*Proof.* Let *c*_{12}=*c*_{12}(N) be such that
*c*_{12}=

∫

*S*

Erf(*|x| −r*

*√c*3*T* )ρ* ^{N,1}*(x)dx.

Let*c*_{13}= lim sup_{N}_{→∞}*c*_{12}(N). Then from**(H1)** and (2.57), we see that for each large*r*
*c*_{13}*≤* lim

*N**→∞*

∫

*S**r*

Erf(*|x| −r*

*√c*3*T* )ρ* ^{N,1}*(x)dx+ lim sup

*N**→∞*

∫

*S**\**S**r*

Erf(*|x| −r*

*√c*3*T* )ρ* ^{N,1}*(x)dx (2.99)

*<∞.*

From **(H1)** we see that *{µ*^{N}*}**N**∈N* converges to *µ* weakly. Hence *{µ*^{N}*}**N**∈N* is tight.

This implies that there exists a sequence of increasing sequences of natural numbers**a*** _{n}*=

*{a*

*n*(m)

*}*

^{∞}*m=1*such that

**a**

*n*

*<*

**a**

*n+1*and that for each

*m*

*n*lim*→∞*lim sup

*N**→∞* *µ** ^{N}*(s(S

*)*

_{m}*≥a*

*(m)) = 0.*

_{n}Without loss of generality, we can take *a** _{n}*(m)

*> m*for all

*m, n∈*N. Then from this, we see that there exists a sequence

*{*p(L)

*}*

*L*

*∈N*converging to

*∞*such that p(L)

*< L*for all

*L∈*Nand that

*L*lim*→∞*lim sup

*N**→∞* *µ** ^{N}*(s(S

_{p(L)})

*≥L) = 0.*(2.100) Recall that the label

*ℓ*

*N*(s) = (s

*i*)

*i*

*∈N*satisfies

*|s*1

*| ≤ |s*2

*| ≤ · · ·*. Using this, we divide the set Sas in such a way that

*{s**L**∈S*_{p(L)}*}*and *{s**L**̸∈S*_{p(L)}*}*.

Then s*∈ {s*_{L}*∈S*_{p(L)}*}* if and only ifs(S_{p(L)})*≥L. Hence we easily see that*

∑

*i>L*

∫

S

Erf(*|s*_{i}*| −r*

*√c*_{3}*T* )µ* ^{N}*(ds)

*≤c*

_{12}(N)µ

*(*

^{N}*{*s(S

_{p(L)})

*≥L}*) +

∫

*S**\**S*_{p(L)}

Erf(*|x| −r*

*√c*_{3}*T* )ρ* ^{N,1}*(x)dx.

Taking the limits on both sides, we obtain

*L*lim*→∞*lim sup

*N**→∞*

∑

*i>L*

∫

S

Erf(*|s*_{i}*| −r*

*√c*_{3}*T* )µ* ^{N}*(ds)

*≤*

*c*13 lim

*L**→∞*lim sup

*N**→∞* *µ** ^{N}*({s(Sp(L))

*≥L}) + lim*

*L**→∞*lim sup

*N**→∞*

∫

*S**\**S*p(L)

Erf(*|x| −r*

*√c*3*T* )ρ* ^{N,1}*(x)dx.

Applying (2.99) and (2.100) to the second term, and (2.57) to the third, we deduce (2.55).

random matrix theory and the interaction Ψ(x) =*−*log*|x|*, the logarithmic function. We
present these in this section. For this we shall confirm the assumptions in Theorem 2.7,
that is, assumptions**(H1)–(H4)** and**(J1)–(J6).**

Assumption **(H1)**is satisfied for the first three examples [43, 67]. As for the last two
examples, we assume**(H1). We also assume** **(H2).** **(H3)**can be proved in the same way
as given in [53]. In all examples, a is always a unit matrix. Hence it holds that **(H4)** is
satisfied and that (2.44) in **(J1)**becomes b* ^{N}* =

^{1}

_{2}d

*. From this we see that SDEs (2.59) and (2.51) become*

^{N}*dX*_{t}* ^{N,i}* =

*dB*

^{N,i}*+1*

_{t}2d* ^{N}*(X

_{t}

^{N,i}*,*X

^{N,}

_{t}

^{⋄}*)*

^{i}*dt*(1

*≤i≤N*), (2.101)

*dX*

_{t}*=*

^{i}*dB*

^{i}*+1*

_{t}2d* ^{µ}*(X

_{t}

^{i}*,*X

^{⋄}

_{t}*)*

^{i}*dt*(i

*∈*N), (2.102) where d

*is the logarithmic derivative of*

^{µ}*µ*given by (2.50). Assumption

**(J6)**for the first three examples with

*β*= 2 can be proved in the same way as [53] as we explained in Remark 2.10. Thus, in the rest of this section, our task is to check assumptions

**(J2)–(J5).**

**2.5.1** **The Airy**_{β}**interacting Brownian motion (β** = 1,2,4)

Let*µ*^{N}_{Airy,β} and *µ*_{Airy,β} be as in Section 2.1. Recall SDEs (2.10) and (2.11) in Section 2.1.

