On global fluctuations of non-colliding processes
Maurice Duits
March 21, Warwick
1.Motivation
Some examples of non-colliding processes or non-intersecting
paths in the class of determinantal point processes
Global fluctuations via multi-time linear statistics
Choose a number of horizontal
sections ⌧m for m
= 1, . . . , N
and consider the linear statistic
N X
n
X
X
n (f ) =
f (⌧m , xj (⌧m ))
m=1 j=1
In this talk we will be interested
in the behavior of the fluctuations
of Xn (f ) as n ! 1
We will discuss N=1 separately.
2. Some example of results for linear
statistics
Linear statistics and the GUE
For a smooth function f consider the linear statistic Xn (f ) =
n
X
f (xj )
j=1
where, for now, the points x1 , . . . , xn are the eigenvalues of a
(properly scaled) GUE matrix.
Then, as n ! 1 , the global distribution is given by 1
1
EXn (f ) !
n
⇡
Z
p
2
p
2
p
f (x) 2
x2 dx
We will be mostly interested in the limiting behavior of the
fluctuations
Xn (f )
EXn (f )
Linear statistics and the GUE
Central Limit Theorem: For the fluctuations we have as n ! 1 , for sufficiently smooth functions f ,
EXn (f ) ! N (0, (f )2 )
Xn (f )
where 2
f
and =
k=1
1
fk =
⇡
1
X
Johansson ’98,….
Z
⇡
k|fk |2
p
f ( 2 cos ✓) cos k✓d✓
0
No normalization! Only for sufficiently smooth functions. One dimensional
section of a 2d Gaussian Free Field.
Linear statistics and the GUE
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?
Stationary DBM
Choose a number of horizontal sections ⌧m for m = 1, . . . , N
Xn (f ) =
N X
n
X
f (⌧m , xj (⌧m ))
m=1 j=1
Multi-time Global CLT: As n ! 1, for sufficiently smooth functions f , EXn (f ) ! N (0, (f )2 )
Xn (f )
where
(f )2 =
1
X
N
X
e
|⌧m1
⌧m2 |k
(m1 ) (m2 )
fk ,
kfk
m1 ,m2 =1 k=1
and
(m)
fk
Borodin, D.
1
=
⇡
Z
⇡
f (m,
0
p
2 cos ✓) cos k✓d✓
3. Biorthogonal Ensembles
Single time fluctuations
Biorthogonal Ensembles
n
R
Consider a probability measure on
defined by 1
det (
Zn
n
n
j (xk ))j,k=1 dµ(x1 ) · · · dµ(xn )
j (xk ))j,k=1 det (
where µ is a Borel measure
functions with respect to µ
j
and
j
are some square integrable
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 Z
j (x) k (x)dµ(x)
This also implies that Zn = n!
=
jk
Recurrence Matrices
Our main assumption will be that the functions j and j are part of
large system of biorthogonal functions { j }j and { j }j , Z
j (x) k (x)dµ(x)
=
jk
for which there exists a banded matrix J such that 0
B
x@
0 (x)
C
1 (x)A
..
.
This means that the
x
1
j (x)
j
0
B
=J@
0 (x)
1
C
1 (x)A
..
.
satisfy a finite term recurrence
=
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 j (x) = j (x) = pj (x) are the orthonormal
polynomials. That is, the polynomials of degree j with positive leading
coefficients satisfying Z
pj (x)pk (x)dµ(x) = jk
There exists {ak }k ⇢ (0, 1) and {bk }k ⇢ R such that
xpk (x) = ak+1 pk+1 (x) + bk+1 pk (x) + ak pk
1 (x)
The recurrence matrix, or Jacobi operator, is symmetric tridiagonal
0
b1
Ba1
B
J =B0
@
..
.
a1
b2
a2
..
.
0
a2
b3
..
.
1
...
. . .C
C
. . .C
A
..
.
