Dynamical Bulk Scaling limit of Gaussian Unitary Ensembles and ...

Dynamical Bulk Scaling limit of Gaussian Unitary Ensembles and Stochastic-Differential-Equation gaps Yosuke Kawamoto ; Hirofumi Osada

arXiv:1610.05969v1 [math.PR] 19 Oct 2016

October 20, 2016

Abstract The distributions of N -particle systems of Gaussian unitary ensembles converge to Sine2 point processes under bulk-scaling limits. These scalings are parameterized by a macroposition θ in the support of the semicircle distribution. The limits are always Sine2 point processes and independent of the macro-position θ up to the dilations of determinantal kernels. We prove a dynamical counter part of this fact. We prove that the solution of the N -particle systems given by stochastic differential equations (SDEs) converges to the solution of the infinite-dimensional Dyson model. We prove the limit infinite-dimensional SDE (ISDE), referred to as Dyson’s model, is independent of the macro-position θ, whereas the N -particle SDEs depend on θ and are different from the ISDE in the limit whenever θ 6= 0. 1

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 N N definition, M N = [Mi,j ]i,j=1 is then an N × N matrix having the form ( ξi if i = j N √ √ √ Mi,j = τi,j / 2 + −1ζi,j / 2 if i < j, where {ξi , τi,j , ζi,j }∞ i 0 and ε > 0. Then   √ N −1 X N 2 . (3.1) ψk (y) = O 2 y − 2N k=0

(3) There is a positive constant c4 such that for all N ∈ N 1

sup |ψN (y)| ≤ c4 N − 12 .

(3.2)

y∈R

Proof. (1) follows from Lemma 5.2 (i) in [6]. (2) follows from Lemma 5.2 (ii) in [6]. From Lemma 6.9 in [17] there exists a constant c4 such that √ c4 1 1 2 |N 12 ψN (2 N + yN − 6 )| ≤ y ∈ [−2N 3 , ∞). 1 , (1 ∨ |y|) 4

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Hirofumi Osada, Yosuke Kawamoto

Hence we have |ψN (y)| ≤

c4 √ 1 1 , 1 N 12 (1 ∨ {N 6 |y − 2 N |}) 4

y ∈ [0, ∞).

(3.3)

Claim (3.2) is immediate from (3.3) and the well-known property such that ψN (y) = ψN (−y) if N is even and that ψN (y) = −ψN (−y) if N is odd.

3.2

Determinantal kernels of N-particle systems

We recall the definition of determinantal point processes. Let K : R2 → C be a measurable kernel. A probability measure µ on S is called a determinantal point process with kernel K if, for each n, its n-point correlation function is given by ρn (x1 , . . . , xn ) = det[K(xi , xj )]ni,j=1 .

(3.4)

If K is an Hermitian symmetric and of locally trace class such that 0 ≤ Spec(K) ≤ 1, then there exists a unique determinantal point process with kernel K [19, 20]. The distribution of the delabeled eigenvalues of GUE associated with (1.1) is a determinantal point process with kernel KN such that KN (x, y) =

N −1 X

ψk (x)ψk (y).

(3.5)

k=0

The Christoffel-Darboux formula and a simple calculation yield the following. r N ψN (x)ψN −1 (y) − ψN −1 (x)ψN (y) N . K (x, y) = 2 x−y

(3.6)

From the scaling (1.3), µN θ is a determinantal point process with kernel

Let xN =

1 N x + Nθ y + Nθ KN , √ ). θ (x, y) = √ K ( √ N N N

(3.7)

√ √ 1 1 LN (x, y) = √ KN (xN , yN ) = √ KN ( N x, N y). N N

(3.8)

√ √ N x and yN = N y. We set

From (3.7) and (3.8) we then clearly see that x y + θ, + θ), N N LN (x, y) = KN θ (N (x − θ), N (y − θ)). N KN θ (x, y) = L (

(3.9)

From (3.6) we deduce √ ′ ′ (xN ) − ψN (xN )ψN LN (x, x) = (1/ 2){ψN −1 (xN )ψN −1 (xN )}.

(3.10)

Using the Schwartz inequality to (3.5) we see from (3.6) and (3.8) that LN (y, z)2 ≤ LN (y, y)LN (z, z).

