (1.22)–(1.24), respectively [15, 33]. Due to the behavior (A.8) of A2πr
N , we can show
that [15, 33] particles in [0, πr] following (1.22) do not arrive at 0, and those following (1.23)
do not arrive at 0 nor πr with probability one, and thus we do not need to impose any boundary
condition at these endpoints of [0, πr] for these systems of SDEs. Hence the proof is com-
pleted. �
26 M. Katori
5 Relaxation to equilibrium processes
in trigonometric Dyson models of types C and D
As a corollary of Theorems 1.1 and 1.2, we obtain the following trigonometric determinantal
processes by taking the temporally homogeneous limit t∗ →∞.
Corollary 5.1. Assume that u ∈ W(0,πr). Then the trigonometric Dyson models of types C
and D given by (1.27) and (1.28), respectively, are determinantal with the spatio-temporal cor-
relation kernels
ǨCN (s, x; t, y) =
N∑
j=1
p[0,πr]aa (s, x|uj)M̌CN
u,uj (t, y)− 1(s > t)p[0,πr]aa (s− t, x|y),
ǨDN (s, x; t, y) =
N∑
j=1
p[0,πr]rr (s, x|uj)M̌DN
u,uj (t, y)− 1(s > t)p[0,πr]rr (s− t, x|y), (5.1)
(s, x), (t, y) ∈ [0,∞)× [0, πr], where
M̌R
u,uj (t, x) = Ẽ
[
Φ̌R
u,uj
(
x+ iB̃(t)
)]
, R = CN , DN
with
Φ̌CN
u,uj (z) =
sin(z/r)
sin(uj/r)
∏
1≤`≤N,
` 6=j
sin((z − u`)/2r)
sin((uj − u`)/2r)
sin((z + u`)/2r)
sin((uj + u`)/2r)
,
Φ̌DN
u,uj (z) =
∏
1≤`≤N,
6̀=j
sin((z − u`)/2r)
sin((uj − u`)/2r)
sin((z + u`)/2r)
sin((uj + u`)/2r)
, j = 1, 2, . . . , N. (5.2)
We find that in the limit t∗ →∞, the equality (2.24) with R = CN and DN gives
det
1≤j,k≤N
[
f̌CNj (uk)
]
= (−1)N(N−1)/22N(N−1)
N∏
`=1
sin
(u`
r
)
×
∏
1≤j<k≤N
sin
(
uk − uj
2r
)
sin
(
uk + uj
2r
)
,
det
1≤j,k≤N
[
f̌DNj (uk)
]
= (−1)N(N−1)/22(N−1)
2
∏
1≤j<k≤N
sin
(
uk − uj
2r
)
sin
(
uk + uj
2r
)
(5.3)
with
f̌CNj (z) = sin
(
jz
r
)
, f̌DNj (z) = cos
(
(j − 1)z
r
)
, j ∈ Z. (5.4)
By Lemma 2.1, we see that (5.2) can be expanded as
Φ̌R
u,uj (z) =
N∑
k=1
φ̌Ru,uj (k)f̌Rk (z), R = CN , DN ,
and
N∑
`=1
f̌Rj (u`)φ̌
R
u,u`
(k) = δjk, j, k ∈ {1, 2, . . . , N}. (5.5)
Elliptic Determinantal Processes and Elliptic Dyson Models 27
We note that the transition probability densities (1.11) and (1.12) are written as (A.12)
and (A.13), respectively. Hence we have
p[0,πr]aa (t, y|x) =
1
πr
∑
n∈Z
e−n
2t/2r2 f̌CNn (y)f̌CNn (x) =
2
πr
∞∑
`=1
e−`
2t/2r2 f̌CN` (y)f̌CN` (x),
p[0,πr]rr (t, y|x) =
1
πr
(
f̌DN1 (y)f̌DN1 (x) + 2
∞∑
`=2
e−(`−1)
2t/2r2 f̌DN` (y)f̌DN` (x)
)
. (5.6)
Now we show relaxation processes to equilibria, which are typical non-equilibrium phenomena.
