Topics in random matrix theory. Lecture 9. Fixed
t multi-point intensity functions for the real
Ginibre ensemble.
Oleg Zaboronski
23.05.2013
As an application of the formalism of spin variables we will show how to rederive the result of Borodin and Sinclair [1] for all the one-dimensional densities
of real eigenvalues of the real Ginibre ensemble in the limit N → ∞. Let us fix
time t = 1 and a set of multiple space points x1 < x2 < . . . < xK . The answer
for an arbitrary time t can be obtained from the t = 1 answer by the diffusive
re-scaling x → √xt . As we have done before, we will first compute the modified
density ρ̃N = ρ̃N (x1 , x2 , . . . , xK ) defined by
)
(K
∏
sxk (M1 )ΛM1 (dxk )
ρ̃N (x1 , x2 , . . . , xK ) dx1 . . . dxK = N E
k=1
As before, we choose right hand limits for the density, that is where the intervals
dxk denote infinitesimal intervals just to the right of the point xk . The answers
are zero for odd K, so we take even K throughout. Moreover it is convenient
to consider only even N throughout (which avoids us tracking various ± signs).
Recall that correlation functions of spins can be computed by integrating ρ̃ with
respect to space variables:
)
(K ∫
(K
)
∏ xk
∏
dyk ρ̃N (y1 , y2 , . . . , yk ).
(1)
E
sxk (M1 ) = (−2)K
k=1
k=1
−∞
Finally, the probability density functions of real eigenvalues can be computed as
explained in Lecture 7.
Equivalence with a correlation function of characteristic polynomials.
The integral over the Gaussian density
∫
K
∏
(2)
ρ̃N (x1 , x2 , . . . , xK ) dx1 . . . dxK =
dM γ1 (M )
sxk (M )ΛM (dxk )
RN 2
1
k=1
can be treated using the Edelman transform for the eigenvalue lying in dxK , and
+
after integrating over the half sphere SN
−1 , we obtain
ρ̃N (x1 , x2 , . . . , xK ) dx1 . . . dxK−1
)
(
K−1
∏
N −1
1
2
=
|SN −1 |π − 2 e−xK EN −1 det (M1 − xK I)
sxk (M1 )ΛM1 (dxk ) ,
2
k=1
where the subscript N − 1 on the EN −1 means that the averaging occurs over the
(N − 1) × (N − 1) Ginibre distribution. Another Edelman transform about the
eigenvalue lying in dxK−1 yields
ρ̃N (x1 , x2 , . . . , xK ) dx1 . . . dxK−2
N −1
N −2
1
2
2
=
|SN −1 ||SN −2 |π − 2 − 2 e−xK −xK−1 (xK−1 − xK )
4
(
)
K−2
∏
EN −2 det (M1 − xK I) det (M1 − xK−1 I)
sxk (M1 )ΛM1 (dxk ) .
k=1
A further (K − 2) applications of Edelman transform will lead to the following
expression for the modified density:
( K
)
K
)
∏
N −k
∆(x) ∏ (
2
det (M1 − xm I) (3)
ρ̃N (x1 , x2 , . . . , xK ) = K
|SN −k |π − 2 e−xk EN −K
2 k=1
m=1
∏
where ∆(x) = 1≤i<j≤K (xj − xi ) is the Van-der-Monde determinant. Therefore,
the problem of computing the modified density has been reduced to the computation of the expectation of the product of characteristic polynomials of the random
matrix M1 .
Integral representation for product of characteristic polynomials. Such
a computation has been carried out in [13] by exploiting Berezin integrals. Firstly,
as we explained in Lecture 8, for any N × N matrix A,
∫
det(A) =
dψdψeψAψ
R0|2N
Representing all determinants in (3) as Gaussian Berezin integrals, we get a Gaussian integral over (N − K) × (N − K) matrices M . This can be computed using
∫
1
T
T
2
T
dM e− 2 T rM M +T rM J = (2π)n /2 eT rJJ
Rn×n
The result has a term
T
eT r(AA ) ,
where A, A are skew-symmetric K ×K matrices constructed out of anti-commuting
vectors:
Aij = ψi · ψj , Aij = ψ i · ψ j .