Let**X*** ^{N}* = (X

*)*

^{N,i}

^{N}*and*

_{i=1}**X**= (X

*)*

^{i}

_{i}*be solutions of*

_{∈N}*dX*

_{t}*=*

^{N,i}*dB*

_{t}*+*

^{i}*β*

2

∑*N*
*j=1, j**̸*=i

1

*X*_{t}^{N,i}*−X*_{t}^{N,j}*dt−β*

2*{N*^{1/3}+ 1

2N^{1/3}*X*_{t}^{N,i}*}dt,* (2.10)
*dX*_{t}* ^{i}*=

*dB*

_{t}*+*

^{i}*β*

2 lim

*r**→∞**{* ∑

*|**X*_{t}^{j}*|**<r,j**̸*=i

1
*X*_{t}^{i}*−X*_{t}^{j}*−*

∫

*|**x**|**<r*

*ϱ(x)*

*−x* *dx}dt* (i*∈*N). (2.11)
**Proposition 2.23.** If *β* = 1,4, then each sub-sequential limit of solutions **X*** ^{N}* of (2.10)
satisfies (2.11). If

*β*= 2, then the full sequence converges to (2.11).

*Proof.* Conditions **(J2)–(J5)** other than (2.48) can be proved in the same way as given
in [54]. In [54], we take *χ** _{s}*(x) = 1

_{S}*(x); its adaptation to the present case is easy.*

_{s}We consider estimates of correlation functions such that

*N*inf*∈N**ρ*^{N,1}_{Airy,β}(x)*≥c*_{14} for all *x∈S** _{r}* (2.103)
sup

*N**∈N**ρ*^{N,2}_{Airy,β}(x, y)*≤c*15*|x−y|* for all *x, y∈S**r**,* (2.104)
where*c*_{14}(r) and*c*_{15}(r) are positive constants. The first estimate is trivial because*ρ*^{N,1}_{Airy,β}
converges to*ρ*^{1}_{Airy,β} uniformly on*S** _{r}*and, all these correlation functions are continuous and
positive. The second estimate follows from the determinantal expression of the correlation
functions and bounds on derivative of determinantal kernels. Estimates needed for the
proof can be found in [54] and the detail of the proof of (2.104) is left to the reader.

Equation (2.48) follows from (2.103) and (2.104). Indeed, the integral in (2.48) is
taken on the bounded domain and the singularity of integral of*g** ^{N}*(x, y) =

*β/(x−y) near*

*{x* = *y}* is logarithmic. Furthermore, the one-point correlation function *ρ*^{N,1}_{Airy,β,x} of the
reduced Palm measure conditioned at*x*is controlled by the upper bound of the two-point
correlation function and the lower bound of one-point correlation function because

*ρ*^{N,1}_{Airy,β,x}(y) = *ρ*^{N,2}_{Airy,β}(x, y)
*ρ*^{N,1}_{Airy,β}(x) *.*
Using these facts, we see that (2.103) and (2.104) imply (2.48).

**2.5.2** **The Bessel**_{2,α} **interacting Brownian motion**

Let*S* = [0,*∞*) and *α∈*[1,*∞*). We consider the Bessel_{2,α} point process*µ*_{bes,2,α} and their
*N*-particle version. The Bessel_{2,α} point process *µ*_{bes,2,α} is a determinantal point process
with kernel

K_{bes,2,α}(x, y) = *J** _{α}*(

*√*

*x)√yJ*_{α}* ^{′}*(

*√y)−√*

*xJ*

_{α}*(*

^{′}*√*

*x)J** _{α}*(

*√y)*

2(x*−y)* (2.105)

=

*√xJ**α+1*(*√*

*x)J**α*(*√*

*y)−J**α*(*√*
*x)√*

*yJ**α+1*(*√*
*y)*

2(x*−y)* *,*

where*J**α*is the Bessel function of order*α*[67, 17]. The densitym^{N}* _{α}*(x)dxof the associated

*N*-particle systems

*µ*

^{N}_{bes,2,α}is given by

m^{N}* _{α}*(x) = 1

*Z*

*α*

^{N}*e*^{−}^{∑}^{N}^{i=1}^{x}^{i}^{/4N}

∏*N*
*j=1*

*x*^{α}_{j}

∏*N*
*k<l*

*|x*_{k}*−x*_{l}*|*^{2}*.* (2.106)

It is known that *µ*^{N}_{bes,2,α} is also determinantal [67, 945p] and [14, 91p] The Bessel_{2,α}
interacting Brownian motion is given by the following [17].

