Inﬁnite-dimensional stochastic diﬀerential equations
related to Airy random ﬁelds
2012/9/25/Tue Okayama• Random matrices and log gasses• Soft edge scaling limit and Airy random ﬁelds• General theory for ISDEs:
quasi-Gibbs property & log derivative• Examples:
Sine RPF,
Bessel RPF,
Airy RPF,
Ginibre RPF,
All canonical Gibbs measures

Gaussian ensembles & semi-circle law• The dist of eigen values of the G(O/U/S)E RandomMatrices are given by (β = 1, 2, 4)∏β ∑N
1− 4 i=1 |xi|2Nβmβ (dxN ) =|xi − xj | edxN ,ZN(1)i<j• The distribution ofN −1N∑δxiunder mN
βi=1

converge the semi-circle law
1
ς(x)dx =2π√4 − x2dx(2)2Sine rpf (Dyson’s model)–Bulk scaling limit∑
1∏|xi|2− β4 N
N
βi=1dxN ,mβ (dxN ) =|xi − xj | eZNi<j1
ς(x)dx =2π√4 − x2dx√• Take xi = si/ N in (1) and set
∏
2 /4N
1∑
−β|s|NβkedsNµsin,β (dsN ) =|si − sj |ZNNi<j(3)k=1• The associated N particle system is given by the SDE:

30p

β1β∑idtiidt−XdXt = dBt +tj
24N
Xi − XNj̸=it(4)t• So the ass ∞ particle system is given by∞

1β∑iidXt = dBt +dtji
2X −X
j̸=itt3

Airy rpf – Soft edge scaling limit

Airy rpf: µAi,β (S = R, β = 1, 2, 4)√Take the scaling xi 7→ 2 N + siN −1/6 in∏β ∑N
1− 4 i=1 |xi|2N
βmβ (dxN ) =|xi − xj | edxN
ZNi<jand set√∏β ∑N
1− 4 i=1 |2 N +N −1/6si|2βN|si − sj | eµAi,β (dsN ) =dsN .
Z
Ni<jThen µAi,β is the TDL of µN
Ai,β :
lim µNAi,β = µAi,βN →∞4

Airy rpf – Soft edge scaling limit• β = 2 ⇒ µAi,β is the det rpf gen by (KAi, dx):
Ai(x)Ai′(y) − Ai′(x)Ai(y)KAi(x, y) =x−yHere Ai(·) the Airy function such that∫3/3)
1i(zk+kAi(z) =dk e,2π Rz ∈ C.

(5)

If β = 1, 4, the correlation func of µAi,β are given bysimilar formula of quaternion determinant.

5

Airy rpf – Soft edge scaling limit• From√∏β ∑N
1− 4 i=1 |2 N +N −1/6si|2N
β|si − sj | eµAi,β (dsN ) =dsNZi<jwe deduce the SDE of the N particle system:
βiidXt = dBt +2• Problem:
DoesN∑
j=1,j̸=i1β1i}dt1/3dt−{N+Xtj1/3i22NX −XttWhat is the limit SDE?

lim {N →∞N∑11/3}−Nji−XXt
j=1,j̸=i t

converge ?

How to solve the limit SDE?
6

Airy rpf – Soft edge scaling limit

Thm 1 (with Tanemura). Let β = 1, 2, 4. Then:• the limit ISDE is∫
∑
ϱ(x)
β1
iidXt = dBt + lim {()−dx}dtj
2 r→∞|x|<r −x
Xti − Xtjj̸=i, |Xt |<r√ϱ(x) =−x1(−∞,0](x)π• The above SDE has a unique, strong solution.
• So far the sto dyn related to Airy RPF was constructed only for
β = 2 by Spohn, Johansson, and others by the method of space-time
cor funs. This sto dyn is same as the above.
• (Xti) is a diﬀusion with state space RN.
∑• Xt = i δX i is reversible w.r.t. µairy,β .
t7General theorems for Inﬁnite-dim SDE: set up