Relevant part of the recurrence matrix (1)
Limit shape: If is a polynomial then 1
lim
(EXn (f )
n!1 n
Tr Pn f (J )Pn ) = 0
This means that the expected value of the moments of the empirical
measure only depend on coefficients in an upper right block of size.
J =
(
(n + O(1)) ⇥ (n + O(1))
)
Relevant parts of the recurrence matrix (2)
Fluctuations: The moments of the fluctuations
E [(Xn (f )
m
EXn (f )) ]
Ẽ
h⇣
Xn (f )
ẼXn (f )
of the moments of the empirical measure only depend on O(1) size
block around n,n entry
More precisly, The m-th moment only depends on the entries
Jn+j,n+k ,
|j|, |k|  M.
where M depends on m and the degree f of but not on n .
J =
(
)
⌘m i
!0
Universality of fluctuations
Theorem (Breuer-D, to appear in JAMS)
Given two biorthogonal ensembles, with recurrence matrices J and J˜
such that for all k and ` we have ⇣
⌘
˜n+k,n+` = 0,
lim
J
J
n+k,n+`
n!1
Then, for any polynomial f (x) and for each m , as n ! 1 ,
E [(Xn (f )
m
EXn (f )) ]
Question: What is the typical
behavior in the red blocks?
Ẽ
h⇣
Xn (f )
J =
ẼXn (f )
(
⌘m i
!0
)
Central Limit Theorem
Theorem (Breuer-D, to appear in JAMS) Suppose that we have a biorthogonal ensemble with a recurrence matrix
satisfying
lim Jn+k,n+` = ak
n!1
`
Then for any polynomial f (x) as n ! 1 ,
Xn (f )
where
EXn (f ) ! N
0,
1
X
kfk f
k=1
k
!
0
1
I
X
1
j A dz
@
fk =
f
aj z
k+1
2⇡i |z|=1
z
j
´In the symmetric case j = j , we also formulate general and weak
conditions to allow C 1 functions of polynomial growth.
Example GUE
The eigenvalues of a GUE matrix have the 1 ⇣
det (pj
n!
n
(x
))
1 k j,k=1
n
⌘2 Y
e
nx2j
dxj
j=1
where pj (x) are rescaled and normalized Hermite polynomials, i.e.
Z
nx2
pj (x)pk (x)e
dx = jk
The (rescaled) Hermite polynomials have there recurrence r
r
k+1
k
xpk (x) =
pk+1 (x) +
pk 1 (x)
2n
2n
Hence we find the following recurrence matrix and the relevant limit
0
p0
B 1
1 B
J = p1 B
B
J =p
2n
nB
B
@
p
1
p0
2
p
2
0
p
3
p
3
..
.
..
.
1
C
C
C
C
.. C
.C
A
(J )n+k,n+l =
(
p1
2
0
|k
l| = 1
otherwise
Remarks on the generality
This theorem gives a criterium for which the model has Gaussian
limiting fluctuation for the linear statistics. Work is still needed to show
that the criteria is practical.
For models involving classical functions / orthogonal polynomials, the
criteria is often satisfied and can be simply looked up on any standard
reference on special functions.
Often, the question of finding leading term in the asymptotics for the
recurrence coefficients is an intrinsically easier question that finding the
full asymptotic description for the correlation kernel.
In certain cases there are general result ensuring that the criterium is
fulfilled for a remarkably wide class of models.
The criterium fails in multi-cut cases since the recurrence coefficients
have quasi-periodic behavior. The fluctuations are not always gaussian.
Pastur ’06, Borot-Guionnet ’13, Shcherbina ’13
Example: OPRL
As the first consider
j
=
j
= pj
1
the OP’s for µ
Y
1
2
(xi
(det pj (xk )) dµ(x1 ) · · · dµ(xn ) ⇠
n!
1i<jn
xj )2 dµ(x1 ) · · · dµ(xn )
Recurrence
xpk (x) = ak+1 pk+1 (x) + bk+1 pk (x) + ak pk
1 (x)
Jacobi operator
0
b1
Ba1
B
J =B0
@
..
.
a1
b2
a2
..