Proposition 5.2. For any initial configuration u ∈ W(0,πr), the trigonometric Dyson models
of types C and D exhibit relaxation to the equilibrium processes. The equilibrium processes are
determinantal with the correlation kernels
ǨCN
eq (t− s, x, y)
=
1
πr
∑
n : |n|≤N
en
2(t−s)/2r2 sin
(nx
r
)
sin
(ny
r
)
, if t > s,
1
2πr
[
sin{(2N + 1)(y − x)/2r}
sin{(y − x)/2r}
− sin{(2N + 1)(y + x)/2r}
sin{(y + x)/2r}
]
, if t = s,
− 1
πr
∑
n : |n|≥N+1
en
2(t−s)/2r2 sin
(nx
r
)
sin
(ny
r
)
, if t < s,
(5.7)
and
ǨDN
eq (t− s, x, y)
=
1
πr
∑
n : |n|≤N−1
en
2(t−s)/2r2 cos
(nx
r
)
cos
(ny
r
)
, if t > s,
1
2πr
[
sin{(2N − 1)(y − x)/2r}
sin{(y − x)/2r}
+
sin{(2N − 1)(y + x)/2r}
sin{(y + x)/2r}
]
, if t = s,
− 1
πr
∑
n : |n|≥N
en
2(t−s)/2r2 cos
(nx
r
)
cos
(ny
r
)
, if t < s,
(5.8)
respectively.
Proof. Here we prove the convergence of correlation kernels. It implies the convergence of the
generating functions of spatio-temporal correlation functions (i.e., the Laplace transformations
of multi-time joint probability densities). Hence the convergence of process is concluded in the
sense of finite-dimensional distributions [22]. Assume u ∈W(0,πr) and let
gCNu (s, x; t, y) =
N∑
j=1
p[0,πr]aa (s, x|uj)M̌CN
u,uj (t, y),
gDNu (s, x; t, y) =
N∑
j=1
p[0,πr]rr (s, x|uj)M̌DN
u,uj (t, y), (s, x), (t, y) ∈ [0,∞)× [0, πr].
It is easy to verify that
Ẽ
[
f̌CNk
(
x+ iB̃(t)
)]
= ek
2t/2r2 f̌CNk (x),
Ẽ
[
f̌DNk
(
x+ iB̃(t)
)]
= e(k−1)
2t/2r2 f̌DNk (x), k ∈ Z,
28 M. Katori
and thus
M̌CN
u,uj (t, y) =
N∑
k=1
φ̌CNu,uj (k)ek
2t/2r2 f̌CNk (y),
M̌DN
u,uj (t, y) =
N∑
k=1
φ̌DNu,uj (k)e(k−1)
2t/2r2 f̌DNk (y), j ∈ {1, 2, . . . , N}.
Then we have
gRu (s, x; t, y) = gReq(t− s, x, y) + rRu (s, x; t, y), R = CN , DN
with
gCNeq (t− s, x, y) =
2
πr
N∑
`=1
e`
2(t−s)/2r2 f̌CN` (x)f̌CN` (y)
=
1
πr
∑
n : |n|≤N
en
2(t−s)/2r2 sin
(nx
r
)
sin
(ny
r
)
,
rCNu (s, x; t, y) =
2
πr
N∑
k=1
ek
2(t−s)/2r2 f̌CNk (y)
∞∑
`=N+1
e−(`
2−k2)s/2r2 f̌CN` (x)
×
N∑
j=1
f̌CN` (uj)φ̌
CN
u,uj (k), (5.9)
and
gDNeq (t− s, x, y) =
1
πr
(
f̌DN1 (x)f̌DN1 (y) + 2
N∑
`=2
e(`−1)
2(t−s)/2r2 f̌DN` (x)f̌DN` (y)
)
=
1
πr
∑
n:|n|≤N−1
en
2(t−s)/2r2 cos
(nx
r
)
cos
(ny
r
)
,
rDNu (s, x; t, y) =
2
πr
N∑
k=1
e(k−1)
2(t−s)/2r2 f̌DNk (y)
×
∞∑
`=N+1
e−{(`−1)
2−(k−1)2}s/2r2 f̌DN` (x)
N∑
j=1
f̌DN` (uj)φ̌
DN
u,uj (k), (5.10)
where (5.5) and (5.6) were used. For any fixed s, t ∈ [0,∞), it is obvious that