*dX*_{t}* ^{N,i}* =dB

_{t}*+*

^{i}*{−*1

8N + *α*

2X_{t}* ^{N,i}* +

∑*N*
*j=1,j**̸*=i

1

*X*_{t}^{N,i}*−X*_{t}^{N,j}*}dt* (1*≤i≤N*), (2.107)
*dX*_{t}* ^{i}* =dB

_{t}*+*

^{i}*{*

*α*

2X_{t}* ^{i}* +

∑*∞*
*j**̸*=i

1

*X*_{t}^{i}*−X*_{t}^{j}*}dt* (i*∈*N). (2.108)

This appears at the hard edge of one-dimensional systems.

**Proposition 2.24.** Assume *α >*1. Then (2.60) holds for (2.107) and (2.108).

*Proof.* **(J2)–(J5)**except (2.55) are proved in [17]. We easily see that the assumptions of
Lemma 2.22 hold and yield (2.55). We thus obtain**(J5).**

**Remark 2.25.** There exist other natural ISDEs and *N*-particle systems related to the
Bessel point processes. They are the non-colliding square Bessel processes and their
square root. The non-colliding square Bessel processes are reversible to the Bessel_{2,α}
point processes, but the associated Dirichlet forms are different from the Bessel_{2,α}
inter-acting Brownian motion. Indeed, the coefficients a* ^{N}* and a in Section 2.2 are taken to be
a

*(x.y) =a(x.y) = 2x. On the other hand, each square root of the non-colliding Bessel*

^{N}processes is not reversible to the Bessel_{2,α}point processes, but has the same type of
Dirich-let forms as the Bessel_{2,α} interacting Brownian motion. In particular, the coefficients a* ^{N}*
and a in Section 2.2 are taken to be a

*(x.y) =a(x.y) = 1/4. That is, they are constant time change of distorted Brownian motion with the standard square field.*

^{N}We refer to [25, 26, 52] for these processes. For reader’s convenience we provide an
ISDE describing the non-colliding square Bessel processes and their square root. We note
that SDE (2.110) is a constant time change of that in [26, 52]. Let **Y*** ^{N}* = (Y

*)*

^{N,i}

^{N}*and*

_{i=1}**Y**= (Y

*)*

^{i}

_{i}*be the non-colliding square Bessel processes. Then*

_{∈N}*dY*_{t}* ^{N,i}* =

√

*Y*_{t}^{N,i}*dB*_{t}* ^{i}*+

*{−Y*

_{t}

^{N,i}8N +*α*+ 1

2 +

∑*N*
*j=1,j**̸*=i

*Y*_{t}^{N,i}

*Y*_{t}^{N,i}*−Y*_{t}^{N,j}*}dt* (1*≤i≤N*), (2.109)
*dY*_{t}* ^{i}* =

√

*Y*_{t}^{i}*dB*_{t}* ^{i}*+

*{α*+ 1

2 +

∑*∞*
*j**̸*=i

*Y*_{t}^{i}

*Y*_{t}^{i}*−Y*_{t}^{j}*}dt* (i*∈*N). (2.110)
Let **Z*** ^{N}* = (Z

*)*

^{N,i}

^{N}*and*

_{i=1}**Z**= (Z

*)*

^{i}*be square root of the non-colliding square Bessel processes. Then applying Itˆo formula we obtain from (2.109) and (2.110)*

_{i∈N}*dZ*_{t}* ^{N,i}* =1

2*dB*^{i}* _{t}*+1

4*{−Z*_{t}^{N,i}

4N +*α*+^{1}_{2}
*Z*_{t}* ^{N,i}* +

∑*N*
*j=1,j**̸*=i

2Z_{t}^{N,i}

(Z_{t}* ^{N,i}*)

^{2}

*−*(Z

_{t}*)*

^{N,j}^{2}

*}dt*(1

*≤i≤N),*(2.111)

*dZ*

_{t}*=1*

^{i}2*dB*^{i}* _{t}*+1

4*{α*+ ^{1}_{2}
*Z*_{t}* ^{i}* +

∑*∞*
*j**̸*=i

2Z_{t}^{N,i}

(Z_{t}* ^{i}*)

^{2}

*−*(Z

_{t}*)*

^{j}^{2}

*}dt*(i

*∈*N). (2.112) We remark that Theorem 2.7 can be applied to the non-colliding square Bessel processes because the equilibrium states are the same as the Bessel interacting Brownian motion and coefficients are well-behaved as a