.
0
a2
b3
..
.
1
...
. . .C
C
. . .C
A
..
.
We allow varying measures so all objects may depend on n
Example: OPRL
For non-varying measures we have the:
Denissov-Rakhmanov Theorem: If µ is a compactly supported
measure for which the essential part of the support is a single
interval [b 2a, b + 2a] and the absolutely continuous part has a
density that is strictly positive a.e. on the essential part. Then an ! a
bn ! b
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.
DBM with random initial condition
Take as the initial configuration the eigenvalues
of a matrix take randomly from a unitary ensemble with a convex potential V
1
e
n
n Tr V (M )
dM
Then run DBM and consider the distribution
of the points at time t . These have the same
distribution as the eigenvalues of t
e MV +
p
1
e
2t M
GUE
(Recall talk by Kuijlaars on Monday)
In this case, we have multiple orthogonal
polynomials with wider band recurrence matrices
but we still find a Toeplitz structure in the limit with relative ease, see Breuer-D. DBM with two initial points
Now consider DBM with two starting points ↵1 and ↵2 . We assume
that the number of paths n is even and that half of the points start at
each point.
(Equivalent picture
of brownian bridges)
↵1
↵2
Multiple Hermite polynomials
The locations at time t form a biorthogonal ensembles constructed by
Multiple Hermite Polynomials (Johansson ’02, Bleher-Kuijlaars ’04)
1
Pk1 ,k2 (x) = p
2⇡i
Z
1
Qk1 ,k2 (x) = p
2⇡2⇡i
(z
k1
a1 ) (z
k2
a2 ) e
1
2 (z
x)2
dz.
iR
I
(w
a1 )
k1
(w
a2 )
k2
e
1
2 (w
x)2
dz.
a1 ,a2
The Pk1 ,k2 is a polynomial of degree k1 + k2.
Then, up to a rescaling and re-parametrization, the points at time t
form a biorthogonal ensemble with
{
n
}
j j=1 = {P0,0 , P1,0 , P1,1 , P2,1 , . . .}
{
n
}
j j=1 = {Q1,0 , Q1,1 , Q2,1 , Q2,2 , . . .}
Recurrence relations for Multiple Hermite Polynomials
One can show that xPk1 ,k2 (x) = Pk1 +1,k2 (x) + a1 Pk1 ,k2 (x) + k1 Pk1
1,k2 (x)
+ k2 Pk1 ,k2
1 (x)
We can exchange indices according to the rule
Pk1 +1,k2 (x) = Pk1 ,k2 +1 (x) + (a2
a1 )Pk1 ,k2 (x).
Then we have the following recurrence relation
xPk1 ,k2 (x) = Pk1 +1,k2 (x) + a1 Pk1 ,k2 (x) + (k1 + k2 )Pk1 ,k2
xPk1 ,k2 (x) = Pk1 ,k2 +1 (x) + a2 Pk1 ,k2 (x) + (k1 + k2 )Pk1
1 (x)
1,k2 (x)
(a2
(a1
a1 )k1 Pk1
a2 )k1 Pk1
1,k2 1 (x).
1,k2 1 (x).
Limit of the recurrence matrix
After some computation by plugging in the recurrences we find
lim
n!1
where
and
⇣
⌘
(J (n) )n+k,n+l
0
A0
B A1
B
J
=
J B
@
A1 = (e
t
A 1
A0
A1
t
e )
✓
(J)n+k,n+l = 0.
A 1
A0
..
.
✓1
2 (a1
◆
A
..
1
..
.
a2 )
.