Let S = Rd, C, [0, ∞).
S: Conﬁguration space over S∑
S = {s =δsi ; si ∈ S, s(|s| < r) < ∞ (∀r ∈ N)}iµ: RPF on S. i.e. prob meas. on S.
Prob: (1) To construct a natural stochastic dynamics
Xt = (Xti)i∈N
related to µ, i.e.
∑Xt =δX ii∈N(labeled dynamics)(unlabeled dynamics)

tis reversible w.r.t. µ.
(2) To ﬁnd the ∞-dim. SDE that Xt satisﬁes.
8

General theorems for Inﬁnite-dim SDE: set up• ρn is called the n-correlation function of µ w.r.t. Radonm. m if∫∫ ∏mn∏s(Ai)!
nm(dxi) =dµρ (xn)k1(s(Ai) − ki)!
A1 ×···×Akmm
Si=1i=1

for any disjoint Ai ∈ B(S), ki ∈ N s.t. k1 + . . . + km = n.
• µ is called the determinantal RPF generated by (K, m)
if its n-correlation fun. ρn is given by
ρn(xn) = det[K(xi, xj )]1≤i,j≤n• Ginibre RPF S = C. µgin is generated by (Kgin,2, g)
Kgin,2(x, y) = exȳ2−1−|x|dxg(dx) = π e
9Gibbs measure• Ψ: Ruelle’s class interaction potential,
Qr = {|x| ≤ r}, πr (s) = s(· ∩ Qr ), πrc(s) = s(· ∩ Qcr )c(s) = π c (ξ))µm(·)=µ(π∈·|s(Q)=m,πrrrrr,ξ

• µ is called (Φ, Ψ)-Gibbs m. if it satisﬁes DLR eq:m∏

1 −Hr (s)−Wr,ξ (s)mdµr,ξ =ee−Φ(sk )dsk
zr,ξ
k=1
Hr =

∑

Ψ(si − sj ), Wr,ξ =si,sj ∈Qr ,i<j∑Ψ(si − ξj )si∈Qr ,ξj ∈Qcr

• Let Ψ(x) = −2 log |x|.
Then, Wr,ξ diverge, so DLR does not make sense10(Φ, Ψ)-Quasi Gibbs measures(Φ, Ψ)-Gibbs m.

∏mmLet νr = k=1 1Qr (sk )e−Φ(sk )dsk

1 −Hr −Wr,ξ m
mdµr,ξ = m edνrzr,ξ

(DLR eq)(Φ, Ψ)-quasi Gibbs m. ∃ cmr,ξ
−1 −Hr mm ≤ cm e−Hr dν mcmedν
≤µr
rr,ξr,ξr,ξ
m diverge. But e−Wr,ξ /z m conv.
• If µ is Airy RPF, Wr,ξ and zr,ξr,ξ−1−Wr,ξ mm/z
≤ccm
≤er,ξr,ξr,ξ

• Quasi-Gibbs is very mild restriction. If µ is (Φ, Ψ)quasi-Gibbs m, then µ is also (Φ + f, Ψ)-quasi Gibbs mfor any loc bdd m’able f .
11

Main theorems: Unlabeled level construction
Let D be the canonical square ﬁeld on S: s =∑i δsi , s = (si).

1∑∇si f˜(s) · ∇si g̃(s)D[f, g](s) =2iµLet D be the set of local smooth fun with E1 (f, f ) < ∞.

∫E µ(f, g) =D[f, g]dµ

SThm 2. [O.96[CMP], 10[JMSJ], 12?[AOP](1) If µ is quasi-Gibbs with upper semi-cont potentials (Φ, Ψ), then(E µ, D, L2(S, µ)) is closable.
(2) If (E µ, D, L2(S, µ)) is closable & all correlation fun are loc bounded,then a diﬀusion Xt associated with the closure (E µ, Dµ) exists.
∑If µ is Poisson rpf with Lebesgue intensity, then Xt = i δB i .
t12Log derivative of µ – SDE representation of the stochastic dynamics• Let µx be the (reduced) Palm m. of µ conditioned at xµx(·) = µ(· − δx|s(x) ≥ 1)
• Let µ1 be the 1-Campbell measure on Rd ×S:∫µ1(A×B) =ρ1(x)µx(B)dx
A• dµ ∈ L1(Rd ×S, µ1) is called the log derivative of µ if∫∫∇xf dµ1 = −f dµdµ1 ∀f ∈ C0∞(Rd) ⊗ D
Rd×S

