Stochastic Differential Equations Related to
Soft-Edge Scaling Limit
Hideki Tanemura
Chiba univ. (Japan)
joint work with Hirofumi Osada (Kyushu Unv.)
2012 March 29
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Hideki Tanemura (Chiba univ.) ()
SDEs related to Soft-Edge scalng limit
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2012 March 29
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1 / 26
Introduction
The Gaussian ensembles are introduced as Hermitian matrices with
independent elements of Gaussian random variables, and joint distribution
invariant under conjugation by appropriate unitary matrices. The ensemble
are divided into classes according to the element being real, complex or
real quaternion, and their invariance under conjugation by
orthogonal matices (GOE),
unitary matrices (GUE),
unitary symplectic matrices (GSE),
respectively.
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Hideki Tanemura (Chiba univ.) ()
SDEs related to Soft-Edge scalng limit
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2012 March 29
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2 / 26
Introduction
The distribution of eigenvalues of the ensembles with size n × n are given
by
{
}
n
∏
∑
1
β
mβn (dxn ) =
|xi − xj |β exp −
|xi |2 dxn ,
Z
4
i<j
i=1
on the configuration space of n particles:
Wn = {xn ∈ Rn : x1 < x2 < · · · < xn }
where the GOE, the GUE, and the GSE correspond with β = 1, 2 and 4,
respectively.
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Hideki Tanemura (Chiba univ.) ()
SDEs related to Soft-Edge scalng limit
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2012 March 29
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3 / 26
Introduction
For the eigenvalues {λn1 , . . . , λnn }
√
√
{ nλn1 , . . . , nλnn } → µsin,β , weakly as n → ∞.
(Bulk scaling limit)
µsin,β is a probability measure on the configuration space of unlabelled
particles:
{
}
M = ξ : ξ is a nonnegative integer valued Radon measures in R
Any element ξ of M can be represented as: ξ(·) =
∑
δxj (·) with some
j∈I
sequence (xj )j∈I of R satisfying ♯{j ∈ I : xj ∈ K } < ∞, for any compact
set K . M is a Polish space with the vague topology.
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Hideki Tanemura (Chiba univ.) ()
SDEs related to Soft-Edge scalng limit
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2012 March 29
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4 / 26
Introduction
For β = 2 (GUE) µsin,2 is the determinantal point process(DPP) , in which
any spatial correlation function ρm is given by a determinant with the sine
kernel
Ksin,2 (x, y ) = Ksin (x, y ) ≡
sin{π(y − x)}
,
π(y − x)
x, y ∈ R.
The moment generating function is given by a Fredholm determinant
∫
{∫
}
[
]
exp
f (x)ξ(dx) µsin,2 (dξ) = Det δx (y ) + Ksin (x, y )χ(y ) ,
M
(x,y )∈R2
R
for f ∈ Cc (R), where χ(·) = e f (·) − 1.
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Hideki Tanemura (Chiba univ.) ()
SDEs related to Soft-Edge scalng limit
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2012 March 29
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5 / 26
Introduction
For β = 1 (GOE) µsin,1 is the quaternion DPP with kernels
[
]
∂
Ksin (x, y )
− ∂y
Ksin (x, y )
∫x
Ksin,1 (x, y ) =
.
− 12 sign(x − y ) + y Ksin,2 (u, y )du
Ksin (x, y )
.
Hideki Tanemura (Chiba univ.) ()
SDEs related to Soft-Edge scalng limit
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2012 March 29
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6 / 26
Introduction
For β = 1 (GOE) µsin,1 is the quaternion DPP with kernels
[
]
∂
Ksin (x, y )
− ∂y
Ksin (x, y )
∫x
Ksin,1 (x, y ) =
.
− 12 sign(x − y ) + y Ksin,2 (u, y )du
Ksin (x, y )
Here we use a natural identification between the 2 × 2 complex matrices
and the quaternions:
(
)
(
)
(
)
(
)
1 0
i 0
0 1
0 i
I =
e1 =
e2 =
e3 =
0 1
0 −i
−1 0
i 0
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Hideki Tanemura (Chiba univ.) ()
SDEs related to Soft-Edge scalng limit
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2012 March 29
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6 / 26
Introduction
For β = 1 (GOE) µsin,1 is the quaternion DPP with kernels
[
]
∂
Ksin (x, y )
− ∂y
Ksin (x, y )
∫x
.
Ksin,1 (x, y ) =
− 12 sign(x − y ) + y Ksin,2 (u, y )du
Ksin (x, y )
Here we use a natural identification between the 2 × 2 complex matrices
and the quaternions:
(
)
(
)
(
)
(
)
1 0
i 0
0 1
0 i
I =
e1 =
e2 =
e3 =
0 1
0 −i
−1 0
i 0
For β = 4 (GSE) µsin,4 is the quaternion DPP with kernels
[
]
∂
Ksin (2x, 2y )
− ∂y
Ksin (2x, 2y )
Ksin,4 (x, y ) = ∫ x
.
Ksin (2x, 2y )
y Ksin (2u, 2y )du
.
Hideki Tanemura (Chiba univ.) ()
SDEs related to Soft-Edge scalng limit
.
.
.
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2012 March 29
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6 / 26
Introduction
The moment generating function is given by
∫
{∫
}
exp
f (x)ξ(dx) µsin,β (dξ)
M
R
[
]1/2
δx (y ) ⊗ I2 + Ksin,β (x, y )χ2 (y )
(x,y )∈R2
[
]
= Pf
δx (y ) ⊗ I2 + Ksin,β (x, y )χ2 (y ) ,
=
Det
(x,y )∈R2
for f ∈ Cc (R), where
(
χ2 (·) =
e f (·) − 1
e f (·) − 1
)
,
and Pf denotes the Fredholm pfaffian.
.
Hideki Tanemura (Chiba univ.) ()
SDEs related to Soft-Edge scalng limit
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2012 March 29
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7 / 26
Introduction
n × n Hermitian matrix valued process (n ∈ N)