C
C
C
A
1
1
2 (a2
0
a1 1
A0 = e
+ (et
0 a2
✓
◆
t 0 0
A 1=e
1 0
t
1
✓
0 0
e )
1 0
t
a1 )
◆
◆
A more general Central Limit Theorem
The CLT of Breuer-D. can be generalized into
Theorem
Suppose there exists and such that ⇣
⌘
(n)
lim
(J
)n+k,n+l (J)n+k,n+l = 0.
n!1
and that for a polynomial we can decompose p(J) = p(J)+ + p(J)
such that 1) p(J)+ is lower triangular,
2) p(J)
is upper triangular,
3) [p(J)+ , p(J) ] is trace class
Then, as n ! 1, we have
Xn (p)
D
EXn (p) ! N (0, Tr[p(J)+ , p(J) ]) ,
Some comments
If J is a Toeplitz matrix, the conditions are satisfied for every
polynomial and therefore the aforementioned theorem in Breuer-D is a
direct consequence of this theorem. This follows by simple identities for
Toeplitz matrices.
If J is a block Toeplitz matrix, then this generically fails, although it may
be true for some polynomials (e.g. in the symmetric two-cut case for
unitary ensemble, even polynomials still satisfy a CLT)
However, in the case of multiple Hermite polynomials, the condition is
true for every polynomial! This is because of a very particular structure
in the blocks.
It is an interesting problem for the theory of multiple orthogonal
polynomials to characterize conditions under which this happens.
CLT for DBM with two starting points
Theorem [D. upcoming]
Let f be a polynomial. Then EXn (f ) ! N
N 0,
0,
Xn (f )
where 1
fk =
2⇡i
and
A1 = (et
I
e t)
✓
✓1
2 (a1
◆
a2 )
0
kTrTr(f(fk kff kk))
k=1
p (zA1 + A0 + A2 /z)
a1 1
A0 = e
+ (et
0 a2
✓
◆
t 0 0
A 1=e
1 0
t
1
X
X
!!
1
1
2 (a2
dz
z k+1
◆
a1 )
✓
◆
0 0
t
e )
1 0
4. Extended Biorthogonal Ensembles
Multi-time fluctuations
Extended Biorthogonal Ensembles
Now consider a probability measure on (Rn )N of the form
N
1
Y
1
n
n
det
(
(x
))
det
(T
(x
,
x
))
j,1
1,k
m
m,i
m+1,j
j,k=1
i,j=1
Zn
m=1
N Y
n
Y
n
⇥ det ( j,N (xN,k ))j,k=1
dµm (xm,j ),
m=1 j=1
We assume that the operators
Tm : L2 (µm ) ! L2 (µm+1 )
defined by
Tm f (y) =
are bounded.
Z
f (x)Tm (x, y)dµm (x),
Recurrence matrices
We assume that the functions are biorthogonal in the sense
Z
j,N (x)TN
1 TN
Then define the functions
j,m
= Tm
1
2
· · · T1
j,m
· · · T1
k,1 (x)dµN (x)
and j,1 ,
=
jk ,
j,m
j,m
= Tm⇤ · · · TN⇤
1
j,N ,
which are biorthogonal as well
Z
j,m (x) k,m (x)dµ(x)
=
jk
The marginals for fixed m gives a biorthogonal ensemble with
respect to the functions j,m and j,m .
Recurrence matrices
We assume that
that
Z
j,1
and
j,N (x)TN
k,N
1 TN
2
can be embedded into families such
· · · T1
k,1 (x)dµN (x)
=
jk ,
and that there exists band matrices
Jm
0
B
B
xB
@
0,m (x)
1
0
C
B
B
C = Jm B
2,m (x)A
@
..
.
1,m (x)C
0,m (x)
1
C
C.
2,m (x)A
..
.
1,m (x)C
The matrices may depend on n , except for the bandwidths.
Universality
Consider linear statistics Xn (f ) =
N X
n
X
f (m, xj,m )
m=1 j=1
Theorem (D. arXiv) Suppose we have two different extended
biorthogonal ensembles such that for each m
⇣
⌘
lim (Jm )n+k,n+l (J˜m )n+k,n+l = 0
n!1
Then for any f (x, m) such that it is a polynomial in
h
E (Xn (f )
as n ! 1
EXn (f ))
k
i
e
E
⇣
Xn (f )
x and
k 2 N
e n (f )
EX
⌘k
! 0,
Central Limit Theorem
Theorem (D. arXiv) Suppose that for each m, k, `
lim (Jm )n+k,n+l =
n!1
(m)
ak l ,
Then for any f (m, x) that is a polynomial in x, as n ! 1 ,
Xn (f )
N
EXn (f ) !