Rd×S

Here ∇x is the nabla on Rd, D is the space of bounded,local smooth functions on S.
• Very informallydµ = ∇x log µ1
13Main theorems: Inﬁnite-dim SDE(A1) µ is a quasi-Gibbs measure.
(closability)∑∞(A2) k=1 kµ(Skr ) < ∞, σrk ∈ L2(Srk , dx)(quasi-regular)
Here Sr = {|x| < r}, Skr = {s(Sr ) = k}, σrk is k-density fun on Sr .
1) exists(A3) The log derivative dµ ∈ L1(µ(SDE rep)loc(A4) {Xti} do not collide each other (non-collision)(A5) each tagged particle Xti never explode (non-explosion)∑N
Let u : S → S such that u((si)) = i δsi .

14

Main theorems: labeled diﬀusions

Thm 3. (O.12(PTRF)) (A1)–(A5) ⇒ ∃S0 ⊂ S such thatµ(S0) = 1,(6)and that, for ∀s ∈ u−1(S0), ∃u−1(S0)-valued pr. (Xti)i∈Nand ∃S N-valued Brownian m. (Bti)i∈N satisfying∑
1iiidXt = dBt + dµ(Xt ,δ j )dt, (X0i )i∈N = s(7)X2tj̸=i15

Main theorems: labeled diﬀusions

1iidXt = dBt + dµ(Xti,2∑δ
j̸=ij )dt,Xt(X0i )i∈N = sThm 4 (O. (JMSJ 10)). The family of processes {(Xti)i∈N}
is a diﬀusion with state space u−1(S0) ⊂ S N.
Remark 1. (1) (A1)–(A5) can be checked for Ginibre RPF (β = 2),
Sine RPFs, Airy RPFs and Bessel RPFs (β = 1, 2, 4).
(2) We can calculate the log derivatives of these measures.
(3) We have general theorems for quasi-Gibbs property and the log
derivatives (O. PTRF12, to appear in AOP, preprint). The statements are too messy to be omitted here.

16

unique, strong solution

H1 = {(x, s) ∈ S × S ; dµ(x, s) is locally Lips cont. }Here “locally” means we regard dµ(x, s) as symmetric fun
on Sr with ﬁxed particles outside Src for ∀r except a capacity zero set (non-single points, say).
Let H = {δx + s ; (x, s) ∈ H1} Assume(A6) Capµ(Hc) = 0.
Thm 5 (with Tanemura). Assume (A1)–(A6). Then theSDE has a unique, strong solution for initial starting∑
Npoints (si) ∈ S such that i δsi ∈ H q.e..
Remark: (1) It is quite likely that all determinantal rpfs in continuous
spaces satisfy (A1)–(A6).
(2) It is likely that ∑the conclusion of Thm 5 holds for all initial pointss = (si) such that i δsi ∈ H. (in progress)
17

Uniqueness of Dirichlet formsµLet Dpoly be the closure of the set of polynomials on Sµsuch that E1 (f, f ) < ∞. ThenµDpoly ⊂ Dµbecause polynomials are local and smooth.
Thm 6 (with Tanemura). Assume (A1)–(A6). Then theµDirichlet form that are extension of (E µ, Dpoly ) is unique.
µIn particular, Dpoly = Dµ, and Lang’s construction andOsada’s construction are same.
Remark 2. If (A5) (non-explosion) does not hold. Then Thm 6 does
not hold. This is very natural theorem that says the uniqueness ofDirichlet forms is related to the non-explosion problem of taggedproblem.
18

Examples: Gibbs measures

All example below satisfy (A1)–(A6). Hence by Thm 5
we have a unique, strong solution.
Gibbs measures :• All Gibbs measures with Ruelle’s class potentials (smoothoutside the origin) satisfy the assumptions (A.1)–(A.6).
Non-collision (A4) does not hold in general. But it always holds ford ≥ 2 and, for repulsive interaction Ψ in d = 1.