*(x.y) =a(x.y) = 2x.*

^{N}**2.5.3** **The Ginibre interacting Brownian motion**

Let *S* =R^{2}. Let*µ*^{N}_{gin} and *µ*gin be as in Section 2.1. Let Φ* ^{N}* =

*|x|*

^{2}and Ψ(x) =

*−*log

*|x|*. Then the

*N*-particle systems are given by

*dX*_{t}* ^{N,i}*=dB

_{t}

^{i}*−X*

_{t}

^{N,i}*dt*+

∑*N*
*j=1,j̸=i*

*X*_{t}^{N,i}*−X*_{t}^{N,j}

*|X*_{t}^{N,i}*−X*_{t}^{N,j}*|*^{2}*dt* (1*≤i≤N*). (2.12)
The limit ISDEs are

*dX*_{t}* ^{i}* =dB

_{t}*+ lim*

^{i}*r**→∞*

∑

*|**X*_{t}^{i}*−**X*_{t}^{j}*|**<r,j**̸*=i

*X*_{t}^{i}*−X*_{t}^{j}

*|X*_{t}^{i}*−X*_{t}^{j}*|*^{2}*dt* (i*∈*N) (2.13)

and

*dX*_{t}* ^{i}* =dB

_{t}

^{i}*−X*

_{t}

^{i}*dt*+ lim

*r**→∞*

∑

*|**X*_{t}^{j}*|**<r,j**̸*=i

*X*_{t}^{i}*−X*_{t}^{j}

*|X*_{t}^{i}*−X*_{t}^{j}*|*^{2}*dt* (i*∈*N). (2.14)

**Proposition 2.26.** (2.60) holds for (2.12) and both (2.13) and (2.14).

*Proof.* **(J2)–(J5)** except (2.55) are proved in [48, 47]. (2.55) is obvious for a Ginibre
point process because their one-correlation functions with respect to the Lebesgue measure
have a uniform bound such that *ρ*^{N,1}_{gin} *≤* 1/π. This estimate follows from (6.4) in [47]

immediately. Let d1 and d2 be the logarithmic derivative associated with ISDEs (2.13)
and (2.14). Thend_{1}=d_{2} a.s. [47]. Hence we conclude Proposition 2.26

**2.5.4** **Gibbs measures with Ruelle-class potentials**

Let*µ*^{Ψ}be Gibbs measures with Ruelle-class potential Ψ(x, y) = Ψ(x*−y) that are smooth*
outside the origin. Let Φ^{N}*∈C** ^{∞}*(S) be a confining potential for the

*N*- particle system.

We assume that the correlation functions of *µ*^{Φ}^{N}* ^{,Ψ}* satisfy bounds sup

_{N}*ρ*

^{N,m}*≤*

*c*

^{m}_{16}for some constants

*c*16; see the construction of [61]. Then one can see in the same fashion as [53] that

*µ*

^{Ψ}satisfy

**(J2)–(J5)**except (2.55). Under the condition sup

_{N}*ρ*

^{N,m}*≤*

*c*

^{m}_{16}, (2.55) is obvious. Moreover, if

*µ*

^{Ψ}is a grand canonical Gibbs measure with sufficiently small inverse temperature

*β, thenµ*

^{Ψ}is tail trivial. Hence we can obtain

**(J6)**in the same way as [53] in this case. We present two concrete examples below.

**2.5.5** **Lennard–Jones 6-12 potentials**

Let *S* = R^{3} and *β >* 0. Let Ψ_{6}_{−}_{12}(x) = *|x|*^{−}^{12}*− |x|*^{−}^{6} be the Lennard-Jones potential.

The corresponding ISDEs are given by the following.

*dX*_{t}* ^{N,i}* =dB

_{t}*+*

^{i}*β*

2*{∇*Φ* ^{N}*(X

_{t}*) +*

^{N,i}∑*N*

*j=1,*
*j**̸*=i

12(X_{t}^{N,i}*−X*_{t}* ^{N,j}*)

*|X*_{t}^{N,i}*−X*_{t}^{N,j}*|*^{14} *−*6(X_{t}^{N,i}*−X*_{t}* ^{N,j}*)

*|X*_{t}^{N,i}*−X*_{t}^{N,j}*|*^{8}*}dt* (1*≤i≤N*),
*dX*_{t}* ^{i}* =dB

_{t}*+*

^{i}*β*

2

∑*∞*
*j=1,j**̸*=i

*{*12(X_{t}^{i}*−X*_{t}* ^{j}*)

*|X*_{t}^{i}*−X*_{t}^{j}*|*^{14} *−*6(X_{t}^{i}*−X*_{t}* ^{j}*)

*|X*_{t}^{i}*−X*_{t}^{j}*|*^{8} *}dt* (i*∈*N).

