By gℓ,N =χℓfk, we deduce that
|E1N(gℓ,N)− E1(f)| (6.41)
≤|E1N(χℓfk)− E1(χℓfk)|+|E1(χℓfk)− E1(χℓf)|+|E1(χℓf)− E1(f)|. From Lemma 6.24 and (6.40) we see that
|E1N(χℓfk)− E1(χℓfk)| ≤γk(ℓ)E1(χℓfk)≤θk. (6.42) By the straightforward calculation we have from (6.35), (6.37), and Lemma 6.24
|E1(χℓfk)− E1(χℓf)|=
∫
S
D[χℓfk]−D[χℓf] +χ2ℓ(fk2−f2)dµ−−−→
k→∞ o(ℓ) (6.43) and
|E1(χℓf)− E1(f)|=
∫
S
D[χℓf]−D[f] + (χ2ℓ −1)f2dµ=o(ℓ). (6.44) Putting (6.42), (6.43), and (6.44) into (6.41) and taking gN from gℓ,N we obtain (6.38) and (6.39).
Proof of Theorem 6.6. From Lemma 6.22 and Lemma 6.25, we obtain the Mosco con-vergence in Definition 6.17. The Mosco concon-vergence implies the concon-vergence of finite-dimensional distributions ofXN toX. (see [34, Section 7]).
Because the initial distribution has a density in Lp(µN) for some 1 < p, we see the tightness of{XN}N∈Nin C([0,∞);S) using Lyons-Zheng decomposition.
Lemma 6.26. (1) We have for eachr∈Nand p
(Er,p,D◦µ)≤(Er,D◦µ)≤(E,D◦µ∩ Brb)≤(E,Dµ◦ ∩ Br,pb ). (6.47) (2) (Er,p,D◦µ) is closable onL2(µ).
(3) (E,Dµ◦ ∩ Br,pb ) is closable onL2(µ).
Proof. By definition, the first inequality in (6.47) is clear because Smr (p) ⊂ Smr and the definition ofDmr,p andEr,pm given by (6.45) and (6.46). The second follows fromEr≤ E and Dµ◦ ⊃ Brb. The third follows from D◦µ∩ Brb⊃ D◦µ∩ Br,pb . We thus obtain (1).
We note that Smr (p) is an open set because σrm is lower semi-continuous for each m, r∈N. Because Smr (p) is an open set, (Er,pm,Dµ◦) is closable onL2(µ). Recall that
(Er,p,D◦µ) = (
∑∞ m=1
Er,pm,D◦µ).
We thus see that (Er,p,Dµ◦) is a countable sum of closable forms onL2(µ). Hence (Er,p,Dµ◦) is closable onL2(µ). We thus prove (2).
From (2) we have closability of (Er,p,D◦µ) onL2(µ). By (6.47) we have had (Er,p,D◦µ)≤ (E,D◦µ∩ Br,pb ). Hence closablity of (E,Dµ◦ ∩ Bbr,p) on L2(µ) follows from that of (Er,p,Dµ◦) on L2(µ). We thus complete the proof of (3).
From Lemma 6.26, we define (Er,p,Dr,p) and (Er,p,Dr,p) as the closures of (Er,p,Dµ◦) and (E,D◦µ∩Br,pb ) inL2(µ), respectively. Letp(n) =p(n)(r, m) be a sequence of functions and satisfy the monotonicity inn∈Nas follows: For each (r, m)∈N2
p(n)(r, m)<p(n+ 1)(r, m) for all n∈N. Lemma 6.27. (1){(Er,p(n),Dr,p(n))}r∈N is increasing for each n∈N. (2){(Er,p(n),Dr,p(n))}r∈N is decreasing for each n∈N.
Proof. We see that Sr(p) ⊂Sr+1(p). Then (1) follows from Er,p≤ Er+1,p and (2) follows from Br,pb ⊂ Bbr+1,p.
By Lemma 6.27, we define the closed form (Ep(n),Dp(n)) as the increasing limit of (Er,p(n),Dr,p(n)) in r, and (Ep(n),Dp(n)) as the decreasing limit of (Er,p(n),Dr,p(n)) in r.