M11 (t) M12 (t) · · · M1n (t)
 M21 (t) M22 (t) · · · M2n (t) 
,
M(t) = 


···
M1 (t) Mn2 (t) · · · Mnn (t)
Mℓk (t) = Mkℓ (t)† .
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Hideki Tanemura (Chiba univ.) ()
SDEs related to Soft-Edge scalng limit
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2012 March 29
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8 / 26
Introduction
n × n Hermitian matrix valued process (n ∈ N)


M11 (t) M12 (t) · · · M1n (t)
 M21 (t) M22 (t) · · · M2n (t) 
,
M(t) = 


···
M1 (t) Mn2 (t) · · · Mnn (t)
Mℓk (t) = Mkℓ (t)† .
R (t), 1 ≤ k ≤ ℓ ≤ n : indep. BMs
(GOE) Bkℓ
1 R
(t), 1 ≤ k < ℓ ≤ n,
Mkℓ (t) = √ Bkℓ
2
R
Mkk (t) = Bkk
(t), 1 ≤ k ≤ n,
.
Hideki Tanemura (Chiba univ.) ()
SDEs related to Soft-Edge scalng limit
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.
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2012 March 29
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8 / 26
Introduction
n × n Hermitian matrix valued process (n ∈ N)


M11 (t) M12 (t) · · · M1n (t)
 M21 (t) M22 (t) · · · M2n (t) 
,
M(t) = 


···
M1 (t) Mn2 (t) · · · Mnn (t)
Mℓk (t) = Mkℓ (t)† .
R (t), 1 ≤ k ≤ ℓ ≤ n : indep. BMs
(GOE) Bkℓ
1 R
(t), 1 ≤ k < ℓ ≤ n,
Mkℓ (t) = √ Bkℓ
2
R
Mkk (t) = Bkk
(t), 1 ≤ k ≤ n,
R (t), B I (t),1 ≤ k ≤ ℓ ≤ n : indep. BMs
(GUE) Bkℓ
kℓ
√
−1 I
1 R
(t), 1 ≤ k < ℓ ≤ n,
Mkℓ (t) = √ Bkℓ (t) + √ Bkℓ
2
2
R
Mkk (t) = Bkk
(t), 1 ≤ k ≤ n,
.
.
.
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Hideki Tanemura (Chiba univ.) ()
SDEs related to Soft-Edge scalng limit
.
2012 March 29
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8 / 26
Introduction
α (t), α = 0, 1, 2, 3, 1 ≤ k ≤ ℓ ≤ n : indep. BMs
(GSE) Bkℓ
1 0
0
Mkℓ
(t) = √ Bkℓ
(t), 1 ≤ k < ℓ ≤ n,
2
0
0
Mkk
(t) = Bkk
(t), 1 ≤ k ≤ n,
For α = 1, 2, 3,
√
−1 α
(t), 1 ≤ k < ℓ ≤ n,
= √ Bkℓ
2
α
Mkk
(t) = 0, 1 ≤ k ≤ n,
α
Mkℓ
(t)
2n × 2n self dual Hermitian matrix valued process
M(t) = M 0 (t) ⊗ I +
3
∑
M α (t) ⊗ eα
α=1
.
Hideki Tanemura (Chiba univ.) ()
SDEs related to Soft-Edge scalng limit
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2012 March 29
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9 / 26
Introduction
The eigen-valued process is Dyson’s Brownian motion model [Dyson 62],
which is a one parameter family of the systems solving the following
stochastic differential equation:
dXj (t) = dBj (t) +
β
2
∑
k:1≤k≤n
k̸=j
dt
, 1≤j ≤n
Xj (s) − Xk (s)
(1)
where Bj (t), j = 1, 2, . . . , n are independent one dimensional Brownian
motions.
Osada[PTRF: online first] constructed the diffusion process which solves
the SDE (1) with n = ∞ by the Dirichlet form technique associated with
the measure µsin,β , β = 1, 2 and 4.
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Hideki Tanemura (Chiba univ.) ()
SDEs related to Soft-Edge scalng limit
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2012 March 29
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10 / 26
Introduction
(Soft edge scaling limit) As n → ∞
{n1/6 λn1 − 2n2/3 , . . . , n1/6 λnn − 2n2/3 } → µAi,β , weakly,
For β = 2, µAi,2 is the determinantal point process with the Airy kernel

 Ai(x)Ai′ (y ) − Ai′ (x)Ai(y )
if x ̸= y
KAi (x, y ) =
x −y

(Ai′ (x))2 − x(Ai(x))2
if x = y ,
where Ai(·) is the Airy function defined by
∫
1
3
Ai(z) =
dk e i(zk+k /3) ,
2π R
z ∈ C,
and Ai′ (x) = dAi(x)/dx.
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Hideki Tanemura (Chiba univ.) ()
SDEs related to Soft-Edge scalng limit
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2012 March 29
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11 / 26
Introduction
For β = 1, µAi,1 is the quaternion DPP with the kernel
[
]
∂
KAi (x, y )
− ∂y
KAi (x, y )
∫x
KAi,1 (x, y ) =
− 12 sign(x − y ) + y KAi (u, y )du
KAi (x, y )

(
)

∫∞
−Ai(x)Ai(y )
1  Ai(x) 1 (− y Ai(u)du )
.
+
(
)
∫x
∫∞
∫∞
2
Ai(u)du
1
−
Ai(u)du
1
−
Ai(u)du
Ai(y
)
y
y
x
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Hideki Tanemura (Chiba univ.) ()
SDEs related to Soft-Edge scalng limit
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.
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2012 March 29
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12 / 26
Introduction
For β = 1, µAi,1 is the quaternion DPP with the kernel
[
]
∂
KAi (x, y )
− ∂y
KAi (x, y )
∫x
KAi,1 (x, y ) =
− 12 sign(x − y ) + y KAi (u, y )du
KAi (x, y )

(
)

