fluctuations for linear statistics for determinantal ......biorthogonal ensembles (1) ... orthogonal...

Fluctuations for linear statistics for determinantal processes

through recurrence coefficients

Maurice Duits, KTH, Stockholm

Based on joint works with J. Breuer and R. Kozhan

1.Introduction

Gaussian Unitary Ensemble (1)

For a smooth function consider the linear statistic where, for now, the points are the eigenvalues of a (properly scaled) GUE matrix.

Then, as , the global distribution is given by

In this talk, we will be interested in the limiting behavior of the fluctuations

Xn(f) =nX

j=1

f(xj)

x1, . . . , xn

f

n ! 1

Xn(f)� EXn(f)

1

n

EXn(f) !1

⇡

Z p2

�p2f(x)

p2� x

2dx


Macroscopic CLT: For the fluctuations we have as , for sufficiently smooth functions , where and Johansson ’98

No normalization! Only for sufficiently smooth functions. One dimensional section of a 2d Gaussian Free Field.

Xn(f)� EXn(f) ! N(0,�(f)2)

�2f =

1X

k=1

k|fk|2

fn ! 1

fk =

1

⇡

Z ⇡

0f(p2 cos ✓) cos k✓d✓


Take , and a function of compact support. Consider*

Mesoscopic CLT: As , wherePastur-Shcherbina, Boutet-de-Monvel-Khorunzhy, Fyodorov-Khoruzhenko-Simm, Bourgade-Erdös-Yau-Yin, Lambert.

Xn,↵(f) =nX

j=1

f(n↵(xj � x

⇤))

0 < ↵ < 1 fx

⇤ 2 (�p2,p2)

Xn,↵(f)� EXn,↵(f) ! N(0,�(f))

�(f)2 =1

4⇡

ZZ ✓f(x)� f(y)

x� y

◆2

dxdy

n ! 1x

⇤

⇠ n�↵

2


Dyson (1962). Consider a Hermitian matrix for which the entries (up to symmetry) evolve according to independent Ornstein-Uhlenbeck processes. Then the ordered eigenvalues form a non-colliding process

The invariant measure is the eigenvalue distribution for the GUE.

What about multi-time fluctuations in the stationary situation?

Gaussian Free Field fluctuations for the random surface.


Choose a number of vertical sections for

Multi-time Global CLT: As , for sufficiently smooth functions where and

�(f)2 =NX

m1,m2=1

1X

k=1

e�|⌧m1�⌧m2 |kkf (m1)k f (m2)

k ,

Xn(f) =NX

m=1

nX

j=1

f(⌧m, xj(⌧m))

n ! 1 f

⌧m m = 1, . . . , N

f (m)k =

1

⇡

Z ⇡

0f(m,

p2 cos ✓) cos k✓d✓

Xn(f)� EXn(f) ! N(0,�(f)2)

Borodin, D.

2. Biorthogonal Ensembles

Biorthogonal Ensembles (1)

Consider a probability measure on defined by where is a Borel measure and are some square integrable functions with respect to

Appear in: Unitary Ensembles of random matrix theory, non-colliding process, random tilings of planar domains…..

We will consider fluctuations of the linear statistics where are random from the biorthogonal ensemble.

1

Zndet (�j(xk))

nj,k=1 det ( j(xk))

nj,k=1 dµ(x1) · · · dµ(xn)

µ �j j

µ

Rn

Xn(f) =nX

j=1

f(xj)

x1, . . . , xn

Biorthogonal Ensembles (2)An important ingredient is Andreiéf’s identity that states

By taking linear combination of the the rows in the determinants and the fact that the above is probability measure, we can assume, without loss of generality, that This also implies that

Further manipulations show that the biorthogonal ensemble is a determinantal point process with correlation kernel and reference measure . But we will not use this!

Z�j(x) k(x)dµ(x) = �jk

Zn = n!

Kn(x, y) =nX

j=1

�j(x) j(y)

µ

Zn =

Z· · ·

Zdet�j(xk) det j(xk)dµ(x1) · · · dµ(xn)

= n! det

Z�j(x) k(x)dµ(x)

Recurrence Matrices

Our main assumption will be that the functions and are part of large system of biorthogonal functions and , for which there exists a banded matrix such that

This means that the satisfy a finite term recurrence

x

0

B@�0(x)�1(x)

...

1

CA = J

0

B@�0(x)�1(x)

...