(3.11)

From here on, we assume − We set

2 1 0, and α > −2/3, we apply (3.1) to obtain c7 > 0 such that,

Z

lim sup N →∞

UN 2

Z

c7 LN (y, y)q

dy dy ≤ lim sup

= 0, N N q 2 q N R R |xθ − y| N →∞ U2 |xθ − y|N (y − 2)

which combined with (4.13) yields (4.12).

Lemma 4.4. (4.8) holds. √ √ 3 Proof. Let y = 2 cos τ . Then N sin3 τ ≥ N 2− 2 (2 2N α − N 2α ) for y ∈ UN 1 . Then applying Lemma 3.2 (1) we deduce that for each r > 0

LN (y, y)

dy − θ N N R N − y x N →∞ Tr,∞ (x)∩U1 θ √ N α r Z 2−N o

n Z xθ − N

1 1p

2 dy − θ = lim sup + 2 − y

√ r R xN N →∞ xN − 2+N α θ −yπ θ +N Z √2 1 1p = P.V. √ 2 − y 2 dy − θ = 0. − 2 θ−y π

Z

lim sup

Combining this with (4.12), we obtain (4.8).

N It is well known that KN θ (x, x) are positive and continuous in x, and {Kθ (x, x)}N ∈N converges √ to Kθ (x, x) = 2 − θ2 /π uniformly on each compact set. Then we see

sup sup N ∈N x∈SR

1 KN θ (x, x)

< ∞.

From this, (3.9), and (4.7), we see that the following constant c8 is finite. c8 := sup sup

N ∈N x∈SR

1 N LN (xN θ , xθ )

< ∞.

(4.14)

16

Hirofumi Osada, Yosuke Kawamoto

Lemma 4.5. (4.15) and (4.16) below hold. In particular, (4.9) holds.

Z

2 LN (xN

θ , y) lim lim sup dy

= 0, N N N N r→∞ N →∞ N (x) |x R Tr,∞ θ − y|L (xθ , xθ )

Z N N L (xθ , y)

dy = 0. lim lim sup N − y|LN (xN , xN ) r→∞ N →∞ R |x N Tr,∞ (x) θ θ θ

(4.15) (4.16)

Proof. From (3.11) and (4.12) we deduce that as N → ∞

Z

Z 2 LN (xN LN (y, y)



θ , y) dy dy → 0. ≤

N − y|LN (xN , xN ) N − y| N |x R R |x R\UN R\U θ θ θ θ 1 1

(4.17)

From (3.16) and (4.14) for each N ∈ N and r > 0

2 LN (xN

θ , y) dy ≤

N N N N |xθ − y|L (xθ , xθ ) R

Z

N (x)∩UN Tr,∞ 1

c26 c8 dy

Z

N 2 |xN θ − y|

N (x)∩UN Tr,∞ 1

c2 c8 ≤ 62 . r



3

R

Hence (4.15) follows from (4.17) and (4.18). This completes the proof of (4.15). We next prove (4.16). From (3.11), (4.12), and (4.14) we see for each r > 0

Z LN (xN

θ , y) dy lim sup

= 0. N − y|LN (xN , xN ) N R |x N N →∞ Tr,∞ (x)\U1 θ θ θ

From (3.16) and (4.14) we see that for each N ∈ N and r > 0

Z

Z c6 c8 dy LN (xN



θ , y) dy ≤



N N N N N 2 N N N (x)∩U N (x)∩U |xθ − y|L (xθ , xθ ) R N |xθ − y| R Tr,∞ Tr,∞ 1 1 ≤

(4.18)

(4.19)

(4.20)

2c6 c8 . r

Combining (4.19) and (4.20) we obtain (4.16). Lemma 4.6. (4.21) and (4.22) below hold. In particular, (4.10) holds.

