Product of large Gaussian random matrices
Z. Burda, R. Janik and B. Waclaw
Brunel Workshop, December 19th, 2009
Outline
IntroductionMacroscopic universalityClassical results for Gaussian ensembles
Main resultEigenvalue density of X = X1X2 . . . XM
Surprising universality
Sketch of derivation
Summary
Macroscopic universality
Let a be an N-by-N symmetric matrix
Let aij for i ≤ j be i.i.d. with 〈aij〉 = 0, 〈a2ij 〉 = σ2
The eigenvalue density of A = a√N
converges for N → ∞ to
ρ(λ) =1
2πσ2
√
4σ2 − λ2 for λ ∈ [−2σ, 2σ]
λ
2σ
2σ
−2σ
ρ(λ) 2πσ2
Universality class
Independent, centered but not identically distributed entriesThe Pastur-Lindeberg condition
limN→∞
1N2
∑
i≤j
∫
|x |>ǫ√
Nx2pij(x) dx = 0
Invariant Gaussian ensembles
dµ(A) ∝ DA e− βN4σ2 trA2
GOE, GUE, etcEigenvalue density
ρ(x) =
⟨
1N
N∑
i=1
δ(x − λi)
⟩
Illustration
-2 -1 0 1 2Λ
0.1
0.2
0.3
0.4
ΡHΛL
Monte-Carlo: 200 matrices 100-by-100
green points: real symmetric; centered uniform distribution;
red points: hermitian gaussian;
solid: Wigner semicircle
Complex Gaussian matrices
Two i.i.d. GUE matrices
dµ(A, B) ∝ DA DB e− N2σ2 trA2
e− N2σ2 trB2
Complex matrices
X =1√2
(A + iB) , X † =1√2
(A − iB)
Girko-Ginibre ensemble
dµ(X , X †) ∝ DX DX † e− Nσ2 TrXX†
Complex eigenvalues z = x + iy
ρ(x , y) =
{ 1πσ2 for x2 + y2 ≤ σ2
0 otherwise
Illustration
Monte-Carlo: 100 complex matrices 100-by-100
points: eigenvalues
solid: unit circle
Elliptic Gaussian measures
Asymmetric mixing
X = cos(φ)A+i sin(φ)B , X † = cos(φ)A−i sin(φ)B , τ = cos(2φ)
Measure
dµ ∝ DX DX † e− N
σ2(1−τ2)(TrXX†− τ
2 Tr(XX+X†X†))
Crisanti, Sommers, Sompolinsky and Stein
ρ(x , y) =
1πσ2(1−τ2)
for x2
σ2(1+τ)2 + y2
σ2(1−τ)2 ≤ 1
0 otherwise
The result holds also for real matrices
Illustration
-2 -1 0 1 2Re z
-2
-1
0
1
2
Im
z
σ = 1; τ = −12
Monte-Carlo: 100 complex matrices 100-by-100(x/a)2 + (y/b)2 = 1 ; a=1/2; b=3/2;
Product of Gaussian matrices
Product of independent matrices X = X1X2 . . . XM
Eigenvalue density
ρ(z, z) =
1Mπσ2 |z|−2+ 2
M for |z| ≤ σ
0 for |z| > σ
Strong universality: Xi ’s do not have to be identical
σ = σ1 . . . σM ; Result is independent of τ1, . . . , τM !!!
For σ=1 and M =2, 3
ρ2(r) =1
2πr, ρ3(r) =
13πr4/3
Illustration
-1 -0.5 0 0.5 1Re z
-1
-0.5
0
0.5
1
Im
z
Product of two GUE matrices X = X1X2
Monte-Carlo: 200 complex matrices 100-by-100points: eigenvaluessolid: unit circle
Illustration
0 0.2 0.4 0.6 0.8 1 1.2r
0
0.2
0.4
0.6
0.8
1
2ΠΡHrL
Product of two GUE matrices X = X1X2
Radial profile 2πrρ(r), where r = |z|Monte-Carlo: 1000 complex matrices 100-by-100
points: eigenvalues
Illustration
0 0.2 0.4 0.6 0.8 1 1.2 1.4r
0
0.2
0.4
0.6
0.8
1
1.2
3Πr4�3ΡHrL
Product of two GUE matrices X = X1X2X3
Radial profile 3πr4/3ρ(r), where r = |z|Monte-Carlo: 1000 complex matrices 100-by-100
points: eigenvalues
Illustration
-1 -0.5 0 0.5 1Re z
-1
-0.5
0
0.5
1
Im
z
-1 -0.5 0 0.5 1Re z
-1
-0.5
0
0.5
1
Im
z
-1 -0.5 0 0.5 1Re z
-1
-0.5
0
0.5
1
Im
z
-1 -0.5 0 0.5 1Re z
-1
-0.5
0
0.5
1
Im
z
X = X1X2
MC 100, 100x100
G-G · G-G
RW · RW (unif. distr.)