2
The integration over ψ, ψ is carried out using Hubbard-Stratonovich transformation:
∫
†
T
†
T r(AT B)
e
=
dHdHe−T r(HH )+T r(HA )+T r(H B)
CK(K−1)/2
, where
dHdH =
∏ dImHij dReHij
π
i<j
and the integral is over the space of complex skew-symmetric K ×K matrices. The
resulting integral over anti-commuting variables is Gaussian which we compute
using
∫
T
dχeχ M χ .
P f (M ) =
R0|2n
After all these steps we arrive at the expression for the modified density as a
K(K − 1)-dimensional integral over Hubbard-Stratonovich variables:
( K
)
(
)N
]
∏
∏ [∫ dzpq dz pq
√1 Z
X
2
2
EN
det (M1 − xm I) =
e−|zpq | P f
. (4)
√1 Z †
−X
π
2
R
2
m=1
1≤p<q≤K
Here X is a diagonal K × K matrix with entries (x1 , x2 , . . . xK ); and Z is a skew
symmetric complex K × K matrix:

 zij i > j
0
i=j
Zij =

−zij i < j
Expression (4) can be neatly written as a matrix integral:
( K
)
(
∫
∏
K(K−1)
1
†
EN
det (M1 − xm ) = π − 2
λ(dZ, dZ † )e− 2 T rZZ P f
Q(K)
m=1
√1 Z
2
−X
X
1
√ Z†
2
)N
where Q(K) = {Z ∈ CK×K | Z T = −Z T } is the space of skew-symmetric complex
matrices, λ(Z, Z † ) is the Lebesgue measure on Q(K) as described above. Note that
the dimension of the integral in the right hand side of (5) is N -independent. The
size of the original matrix only enters the integral as the power of the Pfaffian in
the integrand. This allows one to calculate the large N -limit of (5) using Laplace
method. To facilitate the application√of asymptotic methods, let us re-scale the
integration variables using (Z, Z † ) → N (Z, Z † ), which gives
( K
)
∏
K(K−1)
K(K−1)
NK
NK
EN
det (M1 − xm ) = π − 2 2− 2 N 2 N 2 JN
(6)
m=1
where
∫
JN =

λ(dZ, dZ † )e− 2 T rZZ P f 
√
†
N
Q(K)
3
−
Z
√
2
X
N
2
X
N
Z†
N
.
(7)
, (5)
The integrand in JN is now of the form exp{N FN (Z, Z † )}, where FN is a slow
function of N .
Applying the Laplace method. We will show that the integral JN localizes
onto the subset
C (K) = {Z ∈ Q(K) | ZZ † = I}.
(8)
To do this it is convenient to split J into two parts,
∫
∫
JN,0 =
JN,1 =
Q(K) ∩S
Q(K) \S
where
S = {Z | µk (Z) ∈ [1/2, 2] for k = 1, . . . , K}
and (µk (Z) : 1 ≤ k ≤ K} are the singular values of Z. Note that S is compact
and contains only non-singular matrices. We first bound the integral JN,1 , aiming
to show that it is of smaller order that JN,0 . For c ≥ 0 write (λk (c) : 1 ≤ k ≤ 2K)
for the singular values of the matrix
(
)
Z
cX
−cX Z †
One may bound the difference of singular values via |µk (A) − µk (B)| ≤ ∥A − B∥
so that
|λk (c) − λk (0)| ≤ ∥cX∥ = cx∗
where x∗ = maxk |xk |. Then
(
)
(
)1/2
Z
cX
Z
cX
P f
= det
†
†
−cX Z
−cX Z
1/2
2K
∏
= λk (c)
k=1
1/2
2K
∏
≤ (λk (0) + cx∗ )
k=1
K
∏
=
(µk (Z) + cx∗ ).
k=1
Using this in JN,1 we find
∫
JN,1 ≤
−N
2
λ(dZ, dZ † )e
T rZZ †
Q(K) \S
∫
=
−T rZZ †
λ(dZ, dZ † )e
Q(K) \S
4
e
N
K
∏ (
)
µk (Z) + (2/N )1/2 x∗ k=1
∑
−N K
k=1 HN (µk (Z))
where
(
HN (z) =
1
1
−
2 N
)
(
1/2 )
2
z 2 − ln z + x∗ .