1

CA

�j j

{�j}j { j}j

Z j(x) k(x)dµ(x) = �jk

J

�j

x�j(x) =X

|k|⇢

Jj,j+k�j+k(x)

Orthogonal polynomials and Jacobi matrices

An example of such a system are the Orthogonal Polynomial Ensembles. In that case are the orthonormal polynomials. That is, the polynomials of degree with positive leading coefficients satisfying

It is classical that these polynomials satisfy a three terms recurrence relation. There exists and such that

The recurrence matrix is tridiagonal and usually referred to as the Jacobi operator/matrix.

Other examples include Multiple Orthogonal Polynomial Ensembles.

�j(x) = j(x) = pj(x)

Zpj(x)pk(x)dµ(x) = �jk

xpk(x) = ak+1pk+1(x) + bk+1pk(x) + akpk�1(x)

{ak}k ⇢ (0,1) {bk}k ⇢ R

j

Moments and cumulants (1)

The moment generating function for a linear statistic is given by

By Andreief’s identity we have

By the recurrence assumption we can write (at least for polynomial f)

EheitXn(f)

i= det

✓Zeitf(x)�

j

(x) k

(x)dµ(x)

◆n

j,k=1

EheitXn(f)

i= det

⇣eitf(J )

⌘n

j,k=1

EheitXn(f)

i= det

⇣I + Pn(e

itf(J ) � I)Pn

⌘

EheitXn(f)

i=

1

n!

Z· · ·

ZnY

`=1

eitf(x`) det�j

(xk

) det j

(xk

)dµ(x1) · · · dµ(xn

)

Moments and Cumulants (2)

The cumulants are defined by the following (formal) expansion

By using the identitywe can write each cumulant as a (slightly complicated) sum:

log det(1 +A) = Tr log(1 +A) =

1X

j=1

(�1)

j+1

jTrAj

mX

j=1

(�1)j+1

j

X

l1+···+lj=m

li�1

TrPnf(J )l1Pnf(J )l2Pn · · · f(J )lmPn

l1!l2! · · · lm!

Cm(Xn(f)) =

logEhe

itXn(f)i=

1X

m=1

(it)m

m!

Cm(Xn(f))

m!

Universality of fluctuations

One can show that if is a polynomial then the cumulants only depend on very few entries of the recurrence matrix J. More precisely, the m-th cumulant only depends on the entrieswhere depends on and the degree of but not on .Theorem (Breuer-D, to appear in JAMS)Given two biorthogonal ensembles, with recurrence matrices and such that for all and we have Then, for any polynomial and for each , as ,

Jn+j,n+k, |j|, |k| M.

J J

limn!1

⇣Jn+k,n+` � Jn+k,n+`

⌘= 0,

k `

E [(Xn(f)� EXn(f))m]� E

h⇣Xn(f)� EXn(f)

⌘mi! 0

m n ! 1f(x)

f(x)

M fm n

Theorem (Breuer-D, to appear in JAMS)

Suppose that we have a biorthogonal ensembles with a recurrence matrix satisfying

Then for any polynomial as ,

where

In the symmetric case , we also formulate general and weak conditions to allow functions of polynomial growth.

Central Limit Theorem

limn!1

Jn+k,n+` = ak�`

Xn(f)� EXn(f) ! N

0,

1X

k=1

kfkf�k

!n ! 1

fk =1

2⇡i

I

|z|=1f

0

@X

j

ajzj

1

A dz

zk+1

f(x)

C1�j = j

3. Special cases

Example: OPRL (1)

As the first consider the OP’s for

Recurrence

Jacobi operator

We allow varying measures so all objects may depend on

1

n!(det pj(xk))

2 dµ(x1) · · · dµ(xn)

xpk(x) = ak+1pk+1(x) + bk+1pk(x) + akpk�1(x)

n

J =

0

BBB@

b1 a1 0 . . .a1 b2 a2 . . .0 a2 b3 . . ....

......

. . .

1

CCCA

⇠Y

1i<jn

(xi � xj)2dµ(x1) · · · dµ(xn)

�j = j = pj�1 µ

Example: OPRL (2)

We say that is a right-limit of if along a subsequence

To every right limit there is a limiting distribution for the fluctuations along the same subsequence. A Jacobi matrix can have different right limits!