∫∞
−Ai(x)Ai(y )
1  Ai(x) 1 (− y Ai(u)du )
.
+
(
)
∫x
∫∞
∫∞
2
Ai(u)du
1
−
Ai(u)du
1
−
Ai(u)du
Ai(y
)
y
y
x
For β = 4, µAi,4 is the quaternion DPP with the kernel
]
∂
KAi (22/3 x, 22/3 y )
− ∂y
KAi (22/3 x, 22/3 y )
KAi,4 (x, y ) = 1/3 ∫ x
KAi (22/3 u, 22/3 y )du
KAi (22/3 x, 22/3 y )
2
y
[
]
∫∞
−Ai(22/3 x) y Ai(22/3 u)du
−Ai(22/3 x)Ai(22/3 y )
1
∫∞
∫∞
∫x
+ 2/3
.
− y Ai(22/3 u)du y Ai(22/3 u)du − x Ai(22/3 u)duAi(22/3 y )
2
1
[
.
Hideki Tanemura (Chiba univ.) ()
SDEs related to Soft-Edge scalng limit
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2012 March 29
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12 / 26
Results
Let β = 1, 2 and 4.
(1) There exists a diffusion process describing an infinite particle system,
which has µAi,β as a reversible measure.
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Hideki Tanemura (Chiba univ.) ()
SDEs related to Soft-Edge scalng limit
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2012 March 29
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13 / 26
Results
Let β = 1, 2 and 4.
(1) There exists a diffusion process describing an infinite particle system,
which has µAi,β as a reversible measure.
(2) The system solves the ISDE given by
{
}
∫
∑
1
ρb(x)dx
β
dXj (t) = dBj (t) +
lim
−
dt,
2 L→∞
Xj (t) − Xk (t)
−x
|x|<L
k̸=j,
|Xk (t)|<L
j ∈ N,
where
√
−x
ρb(x) =
1(x < 0).
π
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Hideki Tanemura (Chiba univ.) ()
SDEs related to Soft-Edge scalng limit
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2012 March 29
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13 / 26
Dirichlet spaces
A function f defined on the configuration space M is local if f (ξ) = f (ξK )
for some compact set K .
∑
A local function f is smooth if f ( nj=1 δxj ) = f˜(x1 , x2 , . . . , xn ) with some
smooth function f˜ on Rn with compact support. Put
D0 = {f : f is local and smooth}.
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Hideki Tanemura (Chiba univ.) ()
SDEs related to Soft-Edge scalng limit
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2012 March 29
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14 / 26
Dirichlet spaces
A function f defined on the configuration space M is local if f (ξ) = f (ξK )
for some compact set K .
∑
A local function f is smooth if f ( nj=1 δxj ) = f˜(x1 , x2 , . . . , xn ) with some
smooth function f˜ on Rn with compact support. Put
D0 = {f : f is local and smooth}.
We put
ξ(K )
1 ∑ ∂ f˜(x) ∂ g̃ (x)
D[f , g ](ξ) =
2
∂xj ∂xj
j=1
and for a probability measure µ we introduce the bilinear form
∫
E µ (f , g ) =
D[f , g ]dµ, f , g ∈ D0 .
M
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Hideki Tanemura (Chiba univ.) ()
SDEs related to Soft-Edge scalng limit
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2012 March 29
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14 / 26
quasi Gibbs measure
Let Φ be a free potential, Ψ be an interaction potential. For a given