**2.5.6** **Riesz potentials**

Let *d < a* *∈* N and *β >* 0. Let Ψ*a*(x) = ^{β}_{a}*|x|*^{−}* ^{a}* the Riesz potential. The corresponding
SDEs are given by

*dX*_{t}* ^{N,i}* =dB

_{t}*+*

^{i}*β*

2*{∇Φ** ^{N}*(X

_{t}*) +*

^{N,i}∑*N*
*j=1,j**̸*=i

*X*_{t}^{N,i}*−X*_{t}^{N,j}

*|X*_{t}^{N,i}*−X*_{t}^{N,j}*|*^{2+a}*}dt* (1*≤i≤N*),
*dX*_{t}* ^{i}* =dB

_{t}*+*

^{i}*β*

2

∑*∞*
*j=1,j**̸*=i

*X*_{t}^{i}*−X*_{t}^{j}

*|X*_{t}^{i}*−X*_{t}^{j}*|*^{2+a}*dt* (i*∈*N).

**3** **Dynamical bulk scaling limit of Gaussian unitary ** **ensem-bles and stochastic-differential-equation gaps**

**3.1** **Introduction**

Gaussian unitary ensembles (GUE) are Gaussian ensembles defined on the space of random
matrices *M** ^{N}* (N

*∈*N) with independent random variables, the matrices of which are Hermitian. By definition,

*M*

*= [M*

^{N}

_{i,j}*]*

^{N}

^{N}*is then an*

_{i,j=1}*N×N*matrix having the form

*M*_{i,j}* ^{N}* =
{

*ξ** _{i}* if

*i*=

*j*

*τ**i,j**/√*
2 +*√*

*−1ζ**i,j**/√*

2 if*i < j,*

where *{ξ**i**, τ**i,j**, ζ**i,j**}*^{∞}* _{i<j}* are i.i.d. Gaussian random variables with mean zero and a half
variance. Then the eigenvalues

*λ*

_{1}

*, . . . , λ*

*of*

_{N}*M*

*are real and have distribution ˇ*

^{N}*µ*

*such that*

^{N}ˇ

*µ** ^{N}*(dx

*) = 1*

_{N}*Z*

^{N}∏*N*
*i<j*

*|x*_{i}*−x*_{j}*|*^{2}

∏*N*
*k=1*

*e*^{−|}^{x}^{k}^{|}^{2}*dx*_{N}*,* (3.1)
where **x*** _{N}* = (x

_{1}

*, . . . , x*

*)*

_{N}*∈*R

*and*

^{N}*Z*

*is a normalizing constant [2]. Wigner’s cele-brated semicircle law asserts that their empirical distributions converge in distribution to a semicircle distribution:*

^{N}*N*lim*→∞*

1
*N{δ*_{λ}

1*/**√*

*N*+*· · ·*+*δ*_{λ}

*N/**√*

*N**}*= 1

*π*1_{(}_{−}^{√}_{2,}^{√}_{2)}(x)√

2*−x*^{2}*dx.*

One may regard this convergence as a law of large numbers because the limit distribution
is a*non-random* probability measure.

We consider the scaling of the next order in such a way that the distribution is
sup-ported on the set of configurations. That is, let*θ*be the position of the macro scale given
by

*−√*

2*< θ <√*

2 (3.2)

and take the scaling*x7→y* such that
*x*= *y*

*√N* +*θ√*

*N .* (3.3)

Let*µ*^{N}* _{θ}* be the point process for which the labeled density

**m**

^{N}

_{θ}*dx*

*is given by*

_{N}**m**

^{N}*(x*

_{θ}*N*) = 1

*Z*^{N}

∏*N*
*i<j*

*|x**i**−x**j**|*^{2}

∏*N*
*k=1*

*e*^{−|}^{x}^{k}^{+θN}^{|}^{2}^{/N}*.* (3.4)
The position*θ* in (3.2) is called the bulk and the scaling in (3.3) the bulk scaling (of the
point processes). It is well known that the rescaled point processes *µ*^{N}* _{θ}* satisfy

*N*lim*→∞**µ*^{N}* _{θ}* =

*µ*

*θ*in distribution, (3.5)

where*µ** _{θ}* is the determinantal point process with sine kernelK

*: K*

_{θ}*(x, y) = sin*

_{θ}*{√*

2*−θ*^{2}(x*−y)}*
*π(x−y)* *.*

By definition *µ** _{θ}* is the point process on Rfor which the

*m-point correlation function*

*ρ*

^{m}*with respect to the Lebesgue measure is given by*

_{θ}*ρ*^{m}* _{θ}* (x1

*, . . . , x*

*m*) = det[K

*θ*(x

*i*

*, x*

*j*)]

^{m}

_{i,j=1}*.*

We hence see that the limit is universal in the sense that it is the Sine2 point process
and independent of the macro-position *θ* up to the dilation of determinantal kernels K* _{θ}*.
This may be regarded as a first step of the universality of the Sine

_{2}point process, which has been extensively studied for general inverse temperature

*β*and a wide class of free potentials (see [5] and references therein).