• In this case, the SDEs become∑11
jdXti = dBti − ∇Φ(Xti)dt −∇Ψ(Xti − Xt )dt.
22(8)j̸=i19Examples:

Ruelle’s class potentials

Lennard-Jones 6-12 potential
Let Φ6,12(x) = c{|x|−12 − |x|−6}, where d = 3 and c > 0is a constant. Φ6,12 is called the Lennard-Jones 6-12potential. The corresponding ISDE is:
ciidXt = dBt +2∞∑j{

j=1,j̸=i12(Xti − Xt )j|Xti − Xt |14

j−6(Xti − Xt )j|Xti − Xt |8}dt(i ∈ N)

Coulomb like potentials (not Coulomb!)
Let a > d and set Φa(x) = (c/a)|x|−a, where c > 0. Thenthe corresponding ISDE is:
ciidXt = dBt +2∞∑jXti − Xt

i − X j |a+2|Xtt
j=1,j̸=idt(i ∈ N).

(9)20

Examples:

Ruelle’s class potentials

Coulomb like potentials (not Coulomb!)
Let a > d and set Φa(x) = (c/a)|x|−a, where c > 0. Thenthe corresponding ISDE is:
ciidXt = dBt +∞∑j

Xti − Xt

i − X j |a+2
2|Xtt
j=1,j̸=idt(i ∈ N).

(10)

At ﬁrst glance the ISDE (10) resembles Ginibre IBMs,
because these corresponds to the case a = 0 in (10).
The sums in the drift terms, however, converge absolutely, unlike Coulomb (log) potentials. We emphasizethat the structures of the dynamics given by the solutions of (10) and Ginibre IBMs are completely diﬀerent
from each other.
21

Examples:

Ginibre rpf

Ginibre rpf: Ψ(x) = −β log |x| d = 2, β = 2. If µ = µgin,2,∑ji
Xt − Xt
iidXt = dBt + lim
dtjr→∞
i − X |2|X
jtt
|X i−X |<r,j̸=it(11)tand also∑ji
Xt − Xtiii
dXt = dBt − Xt dt + lim
dt.
jr→∞|Xti − Xt |2j|X |<r j̸=i(12)tThis comes from the plural expressions of dµgin,2 .
For ﬁnite N , these SDEs give diﬀerent solution.
But in the limit N → ∞ give the same solution if theinitial distribution is closed to Ginibre rpf.
22

Examples: Bessel rpf–hard edge scaling limit

Bessel RPF (joint work with Honda):
S = [0, ∞), β = 2, a > 1dXti = dBti +β ∑a1
dt + limdtijr→∞ 22Xt
Xti − Xt
j|Xt |<rj̸=iβ = 1, 4 are in progress.

23

Examples: sine rpf (Dyson’s model)–bulk scaling limit

Sineβ RPF: S = R, β = 1, 2, 4βiidXt = dBt + lim
2 r→∞

∑

1i − Xj
Xjt|X i−X |<r, j̸=i tt

dtt

Spohn (1987) considered the case β = 2:
∑1iidXt = dBt +dt
ji
j̸=i Xt − Xt
He constructed the dynamics as a Markov semigr by Dirichlet form.
The def of µ = µsin,β :β = 2 ⇒ µsin,β is the det rpf generated by (Ksin, dx):sin(π(x − y))Ksin(x, y) =π(x − y)β = 1, 4 ⇒ the correlation funs are given by quaternion det.
24