sequence {br } of N we introduce a Hamiltonian on Ir = (−br , br ):
∑
∑
Hr (ξ) = HrΦ,Ψ (ξ) =
Φ(xj ) +
Ψ(xj , xk )
xj ∈Ir
xj ,xk ∈Ir ,j<k
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Hideki Tanemura (Chiba univ.) ()
SDEs related to Soft-Edge scalng limit
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2012 March 29
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15 / 26
quasi Gibbs measure
Let Φ be a free potential, Ψ be an interaction potential. For a given
sequence {br } of N we introduce a Hamiltonian on Ir = (−br , br ):
∑
∑
Hr (ξ) = HrΦ,Ψ (ξ) =
Φ(xj ) +
Ψ(xj , xk )
xj ∈Ir
xj ,xk ∈Ir ,j<k
Definition A probability measure µ is said to be a (Φ, Ψ)-quasi Gibbs
measure if there exists an increasing sequence {br } of N and measures
{µm
r ,k } such that for each r , m ∈ N satisfying
m
µm
r ,k ≤ µr ,k+1 ,
k ∈ N,
lim µm
r ,k = µ(· ∩ {ξ(Ir ) = m}), weekly
k→∞
and that for all r , m, k ∈ N and for µm
r ,k -a.s. ξ ∈ M
−Hr (ξ)
1{ξ(Ir )=m} Λ(dζ)
c −1 e −Hr (ξ) 1{ξ(Ir )=m} Λ(dζ) ≤ µm
r ,k (πIr ∈ dζ|ξIrc ) ≤ ce
Here Λ is the Poisson random measure with intensity
measure
dx.
.
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Hideki Tanemura (Chiba univ.) ()
SDEs related to Soft-Edge scalng limit
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2012 March 29
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15 / 26
quasi regular Dirichlet space
Theorem (Bulk) [Osada:to appear in AOP]
Let β = 1, 2, 4.
(1) The probability measure µsin,β is a quasi Gibbs measure with
Φ(x) = 0 and Ψ(x) = −β log |x − y |.
(2) The closure of (E µsin,β , D0 , L2 (M, µsin,β )) is a quasi Dirichlet space,
and there exists a µsin -reversible diffusion process (Ξsin,β (t), P) associated
with the Diriclet space.
.
Hideki Tanemura (Chiba univ.) ()
SDEs related to Soft-Edge scalng limit
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2012 March 29
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16 / 26
quasi regular Dirichlet space
Theorem (Bulk) [Osada:to appear in AOP]
Let β = 1, 2, 4.
(1) The probability measure µsin,β is a quasi Gibbs measure with
Φ(x) = 0 and Ψ(x) = −β log |x − y |.
(2) The closure of (E µsin,β , D0 , L2 (M, µsin,β )) is a quasi Dirichlet space,
and there exists a µsin -reversible diffusion process (Ξsin,β (t), P) associated
with the Diriclet space.
Theorem 1 (Soft-Edge) [Osada-T]
Let β = 1, 2, 4.
(1) The probability measure µAi,β is a quasi Gibbs measure with
Φ(x) = 0 and Ψ(x) = −β log |x − y |.
(2) The closure of (E µAi,β , D0 , L2 (M, µAi,β )) is a quasi Dirichlet space,
and there exists a µAi -reversible diffusion process (ΞAi,β (t), P) associated
with the Diriclet space.
.
Hideki Tanemura (Chiba univ.) ()
SDEs related to Soft-Edge scalng limit
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2012 March 29
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16 / 26
Extended Airy kerenel
Let (ξ Ai (t), P) be the determinantal process with the extended Airy kernel
K (s, x; t, y ) =
Ai
 ∫