Moreover, (Ep(n),Dµ◦) is closable and (Ep(n),Dp(n)) coincides with the closure of the max-imal closable part of (Ep(n),Dµ◦).
Lemma 6.28. (1){(Ep(n),Dp(n))}n∈N is increasing.
(2){(Ep(n),Dp(n))}n∈N is decreasing.
(3) For each n∈N we have
(Ep(n),Dp(n))≤(E,D)≤(E,D)≤(Ep(n),Dp(n)). (6.48) Proof. BySr(p(n))⊂Sr(p(n+ 1)) and (6.45) we have (Er,p(n),Dµ◦)≤(Er,p(n+1),D◦µ). By taking closure we have (Er,p(n),Dr,p(n))≤(Er,p(n+1),Dr,p(n+1)). Then taking a increasing limit inr, we obtain (1). By definition, Br,p(n)b ⊂ Br,p(n+1)b . Hence we have
(E,D◦µ∩ Br,p(n)b )≥(E,Dµ◦ ∩ Bbr,p(n+1)) for anyr.
Therefore, we get (Er,p(n),Dr,p(n))≥(Er,p(n+1),Dr,p(n+1)), which shows (2). By (6.47) (Er,p(n),Dr,p(n))≤(Er,Dr)≤(Er,Dr)≤(Er,p(n),Dr,p(n)).
Taking r→ ∞ and recalling
rlim→∞(Er,p(n),Dr,p(n)) = (Ep(n),Dp(n))
rlim→∞(Er,p(n),Dr,p(n)) = (Ep(n),Dp(n)), we obtain (6.48).
By Lemma 6.28 we obtain the limit closed forms (Ep(∞),Dp(∞)) and (Ep(∞),Dp(∞)) of {(Ep(n),Dp(n))}n∈Nand {(Ep(n),Dp(n))}n∈N asngoes to infinity, respectively. We remark that with the same reason as before (Ep(∞),Dp(∞)) is the closed form that coincides with the closure of the maximal closable part of the decreasing limit of (Ep(n),Dp(n)) asn→ ∞.
Then from Lemma 6.28 (1) we have
(Ep(∞),Dp(∞))≤(E,D)≤(E,D)≤(Ep(∞),Dp(∞)). (6.49) Lemma 6.29. Assume (6.14). Then the following hold.
(1) Dp(∞) is dense in Dwith respect toE. (2) Dis dense in Dp(∞) with respect toEp(∞) Proof. LetN =∪∞
m,r=1{s∈Smr ;σmr (s) = 0}. Then by (6.15) we have
Cap(N) = 0. (6.50)
From (6.50) we see thatµ(N) = 0. Thus, there exists a sequence {φq}inD satisfying
φq= 1 on N, (6.51)
0≤φq(s)≤1 for all s∈S,
qlim→0φq(s) = 0 forµ-a.s.s,
qlim→0E1(φq) = 0.
Because of (6.51), we have φq∈ Dp(∞). Hence
(1−φq)∈ Dp(∞). (6.52)
Let f ∈ D ∩L∞(µ). Then from (6.51) and (6.52) we easily deduce that f(1−φq)∈ Dp(∞)∩L∞(µ).
Moreover, whenf, φq∈ D◦µ, we obtain E(f −f(1−φq)) =Ep(∞)(f φq)
= lim
n→∞ lim
r→∞
∫
Sr(p(n))
D[f φq]dµ
≤ lim
n→∞ lim
r→∞2
∫
Sr(p(n))
D[f]|φq|2+|f|2D[φq]dµ
= lim
n→∞ lim
r→∞2
∫
Sr(p(n))
D[f]|φq|2+|f|2D[φq]dµ
=o(q) asq→ ∞.
Hence by approximation, Dp(∞)∩L∞(µ) is dense inD ∩L∞(µ) with respect to E. This completes the proof of (1) becauseD ∩L∞(µ) is dense inD with respect toE.
Remark thatφq ∈ D sinceD ⊂ D. Then forf ∈ Dp(∞)∩L∞(µ), we have f(1−φq)∈ D ∩L∞(µ).
We conclude (2) in a similar way to the proof of (1).