lim
T→∞
rRu (s+ T, x; t+ T, y) = 0, R = CN , DN ,
uniformly on any subset of (x, y) ∈ (0, πr)2, since the summations of k are taken for 1 ≤ k ≤ N ,
while those of ` are taken for ` ≥ N + 1 in (5.9) and (5.10). Hence
lim
T→∞
ǨCN
u (s+ T, x; t+ T, y) = gCNeq (t− s, x, y)− 1(s > t)p[0,πr]aa (s− t, x|y),
lim
T→∞
ǨDN
u (s+ T, x; t+ T, y) = gDNeq (t− s, x, y)− 1(s > t)p[0,πr]rr (s− t, x|y),
in the same sense. It is easy to confirm that the limit kernels are represented by (5.7) and (5.8), if
we use (5.6). The limit kernels (5.7) and (5.8) depend on time difference t−s, which implies that
Elliptic Determinantal Processes and Elliptic Dyson Models 29
the determinantal processes defined by them are temporally homogeneous. The determinantal
processes with the spatio-temporal correlation kernels (5.7) and (5.8) are equilibrium processes,
which are reversible with respect to the determinantal point processes with the spatial correlation
kernels
ǨCN
eq (x, y) =
1
2πr
[
sin{(2N + 1)(y − x)/2r}
sin{(y − x)/2r}
− sin{(2N + 1)(y + x)/2r}
sin{(y + x)/2r}
]
,
ǨDN
eq (x, y) =
1
2πr
[
sin{(2N − 1)(y − x)/2r}
sin{(y − x)/2r}
+
sin{(2N − 1)(y + x)/2r}
sin{(y + x)/2r}
]
, (5.11)
(x, y) ∈ [0, πr]2, respectively. The convergence of processes is irreversible. Thus all statements
of the present proposition have been proved. �
We note that
ρ̌CNeq (x) = lim
y→x
ǨCN
eq (x, y) =
2
πr
N∑
n=1
sin2
(nx
r
)
,
ρ̌DNeq (x) = lim
y→x
ǨDN
eq (x, y) =
1
πr
{
1 + 2
N−1∑
n=1
cos2
(nx
r
)}
,
and hence∫ πr
0
ρ̌CNeq (x)dx =
∫ πr
0
ρ̌DNeq (x)dx = N,
as required.
6 Concluding remarks and open problems
In the previous paper [16, 17] and in this paper (Theorem 1.1), we have introduced seven families
of interacting particle systems ΞR(t) =
N∑
j=1
δXR
j (t), t ∈ [0, t∗) governed by the probability laws PRu
associated with the irreducible reduced affine root systems denoted by R = AN−1, BN , B
∨
N , CN ,
C∨N , BCN , DN . When we proved that they are determinantal processes, we showed that without
change of expectations of symmetric functions of
{
XR
j (·)
}N
j=1
, PRu can be replaced by the signed
measures P̂Ru . Define F̂Rt = σ
(
Ξ̂R(s) : 0 ≤ s ≤ t
)
, t ∈ [0, t∗). As the simplest corollary of this
fact, we can conclude that at any time 0 < t < t∗,
ÊRu
[
1
(
Ξ̂R(t) ∈ F̂Rt
)]
= 1 for R = AN−1, BN , B
∨
N , CN , C
∨
N , BCN , DN .