N
Note that HN (z) → H(z) = 21 z 2 − ln(z) and that H(z) has a minimal value of
H(1) = 21 . On Q(K) \ S there must exist at least one singular value µ lying outside
[ 12 , 2], and for this value H(µ) ≥ 12 + 2δ for an easily calculated δ > 0. For large
N , when Z ∈ Q(K) \ S, we have
K
∑
HN (µk (Z)) ≥
k=1
K
+δ
2
and the value of JN,1 is bounded by Ck e−N K/2 e−N δ . This is is exponentially smaller
than that of JN,0 , which we will see is, to leading exponential order, O(e−N K/2 ).
Next, we will calculate the asymptotic expansion of JN,0 for large N . The N th
power of the Pfaffian in the integrand of JN,0 can be simplified using the Taylor
expansion for the Pfaffian:
)
(
P f A + √1N B
1
= 1 + √ T rBA−1
(9)
P f (A)
2 N
(
)
)
3
1 (
+
T rBA−1 T rBA−1 − 2T rBA−1 BA−1 + O N − 2
8N
While we cannot pinpoint the exact reference for the original derivation of the
above expansion, it can be easily derived using the Berezin integral representation
of the Pfaffian. It is interesting to note that unlike the analogous determinant
expansion formula, the series (9) contains finitely many terms. With the help
of (9), and the fact that the terms with T r(BA−1 ) are zero in our case, we can
re-write the N th power of the Pfaffian in the integrand of JN,0 as follows:

√

(
)N
2
Z
X
( †
)
N
1
N
N 
†
−2
 = det 2 (ZZ ) · 1 + T r Z XZX + O(N )
√
Pf
2
†
N
−
X
Z
N
= det 2 (ZZ † ) · eT r(Z
N
† XZX
) (1 + O(N −1 ))
This allows us to express JN,0 in a form well suited for the application of Laplace
formula:
∫
N
†
†
†
JN,0 =
Λ(dZ, dZ † )e− 2 (T rZZ −ln det(ZZ )) eT r(Z XZX ) (1 + O(N −1 )). (10)
Q(K) ∩S
The fact that S does not contain degenerate matrices, and the compactness of S
allows one to pass the correction term through the integral. In the limit N → ∞,
5
the main contribution to (10) comes from the neighborhood of the points of global
minimum of the function
(
†
F (Z) = T rZZ − ln det ZZ
†
)
K
∑
(
=
)
µ2k (Z) − 2 ln(µk (Z)) .
(11)
k=1
The global minimum value of F is K and it is attained on the set C (K) of skewsymmetric unitary K × K matrices, which is a smooth sub-manifold of Q(K) . We
will show that C (K) is a non-degenerate critical set, which means that the Hessian
of F has the maximal possible rank at every point of C (K) . Therefore we can use
Laplace theorem [4] to calculate the asymptotic expansion of JN,0 : let (w, y) be
local co-ordinates on Q(K) such that the sub-manifold C (K) is locally determined
by the set of equations y = 0; then
∫
N
†
†
†
Λ(dZ, dZ † )e− 2 (T rZZ −ln det(ZZ )) eT r(Z XZX )
Q(K) ∩S
−N F |C (K)
= e
(
1
√
2πN
)dim(Q(K) )−dim(C (K) ) ∫
µ(dw)eT r(Z
† XZX
) . (12)
C (K)
Here µ(dw) is the measure on C (K) generated by the embedding C (K) ⊂ Q(K) and
integration over transverse co-ordinates y. Explicitly,
dim(C (K) )
∏
ρ(w, y = 0)
dµ(w) = √
(K)
|C
det Hess(F )
(w)
dwk ,
(13)
k=1
where ρ(w, y) is the density of Lebesgue measure Λ(dZ, dZ † ) with respect to local
co-ordinates (w, y), the Hessian is defined as the matrix of second derivatives with
respect to transverse co-ordinates y. In writing (12) we used the fact that the
critical manifold lies a positive distance away from the boundary of Q(K) ∩ S.
