1 Random Levy· Matricesweb.mit.edu/sea06/agenda/talks/Burda.pdfRandom Levy·Matrices 1 Not...

Random Levy Matrices 1

Not everything is Gaussian !!!

M.A. Nowak, J. Jurkiewicz (Jagellonian University)

G. Papp (Eotvos Lorand University)

I. Zahed (Stoony Brook, SUNY)

Outline:

• stable distributions (classical probability)

• large random matrices (symmetric real)

• Gaussian domain of attraction (GOE, Wigner)

• Levy’s domain of attraction

– free Levy random variables

– Wigner-Levy ensemble

• summary

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Laws of addition in classical probability

Let x1, x2 be independent random variables with pdfs: p1(x), p2(x)

What is pdf for: x1+2 = x1 + x2 ?

p1+2(x) =∫

dx1p1(x1)p2(x − x1)

Characteristic function:

φ(k) =∫

dx p(x)eikx

φ1+2(k) = φ1(k) · φ2(k)

Cumulants’ generating function:

c(k) = log(φ(k))

c1+2(k) = c1(k) + c2(k)

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Normal distribution:

x ∼ N(0, σ2) , φ(k) = exp− 12σ2k2 , c(k) = −σ2k2

2

Sum of independent normal variables

x1 ∼ N(0, σ21) , x2 ∼ N(0, σ2

2) =⇒ x1+2 ∼ N(0, σ21 + σ2

2)

Sum of iid normal variables

σ21 = σ2

2 ≡ σ2

x1+2√2

∼ N(0, σ2)

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Stability

xj iid N(0, σ2)

X = x1+···+xn√n

∼ N(0, σ2) (fixed point)

Stable laws are important !!

Let xj be iid and E(x) = 0, var(x) = σ2

X = x1+···+xn√n

n→∞=⇒ C(k) = nc

(

k√n

)

→ −σ2k2

2

X ∼ N(0, σ2) (Gaussian domain of attraction)

Universality !!

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Levy domain of attraction

Heavy tails in pdf p(x) ∼ A|x|1+µ for x → ±∞

Levy stable law for addition of iid: X = x1+···+xn

n1/µ

Lµ(x) = 12π

∫

dkL(k)eikx

c(k) = log L(k) = −γµ|k|µ , µ ∈ (0, 2]

c1+2(k) = c1(k) + c2(k) =⇒ γµ1+2 = γµ

1 + γµ2

Assymetry:

p(x) ∼ A±

|x|1+µ for x → ±∞

β = A+−A−

A++A−β ∈ [−1, 1]

Lγ, βµ (x) = 1

2π

∫

dkL(k)eikx

log L(k) = −γµ|k|µ(1 + iβ sgn(k) tan(πµ/2))

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Summary 1/3

• Law of addition in classical probability c1+2(k) = c1(k) + c2(k)

• Importance of stable distributions: domains of attractions

• Classification of stable distributions: Lγ, βµ (x)

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Random Matrices

N×N real symmetric matrices H , N → ∞

GOE

Measure: DH exp(

− N2σ2 trH2

)

where DH =∏

i

dHii

∏

i>j

dHij

Orthogonal invariance: H → OHOτ

Diagonal decomposition: H = O diag[λ1, . . . , λN ] Oτ

Spectral density function (sdf) for N → ∞:

ρ(λ) = 12πσ2

√4σ2 − λ2

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Product measure

GOE: Hij ∼ N(

0, σ2

2N (1 + δij))

Slightly modified ensemble: Hij ∼ N(

0, σ2

2N

)

Wigner ensemble

Hij are iid:

Measure:∏

i≥j dHijp(Hij), E(Hij) = 0 , var(Hij) = σ2

2N

sdf for N → ∞ρ(λ) = 1

2πσ2

√4σ2 − λ2

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Summary 2/3

Two classes of matrices in the Gaussian domain of attraction:

• H = O diag[λ1, . . . , λN ] Oτ ,

• Hij iid: E(Hij) = 0 , var(Hij) = σ2

2N

Remark:

H = H1+···+Hn√n

The two classes have the same limit !!

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Summary 2/3

Two classes of matrices in the Gaussian domain of attraction:

• H = O diag[λ1, . . . , λN ] Oτ

• Hij iid: E(Hij) = 0 , var(Hij) = σ2

2N

Remark:

H = H1+···+Hn√n

The two classes have the same limit !!