GUE · G-G
GUE · Elliptic(τ =−1/2)
Illustration
0 0.2 0.4 0.6 0.8 1 1.2 1.4r
0
0.2
0.4
0.6
0.8
1
1.2
2ΠrΡHrL
X = X1X2; MC 1000 matrices 100x100
red: G-G · G-G
green: RW · RW (uniform distribution)
blue: GUE · G-G
violet: GUE · AC (τ =−1/2)
Illustration
0 0.5 1 1.50
0.25
0.5
0.75
1
1.25
0 0.5 1 1.50
0.25
0.5
0.75
1
1.25
0 0.5 1 1.50
0.25
0.5
0.75
1
1.25
|z||z||z|
Mπ|z|2−
2 Mρ(|z
|)
left: X = X1X2 for GUE · GUE; G-G · G-G; Elliptic · GUE;
middle: X = X1X2 for N = 50, 100, 200, 400;
right: X = X1 . . . XM for M = 2, 3, 4;
Universality
Product of independent matrices X = X1X2 . . . XM
Eigenvalue density of X is rotationally symmetric even ifdensities of Xi ’s are elliptic !!
Eigenvalue distribution is concentrated inside a circle ofradius σ
ρ(r) =1
Mπσ2 r−2+ 2M
Green’s function
Eigenvalue density
ρ(x) =
⟨
1N
N∑
i=1
δ(x − λi)
⟩
Green’s function
g(z) =
⟨
1N
Tr (z1− A)−1⟩
=
⟨
1N
N∑
i=1
1z − λi
⟩
Main relation
−1π
Im1
x + iǫǫ→0+
−→ δ(x) =⇒ ρ(x) = −1π
limǫ→0+
g(x + iǫ)
Large N limit (N → ∞)
Coalescence of poles into a branch cut
g(z) =
⟨
1N
N∑
i=1
1z − λi
⟩
=
∫
dxρ(x)
z − x
Re z Re z
Im z Im zN g(x+i0 )+
Moving along the cut
ρ(x) = −1π
Im g(
x + i0+)
Feynman diagrams
Convention (normalized trace) g(z) = 1N Tr G(z)
Geometric expansion
G(z) =⟨
(Z − A)−1
⟩
=⟨
Z−1+Z−1A Z−1+Z−1A Z−1A Z−1+. . .⟩
Propagators Z−1bc b c
〈AabAcd 〉 =1N
δadδbc a db c
Generating function for two-point diagrams
= + +
+ + + ....
G
Feynman diagrams
Convention (normalized trace) g(z) = 1N Tr G(z)
Geometric expansion
G(z) =⟨
(Z − A)−1
⟩
=⟨
Z−1+Z−1A Z−1+Z−1A Z−1A Z−1+. . .⟩
Propagators Z−1bc b c
〈AabAcd 〉 =1N
δadδbc a db c
Generating function for two-point diagrams
= + +
+ + + ....
G
Planar limit; N → ∞
Generating function for one-line irreducible diagrams Σ
Dyson-Schwinger equations
G = (z1−Σ)−1= + + + ...