0
du e (t−s)u/2 Ai(x − u)Ai(y − u),
−∞
∫ ∞


 −
if s ≤ t,
du e (t−s)u/2 Ai(x − u)Ai(y − u), if s > t.
0
The process is reversible process with reversible measure µAi,2 .
[Forrester-Nagao-Honner (1999)],[Prähofer-Spohn (2002)]
Theorem [Katori-T: MPRF(2011)]
The process (ξ Ai (t), P) is a continuos reversible Markov process.
Open problem.
(ξ Ai (t), P) = (ΞAi,2 (t), P) ?
Proposition [Katori-T: JSP(2007)]
The Dirichlet space of the process (ξ Ai (t), P) is a closed extension of
(E µAi , D0 , L2 (M, µAi )).
.
.
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.
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Hideki Tanemura (Chiba univ.) ()
SDEs related to Soft-Edge scalng limit
2012 March 29
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17 / 26
Infinite dimensional Stochstic Differential Equations
Theorem 2[Osada-T]
Let β = 1, 2, 4. The process ΞAi,β (t) =
dXj (t) = dBj (t) +
β
lim
2 L→∞
{
∑
j∈N δXj (t)
satisfies
∑
1
Xj (t) − Xk (t)
k̸=j:|Xk (t)|<L
}
∫
ρb(x)
−
dx dt,
|x|<L −x
j ∈ N,
where Bj (t), j ∈ N are independent Brownian motions and
√
−x
ρb(x) =
1(x < 0).
π
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Hideki Tanemura (Chiba univ.) ()
SDEs related to Soft-Edge scalng limit
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2012 March 29
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18 / 26
Log derivative
Let µx be the Palm measure conditioned at x = (x1 , . . . , xk ∈ Rk
(
)
k
∑
µx = µ · −
δxj ξ(xj ) ≥ 1 for j = 1, 2, . . . , k .
j=1
Let µk be the Campbell measure of µ:
∫
µk (A × B) =
µx (B)ρk (x)dx,
A ∈ B(Rk ), B ∈ B(M).
A
We call dµ ∈ L1loc (R × M, µ1 ) the log derivative of µ if dµ satisfies
∫
∫
µ
1
d (x, η)f (x, η)dµ (x, η) = −
∇x f (x, η)dµ1 (x, η),
R×M
R×M
f ∈ Cc∞ (R)
⊗D
0.
.
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Hideki Tanemura (Chiba univ.) ()
SDEs related to Soft-Edge scalng limit
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.
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2012 March 29
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19 / 26
ISDE
Theorem [Osada, PTRF (on line first)]
Assume that there exists a log derivative dµ (and some ∑
conditions). There
exists M0 ⊂ M such that µ(M0 ) = 1, and for any ξ = j∈N δxj ∈ M0 ,
there exists RN -valued continuous process X(t) = (Xj (t))∞
j=1 satisfying
X(0) = x = (xj )∞