Noting that F takes the value K on C (K) and that
dim(Q(K) ) − dim(C (K) ) = K(K − 1) − (dim(U (K)) − dim(Sp(K)))
1
= K(K − 1) − K 2 + K(K + 1)
(14)
2
1
K(K − 1),
=
2
we reach
JN,0 = e
− N2K
−
(2πN )
K(K−1)
4
∫
µ(dw)eT r(Z
† XZX
) (1 + O(N −1 )).
(15)
C (K)
Collecting together (3), (6) and (15) we find
ρ̃N (x1 , x2 , . . . , xK ) = c2 (N, K)∆(x)
K
∏
−x2k
∫
e
k=1
6
C (K)
µ(dw)eT r(Z
† XZX
) (1 + o(1)) ,
where
c2 (N, K) = C(K)
K (
∏
|SN −k |π −
N −k
2
)
π−
K(K−1)
2
2−
(N −K)K
2
k=1
(N − K)
(N −K)K
2
(N − K)
K(K−1)
2
e−
(N −K)K
2
(2π(N − K))−
K(K−1)
4
and C(K) denotes a constant only depending on K. It is lengthy but straightforward to check that c2 (N, K) → c3 (K) > 0 as N → ∞ and hence that limiting
modified density ρ̃(x1 , x2 , . . . , xK ) = limN →∞ ρ̃N (x1 , x2 , . . . , xK ) exists and is given
by
∫
K
∏
†
−x2k
ρ̃(x1 , x2 , . . . , xK ) = c3 (K)∆(x)
e
µ(dw)eT r(Z XZX ) .
(16)
k=1
C (K)
The explicit value of c3 (K) will be determined later by using properties of the
densities ρ̃. In the next subsection we will find a parameterisation of the integral
in the right hand side of (16), which will allow us to calculate it very efficiently
using the standard tools of random matrix theory.
Recasting as an integral over the unitary group. An important property of
the function F is its invariance with respect to the following action of the unitary
group U (K) on Q(K) :
U (K) × Q(K) → U (K)
(U, Z) 7→ U ZU T
(17)
(18)
Namely, for any A ∈ U (K),
F (AZAT ) = F (Z).
The decomposition theorem for skew symmetric unitary matrices [8] states that
Z = U JU T ,
(19)
where U is a unitary matrix, J is the canonical symplectic matrix. Notice that
(19) does not determine the unitary matrix U uniquely: indeed Z → Z if U → U S,
where S is a unitary matrix: SJS T = J. The set of such matrices is a subgroup
of U (K) called symplectic group Sp(K):
Sp(K) = {S ∈ U (K) | SJS T = J}.
(20)
It can be checked that the critical manifold C (K) can be identified with the factor
space of U (K) with respect to the action of Sp(K) on U (K) via right multiplications:
C (K) ∼
= U (K)/Sp(K)
7
(21)
The U (K)-action (17) on Q(K) preserves the critical manifold and induces the
U (K)-action on C. Using parameterisation (19) of C (K) this induced action can
be written explicitly:
U (K) × C (K) → C (K) ,
(A, [U ]) 7→ [AU ],
(22)
where [U ] is an equivalence class of U ∈ U (K) with respect to right multiplications
by elements of Sp(K) ⊂ U (K). In the vicinity of a critical point Zc ∈ C (K) ,
1
F (Zc + δZ) = K + T r(δZZc† + Zc δZ † )2 + . . .