Once a static universality is established, then it is natural to enquire of its
dynam-ical counter part. Indeed, we shall prove the dynamdynam-ical version of (3.5) and present a
phenomenon called stochastic-differential-equation (SDE) gaps for*θ̸*= 0.

Two natural *N*-particle dynamics are known for GUE. One is Dyson’s Brownian
mo-tion corresponding to time-inhomogeneous *N*-particle dynamics given by the time
evolu-tion of eigenvalues of time-dependent Hermitian random matrices *M** ^{N}*(t) for which the
coefficients are Brownian motions

*B*

_{t}*[43].*

^{i,j}The other is a diffusion process**X*** ^{θ,N}* = (X

*)*

^{θ,N,i}

^{N}*=*

_{i=1}*{*(X

_{t}*)*

^{θ,N,i}

^{N}

_{i=1}*}*

*t*given by the SDE such that for 1

*≤i≤N*

*dX*_{t}* ^{θ,N,i}* =

*dB*

_{t}*+*

^{i}∑*N*
*j**̸*=i

1

*X*_{t}^{θ,N,i}*−X*_{t}^{θ,N,j}*dt−* 1

*NX*_{t}^{θ,N,i}*dt−θ dt,* (3.6)
which has a unique strong solution for **X**^{θ,N}_{0} *∈* R^{N}*\N* and **X*** ^{θ,N}* never hits

*N*, where

*N*=

*{*

**x**= (x

*k*)

^{N}*;*

_{k=1}*x*

*i*=

*x*

*j*for some

*i̸*=

*j}*[19].

The derivation of (3.6) is as follows: Let ˇ*µ*^{N}* _{θ}* (dx

*) =*

_{N}**m**

^{N}*(x*

_{θ}*)dx*

_{N}*be the labeled symmetric distribution of*

_{N}*µ*

^{N}*. Consider a Dirichlet form on*

_{θ}*L*

^{2}(R

^{N}*,µ*ˇ

^{N}*) such that*

_{θ}*E*^{µ}^{ˇ}^{N}* ^{θ}* (f, g) =

∫

R^{N}

1 2

∑*N*
*i=1*

*∂f*

*∂x*_{i}

*∂g*

*∂x*_{i}*µ*ˇ^{N}* _{θ}* (dx

*).*

_{N}Then using (3.4) and integration by parts, we specify the generator*A** ^{N}* of

*E*

^{µ}^{ˇ}

^{N}*on*

^{θ}*L*

^{2}(R

^{N}*,µ*ˇ

^{N}*) such that*

_{θ}*A** ^{N}* = 1
2∆ +

∑*N*
*i=1*

*{*

∑*N*
*j**̸*=i

1
*x*_{i}*−x*_{j}*}* *∂*

*∂x*_{i}*−*

∑*N*
*i=1*

*{x**i*

*N* +*θ}* *∂*

*∂x*_{i}*.*
From this we deduce that the associated diffusion**X*** ^{θ,N}* is given by (3.6).

Taking the limit *N* *→ ∞* in (3.6), we *intuitively* obtain the infinite-dimensional SDE
(ISDE) of**X*** ^{θ}*= (X

*)*

^{θ,i}

_{i}*such that*

_{∈N}*dX*_{t}* ^{θ,i}*=

*dB*

^{i}*+*

_{t}∑*∞*
*j**̸*=i

1

*X*_{t}^{θ,i}*−X*_{t}^{θ,j}*dt−θ dt,* (3.7)

which was introduced in [68] with *θ* = 0. For each *θ, we have a unique, strong solution*
**X*** ^{θ}* of (3.7) such that

**X**

^{θ}_{0}=

**s**for

*µ*

_{θ}*◦*l

*-a.s.*

^{−1}**s, where**lis a labeling map. Although only the

*θ*= 0 ISDE of

**X**

^{0}=:

**X**= (X

*)*

^{i}*i*

*∈N*is studied in [53, 76], the general

*θ*

*̸*= 0 ISDE is nevertheless follows easily using the transformation

*X*_{t}* ^{θ,i}*=

*X*

_{t}

^{i}*−θt.*

Let X^{θ}* _{t}* = ∑

*i**δ*_{X}*θ,i*

*t* be the associated delabeled process. Then X* ^{θ}* =

*{*X

^{θ}

_{t}*}*takes

*µ*

*as an invariant probability measure, and is*

_{θ}*notµ*

*-symmetric for*

_{θ}*θ̸*= 0.