0, 2
N
X
N
X
1
X
(m1 ) (m2 )
f k
kfk
m1 =1 m2 =m1 +1 k=1
where
(m)
fk
1
=
2⇡i
I
f (m,
|z|=1
+
N X
1
X
m=1 k=1
X
j
(m) (m)
f k
kfk
dz
(m) j
aj z ) k+1 .
z
!
,
Example: Stationary DBM
In stationary DBM we have
1
det (
Zn
n
j,1 (x1,k ))j,k=1
det Ptm+1
n
j,N (xN,k ))j,k=1
⌧1 (j 1)
=e
⌧N (j
=
e
j,N
j,1
n
i,j=1
N Y
n
Y
dµ
(xm,j ),
dxmm,j
m=1 j=1
where
tm (xm,i , xm+1,j )
m=1
⇥ det (
N
Y1
Hj
1)
Hj
p
1( nx)
p
1 ( nx)
e
nx2
where Hj stands for the j-th Hermite polynomial and
1
Pt (x, y) = p
e
2⇡n
1
n
e 2t
(e
t
x y)2
is the transition probability for the OU process.
Example: Dyson’s Brownian motion (2)
Then j,m (x)
= Tm
1
· · · T1
j,1 (x)
j,m
=e
⌧m (j 1)
Hj
1(
p
nx)
and hence we obtain the recurrence matrices
)jk = e
(J(Jmm)jk
⌧m (j k)
(JHermite )jk
Multi-time Global CLT: As n ! 1, for sufficiently smooth functions f EXn (f ) ! N (0, (f )2 )
Xn (f )
where
(f )2 =
N
X
1
X
e
|⌧m1
⌧m2 |k
(m1 ) (m2 )
fk ,
kfk
m1 ,m2 =1 k=1
and
(m)
fk
1
=
⇡
Z
⇡
f (m,
0
p
2 cos ✓) cos k✓d✓
Further comments
This construction can be generalized in a straightforward way. The
Hermite functions are eigenfunctions for the generator of the OrnsteinUhlenbeck processes. Similar statements are true other process for
which the generator has the classical orthogonal polynomials as
eigenfunctions. Such as Laguerre and Jacobi polynomials in the
continuous case and in the discrete case one can turn to birth/death
processes, Meixner, Charlier, Krawtchouk.
It can also be applied to tiling models, such as lozenge tilings of a
regular hexagon.
One can rigorously show that the CLT explains a convergence of the
fluctuation of the associated random height function to the Gaussian
Free Field.
By an extension we can also handle certain cases involving multiple
orthogonal polynomials:
Multi-time CLT for DBM with two starting points
Theorem [D. upcoming]
Let f (⌧, x) be a polynomial in x . Then Xn (f )
N
0, 2
N
X
EXn (f ) ! N
N
X
1
X
Tr (fk f
k=1
⇣
(m1 ) (m2 )
f k
k Tr fk
m1 =1 m2 =m1 +1 k=1
0,
X
⌘
+
k)
!
N X
1
X
⇣
(m) (m)
f k
k Tr fk
m=1 k=1
⌘
!
where (m)
fk
1
=
2⇡i
I
f
|z|=1
and
(m)
A1
(m)
A0
A
(m)
1
= (e
⌧m
⇣
(m)
⌧m , A1 z
e
✓
)
1
a2
✓
◆
0 0
= e ⌧m
1 0
=e
⌧m
a1
0
⌧m
✓1
2 (a1
◆
+
(m)
A0
a2 )
0
+ (e⌧m
+
(m)
A 1z 1
1
1
2 (a2
◆
a1 )
✓
◆
0 0
e ⌧m )
1 0
⌘ dz
z k+1
Thank you for you attention!