and
j=1
(
)
∑
1
dXj (t) = dBj (t) + dµ Xj (t),
δXk (t) dt,
2
j ∈ N.
k:k̸=j
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Hideki Tanemura (Chiba univ.) ()
SDEs related to Soft-Edge scalng limit
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.
.
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2012 March 29
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20 / 26
ISDE
Theorem [Osada, PTRF (on line first)]
Assume that there exists a log derivative dµ (and some ∑
conditions). There
exists M0 ⊂ M such that µ(M0 ) = 1, and for any ξ = j∈N δxj ∈ M0 ,
there exists RN -valued continuous process X(t) = (Xj (t))∞
j=1 satisfying
X(0) = x = (xj )∞
and
j=1
(
)
∑
1 µ
dXj (t) = dBj (t) + d Xj (t),
δXk (t) dt, j ∈ N.
2
k:k̸=j
Lemma
∑ (Bulk) [Osada, PTRF online first]
η=
δyj with η({x}) = 0,
Let β = 1, 2, 4. For x ∈ R and
j∈N
dµsin,β (x, η) = β lim
L→∞
∑
j:|x−yj |≤L
.
Hideki Tanemura (Chiba univ.) ()
SDEs related to Soft-Edge scalng limit
1
x − yj
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2012 March 29
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20 / 26
Key lemma
The key part in the proof of Theorem 2 is to determine the log derivative
of µAi,β .
∑
δyj
Key lemma (Soft-Edge) Let β = 1, 2, 4. For x ∈ R and η =
j∈N
with η({x}) = 0,
dµAi,β (x, η) = β lim


L→∞ 
∑
j:|x−yj |≤L
1
−
x − yj
.
Hideki Tanemura (Chiba univ.) ()
SDEs related to Soft-Edge scalng limit


∫
|u|≤L
.
ρb(u)
du .
−u 
.
.
.
2012 March 29
.
21 / 26
Key lemma
The key part in the proof of Theorem 2 is to determine the log derivative
of µAi,β .
∑
δyj
Key lemma (Soft-Edge) Let β = 1, 2, 4. For x ∈ R and η =
j∈N
with η({x}) = 0,
dµAi,β (x, η) = β lim


L→∞ 
∑
j:|x−yj |≤L
1
−
x − yj
Lemma
∑ (Bulk) [Osada, PTRF online first]
η=
δyj with η({x}) = 0,
j∈N
dµsin,β (x, η) = β lim


L→∞ 
Hideki Tanemura (Chiba univ.) ()
∑
j:|x−yj |≤L


∫
|u|≤L
ρb(u)
du .
−u 
Let β = 1, 2, 4. For x ∈ R and
1
−
x − yj
.
SDEs related to Soft-Edge scalng limit


∫
|u|≤L
.
ρ
du .
−u 
.
.
.
2012 March 29
.
21 / 26
Proof
To prove the key lemma, we use n particle sysytem:
{
}
n
1∏
β∑
n
β
2
mβ (dun ) =
|ui − uj | exp −
|ui | dun ,
Z
4
i<j
√
We put uj = 2 n +
i=1
xj
n1/6
and intrduce the measure defined by
{
}
n
∏
∑
√
β
1
|xi − xj |β exp −
|2 n + n−1/6 xi |2 dxn ,
µnA,β (dxn ) =
Z
4
i<j
i=1
The log derivative dn of the measure µnA,β is given by