The precise meaning of the drift term in (3.7) is the substitution of **X**^{θ}* _{t}* = (X

_{t}*)*

^{θ,i}

_{i}*for the function*

_{∈N}*b(x,*y) given by the conditional sum

*b(x,*y) = lim

*r**→∞**{* ∑

*|**x**−**y**i**|**<r*

1

*x−y*_{i}*} −θ* in*L*^{1}_{loc}(µ^{[1]}* _{θ}* ), (3.8)
where y=∑

*i**δ*_{y}* _{i}* and

*µ*

^{[1]}

*is the one-Campbell measure of*

_{θ}*µ*

*(see (3.17)). We do this in such a way that*

_{θ}*b(X*

_{t}

^{θ,i}*,*∑

*j**̸*=i*δ*_{X}*θ,j*

*t* ). Because *µ** _{θ}* is translation invariant, it can be easily
checked that (3.8) is equivalent to (3.9):

*b(x,*y) = lim

*r**→∞**{*∑

*|**y**i**|**<r*

1

*x−y**i**} −θ* in*L*^{1}_{loc}(µ^{[1]}* _{θ}* ). (3.9)
Let l

*and lbe labeling maps. We denote by l*

_{N}*and l*

_{N,m}*the first*

_{m}*m-components of*l

*N*and l, respectively. We assume that, for each

*m∈*N,

*N*lim*→∞**µ*^{N}_{θ}*◦*l^{−}_{N,m}^{1} =*µ*_{θ}*◦*l^{−1}* _{m}* weakly

*.*(3.10) Let

**X**

*= (X*

^{θ,N}*)*

^{θ,N,i}

^{N}*and*

_{i=1}**X**= (X

*)*

^{i}

_{i}*be solutions of SDEs (3.6) and (3.11), respec-tively, such that*

_{∈N}*dX*_{t}* ^{θ,N,i}* =

*dB*

_{t}*+*

^{i}∑*N*
*j**̸*=i

1

*X*_{t}^{θ,N,i}*−X*_{t}^{θ,N,j}*dt−* 1

*NX*_{t}^{θ,N,i}*dt−θ dt,* (3.6)
*dX*_{t}* ^{i}* =

*dB*

_{t}*+ lim*

^{i}*r**→∞*

∑*∞*
*j**̸*=i,*|**X*_{t}^{i}*−**X*_{t}^{j}*|**<r*

1

*X*_{t}^{i}*−X*_{t}^{j}*dt.* (3.11)

We now state the first main result of the present paper.

**Theorem 3.1.** Assume (3.2) and (3.10). Assume that **X**^{θ,N}_{0} = *µ*^{N}_{θ}*◦*l^{−}_{N}^{1} in distribution
and **X**0 =*µ*_{θ}*◦*l^{−}^{1} in distribution. Then, for each *m∈*N,

*N*lim*→∞*(X^{θ,N,1}*, X*^{θ,N,2}*, . . . , X** ^{θ,N,m}*) = (X

^{1}

*, X*

^{2}

*, . . . , X*

*) (3.12) weakly in*

^{m}*C([0,∞*),R

*). In particular, the limit*

^{m}**X**= (X

*)*

^{i}

_{i}*does not satisfy (3.7) for any*

_{∈N}*θ*other than

*θ*= 0.

We next consider non-reversible initial distributions. Let **X*** ^{N}* = (X

*)*

^{N,i}

^{N}*and*

_{i=1}**Y**

*= (Y*

^{θ}*)*

^{θ,i}

_{i}*be solutions of (3.13) and (3.14), respectively, such that*

_{∈N}*dX*_{t}* ^{N,i}* =

*dB*

^{i}*+*

_{t}∑*N*
*j**̸*=i

1

*X*_{t}^{N,i}*−X*_{t}^{N,j}*dt−* 1

*NX*_{t}^{N,i}*dt,* (3.13)

*dY*_{t}* ^{θ,i}*=

*dB*

^{i}*+ lim*

_{t}*r**→∞*

∑*∞*
*j**̸*=i,*|**Y*_{t}^{θ,i}*−**Y*_{t}^{θ,j}*|**<r*

1

*Y*_{t}^{θ,i}*−Y*_{t}^{θ,j}*dt*+*θ dt.* (3.14)
Note that**X*** ^{N}* =

**X**

^{0,N}and that

**X**

*is not reversible with respect to*

^{N}*µ*

^{N}

_{θ}*◦l*

^{−}

_{N}^{1}for any

*θ̸= 0.*

We remark that the delabeld process Y* ^{θ}* =

*{*∑

*i**∈N**δ*_{Y}*θ,i*

*t* *}* of **Y*** ^{θ}* has invariant probability
measure

*µ*

*and is*

_{θ}*not*symmetric with respect to

*µ*

*for*

_{θ}*θ̸*= 0. We state the second main theorem.