Z

LN (y, y)

lim lim sup dy = 0, N 2 r→∞ N →∞ R N N |x − y| Tr,∞ (x) θ

Z LN (y, z)2

dydz lim lim sup

= 0. N − y||xN − z| r→∞ N →∞ R N (x)2 |x Tr,∞ θ θ

(4.21) (4.22)

N Proof. Note that LN (y, y) ≤ c5 on UN by (3.15). Then by the definition of Tr,∞ (x),

Z

N (x)∩UN Tr,∞

c 2N 2c LN (y, y) dy ≤ 5 = 5. 2 N r r N |xN − y| θ

(4.23)

1

By (3.14) we see LN (y, y) ≤ c5 N 3 on R. Recall that |BN | = 4N α by construction. Furthermore, −2 c9 := lim supN →∞ supy∈BN k|xN kR < ∞. Hence for each r > 0 θ − y| lim sup N →∞

Z

N (x)∩BN Tr,∞

1

c5 N 3 4N α c9 LN (y, y) dy ≤ lim sup = 0. 2 N N |xN N →∞ θ − y|

(4.24)

17

Hirofumi Osada, Yosuke Kawamoto

Here we used α < −1/2. We thus obtain (4.21) from (4.23) and (4.24). We proceed with the proof of (4.22). We first consider the integral away from the diagonal line. By (3.16) and the Schwartz inequality, we see that

Z LN (y, z)2

dydz

N − y||xN − z| R N (x)∩UN )2 ∩{|y−z|≥ 1 } |x (Tr,∞ θ θ N 2

Z

c6

≤ dydz

2 2 N − y||xN − z| R N (x)∩UN )2 ∩{|y−z|≥ 1 } N |y − z| |x (Tr,∞ θ θ N 2 1

n Z o c6 2

≤ dydz 2 2 N − y|2 N (x)2 ∩{|y−z|≥ 1 } N |y − z| |x Tr,∞ θ N nZ o 21 c26

dydz

2 |y − z|2 |xN − z|2 1 N 2 R N Tr,∞ (x) ∩{|y−z|≥ N } θ Z 2

c6

dydz = N 2 2 2 1 R N |y − z| |x − y| N 2 Tr,∞ (x) ∩{|y−z|≥ N } θ 2 2N n 2N o 4c6 = . ≤ c26 2 N r r

The last line follows from a straightforward calculation. Indeed, first integrating z over {|y − z| ≥ 1 N N }, and then integrating y over Tr,∞ (x), we obtain the inequality in the last line. We therefore see that

Z LN (y, z)2

dydz (4.25) lim lim

= 0. N − y||xN − z| r→∞ N →∞ 1 R |x N N 2 (Tr,∞ (x)∩U ) ∩{|y−z|≥ N } θ θ We next consider the integral near the diagonal. From (3.15), we see that

Z

LN (y, z)2

dydz

N − y||xN − z| 1 R |x N N 2 (Tr,∞ (x)∩U ) ∩{|y−z|≤ N } θ θ Z

c25

≤ dydz

N N 1 N (x)∩UN )2 ∩{|y−z|≤ R |xθ − y||xθ − z| (Tr,∞ N} 2

Z

c5 1 1

{ N ≤ + }dydz

N 2 2 1 2 |xθ − y| R |xθ − z| N (x)2 ∩{|y−z|≤ Tr,∞ N} 2 2 Z 2

4c5 2c 2c 2N 1

= . = 5 dy = 5 N 2 N N r r R N (x) |x Tr,∞ θ − y|

From (4.25) and (4.26), we have

Z

lim lim r→∞ N →∞

N (x)∩UN )2 (Tr,∞

LN (y, z)2

dydz

= 0. N − z| R |xN − y||x θ θ

We next consider the integral on BN × BN . Let c10 = lim sup

sup

N →∞ x∈SR ,y∈BN

(4.27)

−1 |xN . θ − y|

Then, we deduce from (3.14) and the definition of BN given by (3.13) that

Z LN (y, z)2

dydz lim sup

N N R N (x)∩BN )2 |x N →∞ (Tr,∞ θ − y||xθ − z| 2

(4.26)

≤ lim c25 c210 N 3 (4N α )2 = 0. N →∞

(4.28)

18

Hirofumi Osada, Yosuke Kawamoto Here we used |BN | = 4N α for the inequality and α < −1/2 for the last equality. We finally consider the case UN × BN . Then a similar argument yields

Z LN (y, z)2

dydz

N − y||xN − z| R |x N N N N (Tr,∞ (x)∩U )×(Tr,∞ (x)∩B ) θ θ Z

LN (y, y)LN (z, z)

≤ dydz N N R |x − y||x − z| N N N N (Tr,∞ (x)∩U )×(Tr,∞ (x)∩B ) θ θ Z

Z LN (z, z) LN (y, y)

dy dz = N N − z| R N (x)∩BN |x N (x)∩UN |x Tr,∞ Tr,∞ θ − y| θ 1

=O(log N )O(N 3 +α ) → 0

(4.29)

as N → ∞.