This is nothing but a rather trivial statement such that the processes
(
Ξ̂R(t)
)
t∈[0,t∗) are well-
normalized, but it provides nontrivial multiple-integral equalities including parameters t ∈ [0, t∗)
and u = (u1, . . . , uN ). For example, for R = DN , we will have∫
[0,πr]N
dx
N∏
`=1
1
2πr
{
ϑ3
(
x` − u`
2πr
;
it
2πr2
)
+ ϑ3
(
x` + u`
2πr
;
it
2πr2
)}
(6.1)
×
∏
1≤j<k≤N
ϑ1((xk − xj)/2πr; τDN (t))
ϑ1((uk − uj)/2πr; τDN (0))
ϑ1((xk + xj)/2πr; τ
DN (t))
ϑ1((uk + uj)/2πr; τDN (0))
=
(
η(τDN (t))
η(τDN (0))
)N(N−2)
,
where η(τ) is the Dedekind modular function (1.10) and τDN (t) = i(N − 1)(t∗ − t)/πr2.
30 M. Katori
In the previous papers [16, 17] and in Theorem 1.2 we have identified the systems of SDEs
which are solved by the four families of determinantal processes,
((
ΞAN−1(t)
)
t∈[0,t∗),P
AN−1
u
)
,
u ∈W(0,2πr)
N , and
((
ΞR(t)
)
t∈[0,t∗),P
R
u
)
, u ∈W(0,πr)
N with R = BN , CN and DN . The systems of
SDEs for other cases R = B∨N , C
∨
N , BCN are not yet clarified. The exceptional cases of reduced
irreducible affine root systems and the non-reduced irreducible affine root systems [30] will be
studied from the view point of the present stochastic analysis.
The classical Dyson models of type A in R given by (1.29) and of types C and D in [0,∞)
given by (1.30) and (1.31) are realized as the eigenvalue processes of Hermitian-matrix-valued
Brownian motions with specified symmetry [6, 18]. It will be an interesting problem to construct
the matrix-valued Brownian motions such that the eigenvalue processes of them provide the
elliptic Dyson models; (1.2) with β = 2 and (1.22)–(1.24).
As shown in Section 4, the factors represented by the Dedekind modular function (1.10) in
the determinantal martingale-functions (1.5)–(1.7) are essential in the proof of Theorem 1.2.
We have noted that NR given by (1.4) is identified with the quantity g given for the reduced
irreducible affine root systems in [30, Appendix 1]. Interpretation of the formulas (1.8) from
the view point of representation theory is desired.
As mentioned after Theorem 1.2 in Section 1, if we take the temporally homogeneous limit
t∗ →∞, the elliptic Dyson models of types A, B, C, and D become the corresponding trigono-
metric Dyson models, and in the further limit r →∞, they are reduced to the Dyson models in R,
in [0,∞) with an absorbing boundary condition at the origin, and in [0,∞) with a reflecting
boundary condition at the origin, respectively. In the limit r → ∞, p
[0,πr]
aa and p
[0,πr]
rr given
by (1.11) and (1.12) become
p[0,∞)
aa (t, y|x) = pBM(t, y|x)− pBM(t, y| − x),
p[0,∞)
rr (t, y|x) = pBM(t, y|x) + pBM(t, y| − x),
and in the double limit t∗ →∞, r →∞, ΦCN
u,uj (z) and ΦDN
u,uj (z) given by (1.19)–(1.21) become
ΦCN
u,uj (z)→
zj
uj
Φ
[0,∞)
u,uj (z), ΦDN
u,uj (z)→ Φ
[0,∞)
u,uj (z)
with
Φ
[0,∞)
u,uj (z) =
∏
1≤`≤N,
6̀=j
z2 − u2`
u2j − u2`
.