Lemma 6.30. Assume (6.14). Then
(Ep(∞),Dp(∞)) = (E,D)≤(E,D) = (Ep(∞),Dp(∞)). (6.53) Proof. (6.53) follows from (6.49) and Lemma 6.29 immediately.
6.5.2 N-particle approximation and the Mosco convergence II.
Next, we define cut off Dirichlet forms associated with µN. Let (ENp(n),DNp(n)) be the in-creasing limit limr→∞(ENr,p(n),DNr,p(n)), here (ENr,p(n),DNr,p(n)) is the closure of (Er,p(n)N ,D◦) onL2(µN). Define (Ep(n)N ,Dp(n)N ) as the maximal closable part less than limr→∞(Er,p(n)N ,Dr,p(n)N ), here (Er,p(n)N ,DNr,p(n)) is the closure of (EN,D◦∩ Br,p(n)b ) onL2(µN). Clearly, we obtain
(ENp(n),DNp(n))≤(EN,DN)≤(Ep(n)N ,Dp(n)N ). (6.54) Lemma 6.31. Assume (M3’). For anyf ∈L2(µ) and n∈N, the following holds.
(1) Assume that limN→∞fN =f weakly. Then Ep(n)(f)≤lim inf
N→∞ ENp(n)(fN). (6.55)
(2) There existfn,N ∈L2(µN) such that limN→∞fn,N =f strongly and that Ep(n)(f) = lim
N→∞Ep(n)N (fn,N) (6.56)
Proof. Recalling that densities ofµare positive onS(p(n)), (M3)holds onS(p(n)) from (6.15) in (M3’). Therefore we obtain (1) by the same argument as in Section 6.4.1.
Similarly, we obtain (2) by the same argument as in Section 6.4.2.
Lemma 6.32. Assume (M3’)and (A3). Assume that fN →f weakly. Then E(f)≤lim inf
N→∞ EN(fN) (6.57)
Proof. By (6.54) we haveENp(n)(fN)≤ EN(fN). From this, (6.53), and (6.55), E(f) =Ep(∞)(f) = lim
n→∞Ep(n)(f) (6.58)
≤lim inf
n→∞ lim inf
N→∞ ENp(n)(fN)
≤lim inf
n→∞ lim inf
N→∞ EN(fN) = lim inf
N→∞ EN(fN).
By (A3)we see E(f) =E(f), which together with (6.58) implies (6.57).
In the next lemma, we prove Definition 6.17 (2).
Lemma 6.33. Assume(M3’)and(A3). Then for anyf ∈L2(µ) there exists a sequence gN ∈L2(µN) (N ∈N) such that gN converges tof strongly and
Nlim→∞EN(gN) =E(f). (6.59) Proof. Letfn,N denote the sequence in (6.56). Note that from (6.53) we obtain
E(f) = lim
n→∞Ep(n)(f) (6.60)
Combining (6.56) and (6.60), we can takegN satisfying (6.59) by choosing a subsequence of {fn,N}n,N∈N.
Proof of Theorem 6.7. Lemma 6.32 and Lemma 6.33 implies the Mosco convergence.
The rest of the proof is same as that of Theorem 6.6.
Proof of Theorem 6.8. (M3”) implies (M3’). Hence Theorem 6.8 follows from Theo-rem 6.7.
6.6 Examples.
In this section, we give examples of dynamical universality. All examples satisfy the assumptions in Theorem 6.6–Theorem 6.10, and main theorems are thus applicable to these examples.
6.6.1 Universality of Airy interacting Brownian motions.
The first example is Airy random point field with β = 2. Let µAi be the Airy random point field with β = 2, that is, µAi is the determinantal random point field on S = R whose kernel with respect to the Lebesgue measure is given by
KAi(x, y) = Ai(x)Ai′(y)−Ai(x)′Ai(y)
x−y .
Here, Ai(x) is the Airy function. The corresponding ISDE is [54]
dXti=dBti+ lim
r→∞
{( ∑
j̸=i, |Xjt|<r
1 Xti−Xtj
)−
∫
|x|<r
ςAi(x)
−x dx }
dt. (6.61) Here we set
ςAi(x) = 1(−∞,0)(x) π
√−x.
It is known that ISDE (6.61) has a pathwise unique, strong solution X = (Xi)i∈N for µAi-a.s.s=∑∞
i=1δXi 0 [54].