Collecting (4.27), (4.28), and (4.29), we conclude (4.22). Lemma 4.7. (4.11) holds. Proof. We shall estimate the three terms in (4.11) beginning with the first. From (3.11) and (4.21) we have

Z

2 LN (xN

θ , y) dy lim lim sup (4.30) N − y|2 LN (xN , xN ) R r→∞ N →∞ N N |x Tr,∞ (x) θ θ θ

Z LN (y, y)dy

≤ lim lim sup = 0. N − y|2 R r→∞ N →∞ N N |x Tr,∞ (x) θ

Next, using the Schwartz inequality, we have for the second term

Z N N LN (y, z)LN (xN

θ , y)L (xθ , z) dydz

N N N N R |xN N (x)2 Tr,∞ θ − y||xθ − z|L (xθ , xθ ) 1 Z

Z

1 N 2 N N 2 L (y, z) dydz 2 L (xθ , y)2 LN (xN

θ , z) dydz 2 ≤

N − y||xN − z| R N − y||xN − z|LN (xN , xN )2 R N (x)2 |x N (x)2 |x Tr,∞ Tr,∞ θ θ θ θ θ θ 1 Z

Z

N N 2 N 2 L (xθ , y) L (y, z) dydz 2

= dy . N − y||xN − z| R N − y|LN (xN , xN ) R N N 2 |x |x Tr,∞ (x) Tr,∞ (x) θ θ θ θ θ Applying (4.22) and (4.15) to the last line, we obtain

Z

lim lim sup

r→∞ N →∞

N (x)2 Tr,∞

N N LN (y, z)LN (xN θ , y)L (xθ , z) dydz

= 0. N N N N R |xN θ − y||xθ − z|L (xθ , xθ )

(4.31)

We finally estimate the third term. From (4.16), as N → ∞, we have

N N LN (xN θ , y)L (xθ , z) dydz N − y||xN − z|LN (xN , xN )2 R N (x)2 |x Tr,∞ θ θ θ θ

n Z o2 LN (xN , y) dy

θ =

N − y|LN (xN , xN ) R N (x) |x Tr,∞ θ θ θ

Z N N 2 L (xθ , y) dy

→ 0 by (4.16). = N − y|LN (xN , xN ) R |x N Tr,∞ (x) θ θ θ

Z

From (4.30), (4.31), and (4.32) we obtain (4.11). This completes the proof.

(4.32)

19

Hirofumi Osada, Yosuke Kawamoto

5

Proof of Theorem 1.1

From Lemma 4.4–Lemma 4.7 we deduce that all the assumptions (2.21)–(2.24) in Lemma 2.6 are satisfied. Hence (2.10) is proved by Lemma 2.6. Then Theorem 1.1 follows from Lemma 2.4, Lemma 2.5, and Lemma 2.6.

6

Proof of Theorem 1.2

In this section we prove Theorem 1.2 using Theorem 1.1. It is sufficient for the proof of Theorem 1.2 to prove (1.15) in C([0, T ]; Rm ) for each T ∈ N. Hence we fix T ∈ N. Let XN = (X N,i )N i=1 be as in (1.13). Let Y θ,N,i = {Ytθ,N,i } such that Ytθ,N,i = XtN,i + θt.

(6.1)

Then from (1.13) we see that Yθ,N = (Y θ,N,i)N i=1 is a solution of dYtθ,N,i = dBti +

N X

1

θ,N,i j6=i Yt

−

Ytθ,N,j

dt −

1 θ,N,i θ Y dt + dt N t N

(6.2)

with the same initial condition as XN . Let P θ,N and Qθ,N be the distributions of XN and Yθ,N on C([0, T ]; RN ), respectively. Then applying the Girsanov theorem [3, pp.190-195] to (6.2), we see that Z TX Z N N θ 1 T X θ 2 dQθ,N i (W) = exp{ dB − (6.3) dt} t dP θ,N 2 0 i=1 N 0 i=1 N = exp{

N θ2 T θ X i BT − }, N i=1 2N

θ,N where we write W = (W i ) ∈ C([0, T ]; RN ) and {B i }N are independent copies of i=1 under P Brownian motions starting at the origin.