**Theorem 3.2.** Assume (3.2) and (3.10). Assume that**X**^{N}_{0} =*µ*^{N}_{θ}*◦*l^{−}_{N}^{1} in distribution and
**Y**_{0}* ^{θ}*=

*µ*

_{θ}*◦*l

^{−}^{1}in distribution. Then for each

*m∈*N

*N*lim*→∞*(X^{N,1}*, X*^{N,2}*, . . . , X*^{N}* ^{,m}*) = (Y

^{θ,1}*, Y*

^{θ,2}*, . . . , Y*

*) (3.15) weakly in*

^{θ,m}*C([0,∞*),R

*).*

^{m}*•* We refer to the second claim in Theorem 3.1, and (3.15) as the SDE gaps. The
convergence in (3.15) of Theorem 3.2 resembles the “Propagation of Chaos” in the
sense that the limit equation (3.14) depends on the initial distribution, although it
is a linear equation. Because the logarithmic potential is by its nature long-ranged,
the effect of initial distributions*µ*^{N}* _{θ}* still remains in the limit ISDE, and the rigidity
of the Sine

_{2}point process makes the residual effect a non-random drift term

*θdt.*

Our result is the dynamical universality of Dyson’s Brownian motion in
infinite-dimension. There are similar result of dynamical universality of Dyson’s Brownian
motion in [36], but finite *N* result, then it is somehow different from ours.

*•* LetS* _{θ}* be a Borel set such that

*µ*

*(S*

_{θ}*) = 1, where*

_{θ}*−√*

2*< θ <√*

2. In [27], the first
author proves that one can choose S* _{θ}* such that S

_{θ}*∩*S

_{θ}*′*=

*∅*if

*θ̸*=

*θ*

*and that for each s*

^{′}*∈*S

*θ*(3.11) has a strong solution

**X**such that

**X**=l(s) and that

X* _{t}*:=

∑*∞*
*i=1*

*δ*_{X}*i*

*t* *∈*S* _{θ}* for all

*t∈*[0,

*∞*).

This implies that the state space of solutions of (3.11) can be decomposed into
uncountable disjoint components. We conjecture that the component S* _{θ}* is ergodic
for each

*θ∈*(

*−√*

2,*√*
2).

*•* For *θ* = 0, the convergence (3.12) is also proved in [52]. The proof in [52] is
alge-braic and valid only for dimension *d* = 1 and inverse temperature *β* = 2 with the
logarithmic potential. It relies on an explicit calculation of the space-time
correla-tion funccorrela-tions, the strong Markov property of the stochastic dynamics given by the
algebraic construction, the identity of the associated Dirichlet forms constructed by

two completely different methods, and the uniqueness of solutions of ISDE (3.7).

Although one may prove (3.10) for *θ* *̸*= 0 using the algebraic method in [52], this
requires a lot of work as mentioned above. We remark that, as a corollary and an
application, Theorem 3.1 proves the weak convergence of finite-dimensional
distri-butions explicitly given by the space-time correlation functions. We refer to [24, 52]

for the representation of these correlation functions.

*•* Tsai proves the pathwise uniqueness and the existence of strong solutions of
*dX*_{t}* ^{i}* =

*dB*

_{t}*+*

^{i}*β*

2 lim

*r**→∞*

∑*∞*
*j**̸*=i,*|**X*_{t}^{i}*−**X*_{t}^{j}*|**<r*

1

*X*_{t}^{i}*−X*_{t}^{j}*dt* (i*∈*N) (3.16)
for general *β* *∈* [1,*∞*) in [76]. The proof uses the classical stochastic analysis and
crucially depends on a specific monotonicity of SDEs (3.16). For *β* = 1,4, we have
a good control of the correlation functions as for *β* = 2. Hence our method can be
applied to*β*= 1,4 and the same result as for *β*= 2 in Theorem 3.1 holds. We shall
return to this point in future.

The key point of the proof of Theorem 3.1 is to prove the convergence of the drift
coefficient *b** ^{N}*(x,y) of the

*N*-particle system to the drift coefficient

*b(x,*y) of the limit ISDE even if

*θ̸*= 0. That is, as

*N*

*→ ∞*,

*b** ^{N}*(x,y) =

*{*

∑*N*
*i=1*

1

*x−y*_{i}*} −θ* =*⇒* *b(x,*y) = lim

*r**→∞**{* ∑

*|**y**i**|**<r*

1
*x−y*_{i}*}.*

Note that support of the coefficients *b** ^{N}*(x,y) and

*b(x,*y) are mutually disjoint, and that the sum in

*b*

*is not neutral for any*

^{N}*θ̸*= 0. We shall prove uniform bounds of the tail of the coefficients using fine estimates of the correlation functions, and cancel out the deviation of the sum in

*b*

*with*

^{N}*θ. Because of rigidity of the Sine*2 point process, we justify this cancellation not only for static but also dynamical instances.

The organization of the paper is as follows: In Section 3.2, we prepare general theories for interacting Brownian motion in infinite dimensions. In Section 3.3, we quote estimates for the oscillator wave functions and determinantal kernels. In Section 3.4, we prove key estimates (3.37)–(3.40). In Section 3.5, we complete the proof of Theorem 3.1. In Section 3.6, we prove Theorem 3.2.