The CLT applies to the case when we have a right limit that is constant along the diagonal, i.e. a Laurent matrix. That is, we need both limits

J R J k, l 2 Z�J R

�kl

= limj!1

Jnj+k,nj+l

nj

an+k = a(n)n+k ! a bn+k = b(n)n+k ! b

Example: OPRL (3)

The CLT theorem particularly applies to orthogonal polynomial ensembles in the one-cut case.

For non-varying measures we have the:Denissov-Rakhmanov Theorem: If is a compactly supported measure for which the essential part of the support is a single interval and the absolutely continuous part has a density that is strictly positive a.e. on the essential part. Then

For varying measure related to Unitary Ensembles, we have limits of the recurrence coefficients for one-cut situations.

Also many discrete examples from tiling models, fall in this class.

[b� 2a, b+ 2a]

an ! a bn ! b

µ

Mesoscopic scales: Take , and

Theorem (Breuer-D, to appear in CMP)If, for some , Then for any , where

Example: OPRL (4)

Xn,↵(f) =nX

j=1

f(n↵(xj � x

⇤))

Xn,↵(f)� EXn,↵(f) ! N(0,�(f))

�(f)2 =1

4⇡

ZZ ✓f(x)� f(y)

x� y

◆2

dxdy

a > 0 b 2 R x

⇤ 2 [b� 2a, b+ 2a]

� > ↵

an = a+O(n��) bn = b+O(n��)

f 2 C1c

0 < ↵ < 1

2

Example: OPUC (1)

Consider now a measure on the unit circle Given a measure on the unit circle, consider the probability measure proportional to

Define the orthonormal functions by applying Gramm-Schmidt to and set

Then the probability measure can be written as

µ

⇠Y

1i<jn

|zi � zj |2dµ(z1) · · · dµ(zn)

j(z) = �j(z)

{1, z, z�1, z2, z�2, . . .}

�j

1

n!det�j(zk) det j(zk)dµ(z1) · · · dµ(zn)

{z 2 C | |z| = 1}

Example: OPUC (2)

Then there exists a penta-diagonal matrix such that This is called the CMV matrix with respect to where and are the Verblunsky coefficients

z

0

B@�1(z)�2(z)

...

1

CA = C

0

B@�1(z)�2(z)

...

1

CA

C

µ

C =

0

BBBBBBBB@

↵0 ↵1⇢0 ⇢0⇢1 0 0 0 · · ·⇢0 �↵0↵1 �↵0⇢1 0 0 0 · · ·0 ↵2⇢1 �↵1↵2 ↵3⇢2 ⇢2⇢3 0 · · ·0 ⇢1⇢2 �↵1⇢2 �↵2↵3 �↵2⇢3 0 · · ·0 0 0 ↵4⇢3 �↵3↵4 ↵5⇢4 · · ·0 0 0 ⇢3⇢4 �↵3⇢4 �↵4↵5 · · ·· · · · · · · · · · · · · · · · · · · · ·

1

CCCCCCCCA

,

⇢j =q

1� |↵j |2 ↵j

Example OPUC (3)

Theorem (D-Kozhan, appearing soon on the arXiv) Suppose that is such that Then for any such that Then where is an explicit expression.Related to Toeplitz determinants on an arc.

↵j ! ↵

µ

f(z) =X

k

fkzk

X

k

p|k||fk| < 1

Xn(f)� EXn(f) ! N(0,�(f,↵)2)

�(f,↵)2

4. Extended Biorthogonal Ensembles

Extended Biorthogonal Ensembles

Now consider a probability measure on of the form

Here we assume that the operatordefined byis bounded.

We can assume:

1

Zndet (�j,1(x1,k))

nj,k=1

N�1Y

m=1

det (Tm(xm,i, xm+1,j))ni,j=1

⇥ det ( j,N (xN,k))nj,k=1

NY

m=1

nY

j=1

dµm(xm,j),

(Rn)N

Tmf(y) =

Zf(x)Tm(x, y)dµm(x),

Tm : L2(µm) ! L2(µm+1)

Z j,N (x)TN�1TN�2 · · · T1�j,1(x)dµN (x) = �jk,

Recurrence matrices

Then define the functions which are biorthogonal as well

The marginals for fixed m gives a biorthogonal ensemble with ,

We assume that there exists a banded matrix such that

�j,m = Tm�1 · · · T1�j,1, j,m = T ⇤m · · · T ⇤

N�1 j,N ,

Z�j,m(x) j,m(x)dµm(x) = �jk,

x

0

BBB@

�0,m(x)�1,m(x)�2,m(x)

...