Lemma 6.1. For each ǫ > 0,  dP θ,N  lim Qθ,N θ,N (W) − 1 ≥ ǫ = 0. N →∞ dQ

(6.4)

Proof. It is sufficient for (6.4) to prove, for each ǫ > 0,

  dQθ,N lim P θ,N θ,N (W) − 1 ≥ ǫ = 0. N →∞ dP

This follows from (6.3) immediately.

Proof of Theorem 1.2. We write Wm = (W 1 , . . . , W m ) ∈ C([0, T ]; Rm ) for W = (W i )N i=1 , where m ≤ N ≤ ∞. Let Qθ be the distribution of the solution Yθ with initial distribution µθ ◦ l−1 . From Theorem 1.1 and (6.1) we deduce that for each m ∈ N lim Qθ,N (Wm ∈ ·) = Qθ (Wm ∈ ·)

N →∞

weakly in C([0, T ]; Rm ). Then from this, for each F ∈ Cb (C([0, T ]; Rm )), Z Z lim F (Wm )dQθ,N = F (Wm )dQθ . N →∞

C([0,T ];RN )

C([0,T ];RN )

(6.5)

20

Hirofumi Osada, Yosuke Kawamoto

We obtain from (6.4) and (6.5) that lim

N →∞

Z

m

F (W )dP

C([0,T ];RN )

N,θ

= lim

N →∞

= lim N →∞ Z =

Z

Z

F (Wm ) C([0,T ];RN )

dP θ,N (W)dQθ,N dQθ,N

F (Wm )dQθ,N C([0,T ];RN )

F (Wm )dQθ .

C([0,T ];RN )

This implies (1.15). We have thus completed the proof of Theorem 1.2.

7

Acknowledgement

H.O. thanks Professor H. Spohn for a useful comment at RIMS in Kyoto University in 2002. H.O. is supported in part by JSPS KAKENHI Grant Nos. 16K13764, 16H02149, 16H06338, and kiban-a 24244010. Y.K. is supported by Grant-in-Aid for JSPS JSPS Research Fellowships (Grant No. 15J03091).

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Hirofumi Osada, Yosuke Kawamoto

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[12] Osada, H., Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials, Ann. of Probab. 41, 1-49 (2013). [13] Osada, H., Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials II : Airy random point field, Stochastic Processes and their Applications 123, 813838 (2013). [14] Osada, H., Osada, S., Discrete approximations of determinantal point processes on continuous spaces: tree representations and tail triviality, arXive:1603.07478-v3. [15] Osada, H., Tanemura, H., Strong Markov property of determinantal processes with extended kernels, Stochastic Processes and their Applications 126, 186-208, no. 1, (2016), doi:10.1016/j.spa.2015.08.003. [16] Osada, H., Tanemura, H., Infinite-dimensional stochastic differential equations and tail σfields, arXiv:1412.8674v5 [math.PR]. [17] Osada, H., Tanemura, H., Infinite-dimensional stochastic differential equations related to Airy random point fields, arXiv:1408.0632. ´ [18] Plancherel, M., Rotach, W., Sur les valeurs asymptotiques des polynomes dHermite Hn (x) = −x2 /2 n x2 /2 dn ), Comment. Math. Helv. 1, 227-254 (1929). (−1) e dxn (e [19] Shirai, T., Takahashi, Y., Random point fields associated with certain Fredholm determinants I: fermion, Poisson and boson point process, J. Funct. Anal. 205, 414-463 (2003). [20] Soshnikov, A., Determinantal random point fields, Russian Math. Surveys 55, 923-975 (2000). [21] Spohn, H., Interacting Brownian particles: a study of Dyson’s model, In: Hydrodynamic Behavior and Interacting Particle Systems, G. Papanicolaou (ed), IMA Volumes in Mathematics and its Applications, 9, Berlin: Springer-Verlag, pp. 151-179 (1987). [22] Tsai, Li-Cheng, Infinite dimensional stochastic differential equations for Dyson’s model, Probab. Theory Relat. Fields (published on line) DOI 10.1007/s00440-015-0672-2 (2015). Yosuke Kawamoto Faculty of Mathematics, Kyushu University, Fukuoka 819-0395, Japan [email protected] Hirofumi Osada Faculty of Mathematics, Kyushu University, Fukuoka 819-0395, Japan [email protected]