2
(23)
We notice that the quadratic form describing the second order term in the above
Taylor expansion of F is U (K)-invariant and has the maximal possible rank equal
to 12 K(K − 1) = dim(Q(K) ) − dim(C (K) ). We rewrite the integral from (16), using
the mapping (21), as
∫
∫
†
T r(Z † XZX )
µ(dw)e
=
µ̂(dU )eT r(Z(U ) XZ(U )X ) )
(24)
C (K)
U (K)/Sp(K)
where Z(U ) is given by (19) and µ̂(dU ) is the pull back of the measure µ on
the critical manifold. We can work out an explicit expression for µ in local coordinates on C (K) using the general formula (13). We will not do that. Instead
we will characterise µ̂ up to a multiplicative constant by establishing its symmetry
with respect to the U (N )-action on C (K) : recall that the measure µ is determined
by the Lebesgue measure on Q(K) and the determinant of the quadratic form in
the right hand side of (23). As it is easy to check,
(i.) The Lebesgue measure Λ and the quadratic form T r(δZZ † + ZδZ † )2 on Q(K)
are invariant with respect to the U (K)-action (17).
(ii.) The critical manifold C (K) is invariant with respect to the U (K)-action.
(iii.) The restriction of the quadratic form T r(δZZ † + ZδZ † )2 on Q(K) to C (K)
has maximal rank.
A calculation employing elementary tools of differential geometry [3] shows that
the above three observations imply the invariance of the measure µ̂ with respect to
the induced action of U (K) on the critical manifold U (K)/Sp(K) defined by (22).
Therefore µ̂ is a Haar measure on the symmetric space U (K)/Sp(K), which is
unique up to normalization. It is generally easier to work with integrals over the
whole unitary group rather than a factor space. As we have established already
the measure µ̂ is invariant with respect to the action of U (K) on C (K) . Note also
8
(
)
that the function T r Z(U )† XZ(U )X which determines the integrand of (24) is
also U (K)-invariant. Therefore, by Weyl’s theorem [6], Chapter X,
∫
∫
†
T r(Z(U )† XZ(U )X )
µ̂(dU )e
=
µH (dU )eT r(Z(U ) XZ(U )X )
(25)
U (K)/Sp(K)
U (K)
where µH is an appropriately normalized Haar measure on the unitary group.
We will determine the normalization factor later using the properties of spin-spin
correlation functions. Substituting (19) into (25) we find that
∫
∫
T
T r(Z(U )† XZ(U )X )
µH (dU )e
=
µH (dU )e−T r(JHJH )
(26)
U (K)
U (K)
where H is a Hermitian matrix with eigenvalues x1 , x2 , . . . , xK given by H =
U XU † . Tracing back we find that the large N behaviour of
E
K
∏
det (M1 − xm )
m=1
- the expected value of the product of K characteristic polynomials in the real
Ginibre ensemble - turns out to be determined by the integration of the symplecticinvariant Gaussian weight exp{−T rJHJH T } with respect to unitary degrees of
freedom. See [12] for a discussion of the origin of the connection between real
Ginibre and symplectic ensembles. We may rewrite the integral in (26) as
∫
R
µH (dU )eT r(HH )
(27)
U (K)
where H = U XU † and
H R = JH T J T
(28)
is a ’symplectic’ involution on the space of complex K × K matrices, see [9] for
details. Following Mehta, we will call matrix M self-dual if M = M R and antiself-dual if M = −M R . It is easy to check that any even-dimensional matrix can
be uniquely represented as a sum of a self-dual and anti-self-dual matrices. Let
ASD(K) be the linear space of all anti-self dual K × K matrices. In order to
perform the integration over the unitary group in the right hand side of (27), we
use use the following transformation found in [11], [10]:
∫
√
2
T rHH R
T rH 2
(29)
e
= ZK e
Λ(dA)eT rA +2 2T rHA ,
ASD (K)
where Λ(dA) is a Lebesgue measure on ASD(K) , ZK is a normalization constant.