Then the spatio-temporal correlation kernels (1.17) are reduced to the following
KCN
u (s, x; t, y) =
N∑
j=1
p[0,∞)
aa (s, x|uj)Ẽ
[
y + iB̃(t)
uj
Φ
[0,∞)
u,uj
(
y + iB̃(t)
)]
− 1(s > t)p[0,∞)
aa (s− t, x|y),
KDN
u (s, x; t, y) =
N∑
j=1
p[0,∞)
rr (s, x|uj)Ẽ
[
Φ
[0,∞)
u,uj
(
y + iB̃(t)
)]
− 1(s > t)p[0,∞)
rr (s− t, x|y),
(s, x), (t, y) ∈ [0,∞) × [0,∞). Consider the D-dimensional Bessel processes BES(D) in [0,∞),
whose transition probability densities pBES(D)
are expressed using the modified Bessel func-
tion Iν(z) with ν = (D − 2)/2 [15, 33]. In particular, we find that
pBES(3)(t, y|x) =
y
x
p[0,∞)
aa (t, y|x), pBES(1)(t, y|x) = p[0,∞)
rr (t, y|x),
Elliptic Determinantal Processes and Elliptic Dyson Models 31
x, y ∈ (0,∞), t ≥ 0. If we put
KBES(3)
u (s, x; t, y) =
N∑
j=1
pBES(3)(s, x|uj)Ẽ
[
y + iB̃(t)
y
Φ
[0,∞)
u,uj
(
y + iB̃(t)
)]
− 1(s > t)pBES(3)(s− t, x|y), (6.2)
KBES(1)
u (s, x; t, y) =
N∑
j=1
pBES(1)(s, x|uj)Ẽ
[
Φ
[0,∞)
u,uj
(
y + iB̃(t)
)]
− 1(s > t)pBES(1)(s− t, x|y), (6.3)
the following equalities are established
KCN
u (s, x; t, y) =
y
x
KBES(3)
u (s, x; t, y), KDN
u (s, x; t, y) = KBES(1)
u (s, x; t, y), (6.4)
(s, x), (t, y) ∈ [0,∞)× [0,∞). The functions (6.2) and (6.3) are the spatio-temporal correlation
kernels of the N -particle systems of BES(3) and BES(1) with noncolliding condition (see [14,
Section 7]). It is easy to verify that the spatio-temporal Fredholm determinant (2.17) is invariant
under the transformation of kernel
K̂u(s, x; t, y)→ a(t, y)
a(s, x)
K̂u(s, x; t, y)
with an arbitrary continuous function a. Then (6.4) implies that the elliptic Dyson models of
types C and D given by Theorem 1.2 are deduced to the noncolliding BES(3) and BES(1) in the
double limit t∗ →∞, r →∞, respectively. In other words, the present elliptic Dyson models of
types C and D can be regarded as the elliptic extensions of the noncolliding BES(3) and BES(1),
respectively. Trigonometric and elliptic extensions of noncolliding BES(D) with general D ≥ 1
[14, 23] will be studied. Moreover, infinite-particle limits of the elliptic determinantal processes
should be studied [16, 17, 22].
Connection between the present elliptic determinantal processes and probabilistic discrete
models with elliptic weights [1, 3, 35, 36, 37] will be also an interesting future problem.
A The Jacobi theta functions and related functions
A.1 Notations and formulas of the Jacobi theta functions
Let
z = evπi, q = eτπi,
where v, τ ∈ C and =τ > 0. The Jacobi theta functions are defined as follows [32, 43]
ϑ0(v; τ) = −ieπi(v+τ/4)ϑ1
(
v +
τ
2
; τ
)
=
∑
n∈Z
(−1)nqn
2
z2n = 1 + 2
∞∑
n=1
(−1)neτπin
2
cos(2nπv),
ϑ1(v; τ) = i
∑
n∈Z
(−1)nq(n−1/2)
2
z2n−1 = 2
∞∑
n=1
(−1)n−1eτπi(n−1/2)
2
sin{(2n− 1)πv},
ϑ2(v; τ) = ϑ1
(
v +
1
2
; τ
)
=
∑
n∈Z
q(n−1/2)
2
z2n−1 = 2
∞∑
n=1
eτπi(n−1/2)
2
cos{(2n− 1)πv},
32 M. Katori
ϑ3(v; τ) = eπi(v+τ/4)ϑ1
(
v +
1 + τ
2
; τ
)
=
∑
n∈Z
qn
2
z2n = 1 + 2
∞∑
n=1
eτπin
2
cos(2nπv). (A.1)
(Note that the present functions ϑµ(v; τ), µ = 1, 2, 3 are denoted by ϑµ(πv, q), and ϑ0(v; τ) by
ϑ4(πv, q) in [43].) For =τ > 0, ϑµ(v; τ), µ = 0, 1, 2, 3 are holomorphic for |v| < ∞ and satisfy
the partial differential equation
∂ϑµ(v; τ)
∂τ
=
1
4πi
∂2ϑµ(v; τ)
∂v2
. (A.2)
They have the quasi-periodicity; for instance, ϑ1 satisfies
ϑ1(v + 1; τ) = −ϑ1(v; τ), ϑ1(v + τ ; τ) = −e−πi(2v+τ)ϑ1(v; τ).