{∑
}
n−1
n−1
∑
n−1/3
1
1/3
n
n

−n −
x .
d (x, η) = d
x,
δyj = β
x − yj
2
j=1
j=1
.
Hideki Tanemura (Chiba univ.) ()
SDEs related to Soft-Edge scalng limit
.
.
.
.
2012 March 29
.
22 / 26
Proof
Lemma 3 is derived from the fact that

 ∑
Ai
dµ (x, η) = lim dn (x, η) = β lim
n→∞
L→∞ 
|x−yj |<L
1
−
x − yj


∫
|u|≤L
ρb(y )
du

−y
(2)
.
Hideki Tanemura (Chiba univ.) ()
SDEs related to Soft-Edge scalng limit
.
.
.
.
2012 March 29
.
23 / 26
Proof
Lemma 3 is derived from the fact that

 ∑
Ai
dµ (x, η) = lim dn (x, η) = β lim
n→∞
L→∞ 
|x−yj |<L
1
−
x − yj


∫
|u|≤L
ρb(y )
du

−y
(2)
To check (2) we divide dn /β into three parts:
∫
∑
ρnA,β,x (u)
1
gnL (x, η) =
−
du,
x − yj
|x−u|<L x − u
|x−yj |<L
wLn (x, η)
=
∫
u n (x) =
R
∑
1
−
x − yj
|x−yj |≥L
ρnA,β,x (u)
x −u
∫
ρnA,β,x (u)
du − n1/3 −
du,
n−1/3
x.
2
.
Hideki Tanemura (Chiba univ.) ()
x −u
|x−u|≥L
SDEs related to Soft-Edge scalng limit
.
.
.
.
2012 March 29
.
23 / 26
Proof
The fact (2) is obtained if the following conditions hold:
lim gn (x, η)
n→∞ L
= gL (x, η), in Lp̂ (µ1Ai,β ) for any L > 0,
∫
lim lim sup
|wLn (x, y)|p̂ dµn,1
A (dxdη) = 0,
L→∞ n→∞
lim u n (x) = u(x),
with
∑
gL (x, η) =
|x−yj |<L
{∫
u(x) = lim
L→∞
|u|≤L
in Lp̂loc (R, dx) ,
1
−
x − yj
ρAi,β,x (u)
du −
x −u
∫
|x−u|<L
∫
|u|≤L
(5)
ρAi,β,x (u)
du,
x −u
ρb(u)
du
−u
.
Hideki Tanemura (Chiba univ.) ()
(4)
[−r ,r ]×M
n→∞
and
(3)
SDEs related to Soft-Edge scalng limit
.
}
∈ Lp̂loc (R, dx).
.
.
.
2012 March 29
.
24 / 26
Proof
In the case β = 2, µnA is the DPP with the correlation kernel
Ψn (x)Ψn−1 (y ) − Ψn−1 (x)Ψn (y )
x −y
(√
)
where Ψn (x) = n1/12 φn 2n + √2nx 1/6 , and φk (x) is the normalized
orthogonal functions on R comprising the Hermite polynomials Hk (x).
n
KA
(x, y ) = n1/3
.
Hideki Tanemura (Chiba univ.) ()
SDEs related to Soft-Edge scalng limit
.
.
.
.
2012 March 29
.
25 / 26
Proof
In the case β = 2, µnA is the DPP with the correlation kernel
Ψn (x)Ψn−1 (y ) − Ψn−1 (x)Ψn (y )
x −y
(√
)
where Ψn (x) = n1/12 φn 2n + √2nx 1/6 , and φk (x) is the normalized
orthogonal functions on R comprising the Hermite polynomials Hk (x).
We also use the function
√ (
)
1
u
n
ρb (u) =
−u 1 + 2/3 , −4n2/3 ≤ u ≤ 0
π
4n
n
KA
(x, y ) = n1/3
and the facts
∫
R
ρbn (u)
du = n1/3 ,
−u
lim ρbn (u) → ρb(u).
n→∞
.
Hideki Tanemura (Chiba univ.) ()
SDEs related to Soft-Edge scalng limit
.
.
.
.
2012 March 29
.
25 / 26
Thanks
Thank you for your attention!
.
Hideki Tanemura (Chiba univ.) ()
SDEs related to Soft-Edge scalng limit
.
.
.
.
2012 March 29
.
26 / 26