1

CCCA= Jm

0

BBB@

�0,m(x)�1,m(x)�2,m(x)

...

1

CCCA.

Jm

�j,m j,m

Example: Dyson’s Brownian Motion (1)

As an example we can take where stands for the Hermite polynomials and the transitions are given by the Mehler kernelwhere and

Tm(x, y) = (2⇡)

�1/2exp

✓��

2m(x

2+ y

2) +�mxy

1��

2m

◆

dµm

(x) = dµ(x) = e�x

2

dx�m = e�(⌧m+1�⌧m)

�j,1 = e�⌧1(j�1)Hj�1 j,N = e⌧N (j�1)Hj�1

Hj

⌧1 ⌧N⌧2

Example: Dyson’s Brownian motion (2)

Then and hence we obtain the recurrence matrix

This construction can be generalized in a straightforward way. The Hermite functions are eigenfunctions for the generator of the Ornstein-Uhlenbeck processes. Similar statements are true for Laguerre and Jacobi polynomials. In the discrete case one can turn to birth/death processes, Meixner, Charlier, Krawtchouk.

�j,m = e�⌧m(j�1)Hj�1 j,m = e⌧m(j�1)Hj�1

(Jm)jk = e�⌧m(j�k) (JHermite)jk

Consider linear statistics

Theorem (D. arXiv) Suppose we have two different extended biorthogonal ensembles such that Then for any such that it is a polynomial in andas

Universality

limn!1

✓(Jm)n+k,n+l �

⇣Jm

⌘

n+k,n+l

◆= 0,

Eh(Xn(f)� EXn(f))

ki� eE

⇣Xn(f)� eEXn(f)

⌘k�! 0,

f(x,m)x

Xn(f) =NX

m=1

nX

j=1

f(m,xj,m)

k 2 N

n ! 1

Theorem (D. arXiv) Suppose that for each Then for any that is a polynomial in , as ,where

Central Limit Theorem

limn!1

(Jm)n+k,n+l = a(m)k�l,

f (m)k =

1

2⇡i

I

|z|=1f(m,

X

j

a(m)j zj)

dz

zk+1.

N

0, 2

NX

m1=1

NX

m2=m1+1

1X

k=1

kf (m1)k f (m2)

�k +NX

m=1

1X

k=1

kf (m)k f (m)

�k

!,

Xn(f)� EXn(f) !

f(m,x)

m, k, `

x n ! 1


Choose a number of vertical sections for

Multi-time Global CLT: As , for sufficiently smooth functions where and

Xn,↵(f)� EXn,↵(f) ! N(0,�(f))

�(f)2 =NX

m1,m2=1

1X

k=1

e�|⌧m1�⌧m2 |kkf (m1)k f (m2)

k ,

Xn(f) =NX

m=1

nX

j=1

f(⌧m, xj(⌧m))

n ! 1 f

⌧m m = 1, . . . , N

f (m)k =

1

⇡

Z ⇡

0f(m,

p2 cos ✓) cos k✓d✓

Further comments

In the symmetric situation, we can allow for more general test functions in the linear statistic under general conditions.

There is also a version where we let N grow with n. Some of the sums are then Riemann sums giving integrals in the limit.

It covers examples of non-colliding processes for which the classical orthogonal polynomial ensembles are the invariant measure. In the stationary case, one can rigorously show that the CLT explains a convergence of the fluctuation of the associated random surface to the Gaussian Free Field.

It can also be applied to tiling models, such as lozenge tilings of a regular hexagon.

Literature

J. Breuer and M. Duits, Central Limit Theorems for biorthogonal ensembles and asymptotics of recurrence coefficients, accepted for publication in Journal of the AMS

J. Breuer and M. Duits, Universality of mesoscopic fluctuation in orthogonal polynomial ensembles, accepted for publication in Comm. Math. Phys.

M. Duits, On global fluctuations for non-colliding process, arXiv:1510:08248

M. Duits and R. Kozhan, Asymptotics of ratios of Toeplitz determinants via Verblunsky coefficients, in preparation

G. Lambert, CLT for biorthogonal ensembles and related combinatorial identities, arXiv:1511.06121

http://arxiv.org/abs/1511.06121

Thank you for you attention!

fluctuations for linear statistics for determinantal ......biorthogonal ensembles (1) ... orthogonal...

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