To introduce theN-particle system, we setl∈Nand κ2l>0. Let V(x) =
∑2l
i=0
κixi. ConsiderµNAi,V such that
µNV,Ai(dsN) = 1 Z
∏N i<j
|si−sj|2
∏N k=1
exp(−N V(N−2l1(cN (
1 + s
αNN23 )
+dN)))dsN. Here,αN,cN,dN are constants depending only on V andN. By integration by parts, we can easily calculate the logarithmic derivative of µNAi,V and show that the corresponding N-dimensional SDE is
dXtN,i =dBti+ ∑
1≤j̸=i≤N
1
XtN,i−XtN,jdt
− N13−2l1cN 2αN
V′(N−2l1(cN(1 + XtN,i αNN23
) +dN))dt.
LetρN,nAi,V be then-correlation function ofµNAi,V. Then the soft edge universality result is proved by [9], which asserts
Nlim→∞ρN,nAi,V =ρnAi compact uniformly.
In [31] (A3) is proved. We can thus apply Theorem 6.8 and Theorem 6.10, which show the convergences (6.13) and (6.19).
6.6.2 Universality of Ginibre interacting Brownian motions at strong non-Hermiticity.
In this section we apply our result to random matrix model with strong non-Hermiticity introduced in [1].
LetS =R2. We naturally regardR2 asCby (x, y)7→x+√
−1y. The Ginibre random point field µgin is the determinantal random point field with kernel with respect to the Lebesgue measure such that
Kgin(x, y) = 1
πexp[−|x|2+|y|2
2 +xy].¯
Then by definition thek-point correlation function ρkgin is given by ρkgin(x1, . . . , xk) = det[Kgin(xi, xj)]ki.j=1.
It is known that the Ginibre random point fieldµgin satisfies(A3) (see [31]).
Let J(N) be the space of the normal matrices of order N. For constants γ ≥ 0, Kp ∈R, and τ ∈[0,1), we consider probability onJ(N) whose density is given by
σ(J) = 1 Z exp
[− N
1−τ2Tr(J J∗− τ
2(J2+J∗2))−γ(TrJ J∗−N Kp)2 ]
Then the joint density of eigen values is proportional to
∏N i̸=j
|zi−zj|2×exp[− N 1−τ2(
∑N i=1
|zi|2−τ 2
∑N i=1
(zi2+ ¯zi2))−γ(
∑N i=1
|zi|2−N Kp)2] (6.62)
Let ρkN be the k-point correlation function of the eigen value density corresponding to (6.62). Then there existsc41,c42,c43>0 depending onKp,γ,τ such that withE ={z∈ C;c41(ℜz)2+c42(ℑz)2 <1} the following universality holds:
Theorem 6.34 ([1, Theorem 1]). For anyζ ∈E,k∈N
Nlim→∞
1
Nρ1N(ζ) = c43
π 1E(ζ).
As for local densities we have 1
(c43N)kρkN(ζ+√z1
c43N, . . . , ζ+ zk
√c43N) =ρkgin(z1, . . . , zk) +O( 1
√N).
From this theorem we obtain (M3”). Hence we obtain (6.14) and (6.19).
From (6.62) we easily calculate the logarithmic derivative of N-particle system, and obtain theN-dimensional SDE corresponding to µNGin as follows:
dXti=dBit+1 2
{ ∑
1≤j̸=i≤N
2(Xti−Xtj)
|Xti−Xtj|2
}− τ N
1−τ2(ζ+ Xti
√c43N) 1
√c43N
+ τ N
1−τ2(ζ+ Xti
√c43N)† 1
√c43N
−(ζ+ Xti
√c43N) 2γ
√c43N {∑N
k=1
ζ+ Xtk
√c43N
2−N Kp }
dt (i=1,. . . ,N).
Here we set (x, y)†= (x,−y)∈R2. The ISDE corresponding toµGinis the following [47]:
dXti =dBti+ lim
r→∞
∑
|Xti−Xtj|<r,j̸=i
Xti−Xtj
|Xti−Xtj|2dt (i∈N).
We thus see the representation of N-particle SDE is quite complicated, and the limit is very simple and universal.
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