By the definition (A.1), when =τ > 0,
ϑ1(0; τ) = ϑ1(1; τ) = 0, ϑ1(x; τ) > 0, x ∈ (0, 1),
ϑ0(x; τ) > 0, x ∈ R. (A.3)
We see the asymptotics
ϑ0(v; τ) ∼ 1, ϑ1(v; τ) ∼ 2eτπi/4 sin(πv), ϑ2(v; τ) ∼ 2eτπi/4 cos(πv),
ϑ3(v; τ) ∼ 1 in =τ → +∞ (i.e., q = eτπi → 0). (A.4)
The following functional equalities are known as Jacobi’s imaginary transformations [32, 43]
ϑ0(v; τ) = eπi/4τ−1/2e−πiv
2/τϑ2
(
v
τ
;−1
τ
)
,
ϑ1(v; τ) = e3πi/4τ−1/2e−πiv
2/τϑ1
(
v
τ
;−1
τ
)
,
ϑ3(v; τ) = eπi/4τ−1/2e−πiv
2/τϑ3
(
v
τ
;−1
τ
)
. (A.5)
A.2 Basic properties of A2πr
N (t∗ − t, x)
For 0 < t∗ <∞, 0 < r <∞, 0 < N <∞, the function A2πr
N (t∗ − t, x) is defined by (1.1), which
is written as
A2πr
N (t∗ − t, x) =
1
2πr
ϑ′1
(
x/2πr; iN (t∗ − t)/2πr2
)
ϑ1
(
x/2πr; iN (t∗ − t)/2πr2
) , t ∈ [0, t∗), (A.6)
where ϑ′1(v; τ) = ∂ϑ1(v; τ)/∂v. As a function of x ∈ R, A2πr
N (t∗ − t, x) is odd,
A2πr
N (t∗ − t,−x) = −A2πr
N (t∗ − t, x), (A.7)
and periodic with period 2πr
A2πr
N (t∗ − t, x+ 2mπr) = A2πr
N (t∗ − t, x), m ∈ Z.
It has only simple poles at x = 2mπr, m ∈ Z, and simple zeroes at x = (2m + 1)πr, m ∈ Z.
Independently of the values of t ∈ [0, t∗) and 0 < N <∞, A2πr
N (t∗ − t, x) behaves as
A2πr
N (t∗ − t, x) ∼
1
x
, as x ↓ 0,
− 1
2πr − x
, as x ↑ 2πr.
(A.8)
Elliptic Determinantal Processes and Elliptic Dyson Models 33
Using Jacobi’s imaginary transformation (A.5), we can show by (A.4) that
A2πr
N (t∗ − t, x) ∼
− x− πr
N (t∗ − t)
, if x > 0,
− x+ πr
N (t∗ − t)
, if x < 0,
as t ↑ t∗. (A.9)
A.3 Dedekind modular function
The Dedekind modular function η(τ) is defined by (1.10), that is,
η(τ) = eτπi/12
∞∏
n=1
(1− e2nτπi), =τ > 0.