Examples: Airy rpf – Soft edge scaling limit

Thm 7 (with Tanemura). Let β = 1, 2, 4. Then:• The log derivative dµAi,β is∫∑
1
ϱ(x)µAi,βd(x, s) = β lim {()−dx}r→∞x − si|x|<r −x|x−si|<rHere√−xϱ(x) =1(−∞,0](x)π• Airy rpf µAi,β satisfy (A1)–(A6) and the limit ISDE is∫∑β1
ϱ(x)iidXt = dBt + lim {(dx}dt)−j
2 r→∞|x|<r −x
Xti − Xtjj̸=i, |Xt |<r

25

Examples: Airy rpf – Soft edge scaling limit• The key idea is to take the rescaled semi-circle law ς,N,1
as the ﬁrst approximation of the 1-correlation fun ρAi,β .
• Our method can be applied to other soft edge scaling.
Our result is the ﬁrst time to clarify the SDE describing
the limit inﬁnite system for the soft edge.

26

Examples: Airy rpf – Soft edge scaling limit

Thm 8 (with Tanemura). Assume β = 2.
Let us label Xti > Xti+1 (∀i).
• The top particle Xt1 is the Airy process A(t) in the senseof Spohn.
• The inﬁnite dim stochastic dynamics constructed bySpohn, Johansson & others by the space-time correlation
fun is a solution of the prescribed SDE:dXti = dBti +βlim {(r→∞2∑1∫)−jXt − Xtj
j̸=i, |Xt |<riϱ(x)dx}dt|x|<r −x27

Examples: Airy rpf – Soft edge scaling limit• The SDE gives a kind of Girsanov formula.
• These examples are the ﬁrst time that the inﬁnite dynamics are constructed for rpf appeared in random matrix
theory with β = 1, 4 even if the bulk and the hard edge
as well as the soft edge scaling
In one dimensional system, the method of space-timecorrelation functions are available (Nagao, Katori-Tanemura,Spohn, and others), but this method is restricted toβ = 2.
• By construction, if the total system start from the Airyβ
rpf µAi,β , then the distribution of the top particle Xt1
equals Fβ,edge(x), the β Tracy-Widom distribution, whereβ = 1, 2, 4.
28

To sum up

Thm 9. Ginibre RPF (β = 2), Sine RPFs, Airy RPFs
(β = 1, 2, 4) and Bessel RPFs (β = 2) are quasi-Gibbs
m. for Ψ(x) = −β log |x|, and the log derivative can becalculated. The associated ISDE has a unique, strongsolution.
Remark 3. Virág et all have been constructed the RPF for
all β on Dyson, Airy and Bessel RPFs (called β ensemble).
It is quite likely that these RPFs satisfy our assumptions(A1)–(A6). But unfortunately, they have not yet provethe existence of correlation functions for these models.
Only an existence of TDL has been established!
It is important to prove these are quasi-Gibbs measuresand to calculate the log derivative.
29

Thank You !

30

To sum up• The key point of the proof is to use the small ﬂuctuation property(SFP) of linear statistics for these measures.
• SFP was established by Soshnikov (Sine, Airy, Bessel RPFs), Shirai(Ginibre RPF).
• Proof consists of several parts:(1) To ﬁnd a good ﬁnite particle approximation {µN }(2) To prove uniform small ﬂuctuation of {µN }(3) To prove uni bounds of 1 & 2 cor funs of {µN }(4) To carry out the limiting procedure of dµN & quasi-Gibbs property
by using general theorems. (O. 11,12)31

Derivation of (4): (EµNsin,βµN
E sin,β (f, g) =, L2(µNsin,β ))∫D[f, g]µNsin,β (dx),µN
E sin,β (f, g) =

1=−2

∫

D[f, g] =N
∑1
∂f ∂g

2i=1∂si ∂si∏2 /4N1∑β−β|s|k|si − sj |edsN
D[f, g]Z∫{∆f +N∑i̸=jN

Ni<jk=1∑ βsk ∂fβ ∂f−}gsi − sj ∂si2N ∂skN

k=1∏2 /4N1∑β−β|s|k|si − sj |edsN
Z∫=−N

Ni<jk=11β∑ ∑ 1si ∂f{ ∆f +[()−]}g µNsin,β (ds)22si − sj
2N ∂siNNi=1

j̸=i32