G Σ Σ Σ
Σad = Gbc1N
δadδbc =⇒ Σ = g1G
Σ =a d a b c d
Solutiong = (z − σ)−1 , σ = g
g =12
(
z ±√
z2 − 4)
→ ρ(x) =1
2π
√
4 − x2
Complex eigenvalue density
Eigenvalue density
ρ(z, z) =
⟨
1N
N∑
i=1
δ(2)(z − λi)
⟩
Dirac’s delta
δ(2)(z − λ) = limǫ→0
1π
ǫ2
(|z − λ|2 + ǫ2)2 = limǫ→0
1π
∂
∂z
[
z − λ
|z − λ|2 + ǫ2
]
Green’s function
g(z, z)= limǫ→0
*
1N
NX
i
z − λi
|z − λi |2 + ǫ2
+
= limǫ→0
fi
1N
Trz1− X †
(z1− X †)(z1− X ) + ǫ21
fl
Relationρ(z, z) =
1π
∂g(z, z)
∂z
Extended form of Green’s function
Method by Janik, Nowak, Papp, Zahed
Matrix 2N-by-2N (four blocks)
G =
(
Gzz Gzz
Gzz Gzz
)
= limǫ→0
⟨
(
z1− X iǫ1iǫ1 z1− X †
)−1⟩
Upper left corner
g(z, z) ≡ gzz(z, z) =1N
Tr Gzz(z, z)
Poles coalescence for N→∞ into a 2d region(ρ = 1
π∂zg 6= 0)
Limit’s order: first N → ∞ and then ǫ → 0
Analogy to symmetry breaking
For finite N there are isolated poles
∂zg(z, z) = 0 almost everywhere
Example: Ising model with Z2 global symmetry
For finite N symmetry is preserved 〈M〉 = 0
For N → ∞ symmetry gets spontaneously broken 〈M〉 6= 0
Weak external field h breaking symmetry for finite N too
Take first the limit N → ∞ and then h → 0
Dyson-Schwinger equation (1)
2N-by-2N extension of matrices
G =
(
Gzz Gzz
Gzz Gzz
)
, Σ =
(
Σzz Σzz
Σzz Σzz
)
Planar Dyson-Schwinger equations N → ∞(
Gzz Gzz
Gzz Gzz
)
=
(
z1− Σzz −Σzz
−Σzz z1− Σzz
)−1
= + + + ...G Σ Σ Σ
Dyson-Schwinger equation (2)
Propagators for the zz, zz, zz and zz sectors:
〈XabXcd〉 = 0 , 〈XabX †cd〉 = 1
N δadδbc
〈X †abXcd〉 = 1
N δadδbc , 〈X †abX †
cd〉 = 0
For each sector separately
Σad = 0 , Σad = 1N δadδbcGbc = δadgzz
Σad = 1N δadδbcGbc = δadgzz , Σad = 0
GΣ =
a d a b c d
Solution
Trace„
σzz σzz
σzz σzz
«
=
„
0 gzz
gzz 0
«
,
„
gzz gzz
gzz gzz
«
=
„
z − σzz −σzz
−σzz z − σzz
«−1
Inserting sigma(
gzz gzz
gzz gzz
)
=1
|z|2 − gzzgzz
(
z gzz
gzz z
)
Solution
g(z, z) =
z for |z| ≤ 1
1/z for |z| > 1
Linearization
ProblemG(z) =
⟨
(z − X1X2 . . . XM)−1⟩
Related resolvent
GY (w) =⟨
(w − Y )−1⟩
where
Y =
0 X1 00 0 X2 0
. . . . . .0 0 XM−1
XM 0
Result
Y M = blockdiag(X1X2 . . . XM , . . . , cyclic, . . .)
Y M has the same eigenvalues as X = X1X2 . . . XM (butM-fold degenerate)
Eigenvalue density of Y
ρY (w , w) =
{ 1π for |w | ≤ 10 for |w | > 1
Eigenvalue density of X = X1 . . . XM : z = wM
ρ(z, z) = M∂w∂z
∂w∂z
ρY (w , w) =1
Mπ|z|−2+ 2
M
Summary
Eigenvalue density of X = X1X2 . . . XM
ρ(r) =1
Mπσ2 r−2+ 2M for r = |z| ≤ σ
Surprising universality (independence of τ1, . . . , τM )
Conjecture: this result also holds for a product of Wignermatrices having independent centered entries with a finitevariance (belonging to the Gaussian universality class);
Towards S-transform (FRV for complex spectra)
arXiv: 0912.3422; Z. B., R. Janik and B. Waclaw, Spectrum ofthe Product of Independent Random Gaussian Matrices