For t ∈ [0, t∗), 0 < N <∞, define
η1N (t∗ − t) =
π2
ω1
(
1
12
− 2
∞∑
n=1
nq2n
1− q2n
)∣∣∣∣∣
ω1=πr, q=e−N (t∗−t)/2r2
. (A.10)
Then the following equality is established [16, 17]
d log η(τR(t))
dt
= NR η
1
NR(t∗ − t)
2πr
. (A.11)
A.4 Transition probability densities of Brownian motions in an interval
Using the definitions of the Jacobi theta functions (A.1) and Jacobi’s imaginary transforma-
tions (A.5), we can obtain the following expressions for the transition probability densities
p
[0,πr]
aa (t, y|x) and p
[0,πr]
rr (t, y|x) defined by (1.11) and (1.12)
p[0,πr]aa (t, y|x) = pBM(t, y|x)ϑ3
(
i(y − x)r
t
;
2πir2
t
)
− pBM(t, y| − x)ϑ3
(
i(y + x)r
t
;
2πir2
t
)
=
1
2πr
{
ϑ3
(
y − x
2πr
;
it
2πr2
)
− ϑ3
(
y + x
2πr
;
it
2πr2
)}
=
1
πr
∑
n∈Z
e−n
2t/2r2 sin
(ny
r
)
sin
(nx
r
)
, (A.12)
and
p[0,πr]rr (t, y|x) = pBM(t, y|x)ϑ3
(
i(y − x)r
t
;
2πir2
t
)
+ pBM(t, y| − x)ϑ3
(
i(y + x)r
t
;
2πir2
t
)
=
1
2πr
{
ϑ3
(
y − x
2πr
;
it
2πr2
)
+ ϑ3
(
y + x
2πr
;
it
2πr2
)}
=
1
πr
∑
n∈Z
e−n
2t/2r2 cos
(ny
r
)
cos
(nx
r
)
, (A.13)
for x, y ∈ [0, πr], t ≥ 0. It should be noted that
∫ πr
0 p
[0,πr]
rr (t, y|u)dy ≡ 1, ∀ t ≥ 0, u ∈ [0, πr],
while
psurvivalu (t) ≡
∫ πr
0
p[0,πr]aa (t, y|u)dy ' 4
π
e−t/2r
2
sin
(u
r
)
→ 0 as t→∞, u ∈ (0, πr).
When both boundaries at 0 and πr are absorbing, psurvivalu (t) gives the probability for a Brownian
motion to survive up to time t in the interval (0, πr), when it starts from u ∈ (0, πr). In other
words, with probability 1−psurvivalu (t), the Brownian motion was absorbed before time t and has
been fixed at one of the boundaries.
34 M. Katori
A.5 Weierstrass ℘ function and zeta function ζ
The Weierstrass ℘ function and zeta function ζ are defined by
℘(z|2ω1, 2ω3) =
1
z2
+
∑
(m,n)∈Z2\{(0,0)}
[
1
(z − Ωm,n)2
− 1
Ωm,n
2
]
,
ζ(z|2ω1, 2ω3) =
1
z
+
∑
(m,n)∈Z2\{(0,0)}
[
1
z − Ωm,n
+
1
Ωm,n
+
z
Ωm,n
2
]
, (A.14)
where ω1 and ω3 are fundamental periods with τ = ω3/ω1 and Ωm,n = 2mω1+2nω3. Put (4.16),
then the following functional equation holds (see [43, Section 20.41] and [16, Lemma 2.1])
(ζNR(t∗ − s, z + u)− ζNR(t∗ − s, z)− ζNR(t∗ − s, u))2
= ℘NR(t∗ − s, z + u) + ℘NR(t∗ − s, z) + ℘NR(t∗ − s, u). (A.15)
Acknowledgements
The author would like to thank the anonymous referees whose comments considerably improved
the presentation of the paper. A part of the present work was done during the participation of
the author in the ESI workshop on “Elliptic Hypergeometric Functions in Combinatorics, Inte-
grable Systems and Physics” (March 20–24, 2017). The present author expresses his gratitude
for the hospitality of Erwin Schrödinger International Institute for Mathematics and Physics
(ESI) of the University of Vienna and for well-organization of the workshop by Christian Krat-
tenthaler, Masatoshi Noumi, Simon Ruijsenaars, Michael J. Schlosser, Vyacheslav P. Spiridonov,
and S. Ole Warnaar. He also thanks Soichi Okada, Masatoshi Noumi, Simon Ruijsenaars, and
Michael J. Schlosser for useful discussion. This work was supported in part by the Grant-in-Aid
for Scientific Research (C) (No. 26400405), (B) (No. 26287019), and (S) (No. 16H06338) of
Japan Society for the Promotion of Science.