The absolute convergence of the above integral can be checked using the decomposition theorem for anti-self-dual matrices, see [8]: for any A ∈ ASD(K) there
9
exists V ∈ U (K) and a diagonal matrix Θ with entries ±θ1 , ±θ2 , . . . ± θ K , where
2
θ1 ≥ 0, θ2 ≥ 0, . . . θK/2 ≥ 0, so that
A = iV ΘV † .
(30)
Substituting (29) into (27) and using the invariance property of the Haar measure
µH we get:
∫
R
µH (dU )eT r(HH )
U (K)
∫
∫
√
∑ 2
2
†
xk
ν(dθ)e−T r(Θ ) ei2 2T rU XU Θ
= CK e
µH (dU )
(31)
K/2
U (K)
R+
where ν(dθ) is the measure on the eigenvalues of unitary matrices induced by the
marginalization of the Lebesgue measure on ASD(K) over the unitary degrees of
freedom, explicitly [9]
(
ν(dθ) = ∆ ±θ1 , ±θ2 , . . . ± θ K
) K/2
∏
2
θk dθk .
(32)
k=1
We now allow the constants CK , depending only on K, to change value from line
to line. Now we can integrate over the unitary group U (K) using Harish-ChandraItzykson-Zuber formula [5], [7]. The result is
∫
R
µH (dU )eT r(HH )
U (K)
∑
= CK e
x2k
−1
∫
∏
[
K/2
∆(x)
K/2
R+
−θk2
θk e
]
√
i2 2xi Θjj
dθk det e
.
(33)
1≤i,j≤K
k=1
The remaining integration over the singular values θ1 , θ2 , . . . , θK/2 is carried out
using de Bruijn formula [2]:
∫
[
]
∑ 2
R
2
µH (dU )eT r(HH ) = CK e xk ∆(x)−1 P f (xi − xj )e−2(xi −xj )
. (34)
1≤i,j≤K
U (K)
Combined with (16) this yields
[
]
2
ρ̃(x1 , x2 , . . . , xK ) = Ck P f (xi − xj )e−2(xi −xj )
.
(35)
1≤i,j≤K
Integrating all variables xk for k = 1, . . . , K as in (1) we find the spin-spin correlation
[∫
]
K
∏
−2z 2
E
skk (M1 ) = Ck P f
e
dz
.
(36)
xi −xj
k=1
10
1≤i,j≤K
Now the constants Ck can be found inductively in k by allowing x2k ↓ x2k−1 , and
noting that ρ(x1 , x1 ) = 1. Expression (36) coincides with the continuous limit of
single time correlation function of spin variables for the system of one dimensional
annihilating Brownian motions under the maximal entrance law, see [14] for details.
Differentiating (36) with respect to spatial variables and using the points according
to
ρt1 ,t2 ,...,tn (x1 , x2 , . . . , xn )
) ( n
)
( )n (∏
n
∏
1
∂
=
−
E
sxm (Mtm ) sxm +ym (Mtm ) 2
∂yk
m=1
k=1
,(37)
yl =+0, l=1,2...,n
we get the first statement of the Corollary 9 of [1].
References
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and its scaling limits. Communications in Mathematical Physics, 291(1):177–
224, 2009.
[2] NG De Bruijn. On some multiple integrals involving determinants. J. Indian
Math. Soc, 19:133–151, 1955.
[3] B.A. Dubrovin, A.T. Fomenko, S.P. Novikov, and R.G. Burns. Modern Geometry - Methods and Applications: Part I: The Geometry of Surfaces, Transformation Groups, and Fields. Graduate Texts in Mathematics. Springer,
1991.
[4] A. Erdélyi. Asymptotic expansions. Number 3. Dover publications, 2010.
[5] Harish-Chandra. Differential operators on a semisimple Lie algebra. American
Journal of Mathematics, pages 87–120, 1957.
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