Product of large Gaussian random matrices · 2019. 6. 7. · Product of large Gaussian random matrices Z. Burda, R. Janik and B. Waclaw Brunel Workshop, December 19th, 2009

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