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1 Introduction and main results
2 Determinantal martingales and determinantal equalities
2.1 Notion of martingale
2.2 Basic equalities and determinantal martingales
2.3 Auxiliary measure and spatio-temporal Fredholm determinant
2.4 Elliptic determinant evaluations of the Macdonald denominators
2.5 Determinantal martingale-functions
3 Proof of Theorem 1.1
4 Proof of Theorem 1.2
5 Relaxation to equilibrium processes in trigonometric Dyson models of types C and D
6 Concluding remarks and open problems
A The Jacobi theta functions and related functions
A.1 Notations and formulas of the Jacobi theta functions
A.2 Basic properties of AN2 r(t-t,x)
A.3 Dedekind modular function
A.4 Transition probability densities of Brownian motions in an interval
A.5 Weierstrass function and zeta function
References
|
| id | nasplib_isofts_kiev_ua-123456789-149273 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T17:45:57Z |
| publishDate | 2017 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Katori, M. 2019-02-19T19:40:23Z 2019-02-19T19:40:23Z 2017 Elliptic Determinantal Processes and Elliptic Dyson Models / M. Katori // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 43 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 60J65; 60G44; 82C22; 60B20; 33E05; 17B22 DOI:10.3842/SIGMA.2017.079 https://nasplib.isofts.kiev.ua/handle/123456789/149273 We introduce seven families of stochastic systems of interacting particles in one-dimension corresponding to the seven families of irreducible reduced affine root systems. We prove that they are determinantal in the sense that all spatio-temporal correlation functions are given by determinants controlled by a single function called the spatio-temporal correlation kernel. For the four families AN₋₁, BN, CN and DN, we identify the systems of stochastic differential equations solved by these determinantal processes, which will be regarded as the elliptic extensions of the Dyson model. Here we use the notion of martingales in probability theory and the elliptic determinant evaluations of the Macdonald denominators of irreducible reduced affine root systems given by Rosengren and Schlosser. This paper is a contribution to the Special Issue on Elliptic Hypergeometric Functions and Their Applications.
 The full collection is available at https://www.emis.de/journals/SIGMA/EHF2017.html.
 The author would like to thank the anonymous referees whose comments considerably improved
 the presentation of the paper. A part of the present work was done during the participation of
 the author in the ESI workshop on “Elliptic Hypergeometric Functions in Combinatorics, Integrable Systems and Physics” (March 20–24, 2017). The present author expresses his gratitude
 for the hospitality of Erwin Schr¨odinger International Institute for Mathematics and Physics
 (ESI) of the University of Vienna and for well-organization of the workshop by Christian Krattenthaler, Masatoshi Noumi, Simon Ruijsenaars, Michael J. Schlosser, Vyacheslav P. Spiridonov,
 and S. Ole Warnaar. He also thanks Soichi Okada, Masatoshi Noumi, Simon Ruijsenaars, and
 Michael J. Schlosser for useful discussion. This work was supported in part by the Grant-in-Aid
 for Scientific Research (C) (No. 26400405), (B) (No. 26287019), and (S) (No. 16H06338) of
 Japan Society for the Promotion of Science. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Elliptic Determinantal Processes and Elliptic Dyson Models Article published earlier |
| spellingShingle | Elliptic Determinantal Processes and Elliptic Dyson Models Katori, M. |
| title | Elliptic Determinantal Processes and Elliptic Dyson Models |
| title_full | Elliptic Determinantal Processes and Elliptic Dyson Models |
| title_fullStr | Elliptic Determinantal Processes and Elliptic Dyson Models |
| title_full_unstemmed | Elliptic Determinantal Processes and Elliptic Dyson Models |
| title_short | Elliptic Determinantal Processes and Elliptic Dyson Models |
| title_sort | elliptic determinantal processes and elliptic dyson models |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/149273 |
| work_keys_str_mv | AT katorim ellipticdeterminantalprocessesandellipticdysonmodels |