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Levy matrices

Class 1: Orthogonally invariant (heavy-tailed) matrices:

H = O diag[λ1, . . . , λN ] Oτ , sdf ρ(λ) ∼ 1λµ+1 0 < µ < 2

Class 2: Wigner-Levy

Hij iid , pdf p(Hij) ∼ 1

Hµ+1ij

0 < µ < 2

H = H1+···+Hn

n1/µ

Class 1: H free random matrices (ortogonally invariant) with stable sdf

Class 2: H pdf p(Hij) → Lγ,βµ (Hij)

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Simple example for 2d-random vectors:

pdfs for (x, y)

p1(x, y) ∼ e−x2

e−y2

p2(x, y) ∼ 11+x2

11+y2

-10

-5

0

5

10

-10 -5 0 5 10

PSfrag replacements

x

y

-10

-5

0

5

10

-10 -5 0 5 10

PSfrag replacements

x

y

xy

p(x, y) = const

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Class 1 = free random variables

Free probability (Voiculescu)

Assymptotic freeness of large matrices from class 1 (Speicher)

Classical probability

p(x) - pdf

φ(k) =∫

dxp(x)e+ikx

c(k) = log φ(k)

c1+2(k) = c1(k) + c2(k)

φ(k) = exp c(k)

p(x) = 12π

∫

dkφ(k)e−ikx

Free probability

ρ(λ) - sdf

G(z) =∫ ρ(λ)dλ

λ−z

G(R(z) + 1z ) = z

R1+2(z) = R1(z) + R2(z)

R(G(z)) + 1G(z) = z

ρ(λ) = − 1π Im G(λ + i0+)

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Free lunch

Bercovici-Pata bijection between classical and free random variable

Stable laws in free probability (Bercovici-Voiculescu)

R(z) = 0

R(z) = bzµ−1

b =

γµ ei( µ2 −1)(1+β)π for 1 < µ ≤ 2

γµ ei[π+ µ2 (1+β)π] for 0 < µ < 1

.

R(z) = −iγ(1 + β) − 2βγπ ln γz for µ = 1

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Two examples

R(G(z)) + 1G(z) = z

Cauchy: µ = 1, β = 0 −→ R(z) = −iγ

−iγ + 1G(z) = z

G(z) = 1z+iγ −→ ρ(λ) = 1

πγ

λ2+γ2

Gauss: µ = 2 −→ R(z) = γ2z

γ2G(z) + 1G(z) = z

G(z) =z−

√z2−4γ2

2γ2 −→ ρ(λ) = 12πγ2

√

4γ2 − λ2

analytically solvable for µ = 1/4, 1/3, 1/2, 2/3, 3/4, 4/3, 3/2

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Class 2 - Wigner-Levy

Bouchaud-Cizeau, (tour de force !!)

ZB, M.A.Nowak, J.Jurkiewicz, G. Papp, I. Zahed

Solution for Hij ∼ L1,βµ (Hij)

γµ/2(z) =∫ +∞−∞ dx|x|−µ/2L

γ(z),β(z)µ/2 (z − x)

β(z) =

R +∞

−∞dx sign(x)|x|−µ/2L

γ(z),β(z)

µ/2(z−x)

R +∞

−∞dx|x|−µ/2L

γ(z),β(z)

µ/2(z−x)

ρ(λ) = Lγ(λ),β(λ)µ/2 (λ)

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Numerical comparison:

methods: J. P. Nolan + our group

-6 -4 -2 0 2 4 6 80

0.05

0.1

0.15

0.2

0.25

PSfrag replacements

λ µ = 1.75

ρ(λ

)

-10 -5 0 5 100

0.05

0.1

0.15

0.2

0.25

PSfrag replacements

λ µ = 1.75

ρ(λ)

λ µ = 1.50

ρ(λ

)

-10 -5 0 5 100

0.05

0.1

0.15

0.2

0.25

0.3

PSfrag replacements

λ µ = 1.75

ρ(λ)

λ µ = 1.50

ρ(λ)

λ µ = 1.25

ρ(λ

)

-10 -5 0 5 100

0.1

0.2

0.3

0.4

0.5

PSfrag replacements

λ µ = 1.75

ρ(λ)

λ µ = 1.50

ρ(λ)

λ µ = 1.25

ρ(λ)

λ µ = 1.00

ρ(λ

)

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From Wigner-Levy to free random Levy

Hi random Wigner-Levy Oi random orthogonal matrices

H = 1n1/µ

∑ni=1 OiHiO

τi

-6 -4 -2 0 2 4 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

PSfrag replacements

λ n = 100, µ = 1.25

ρ(λ

)

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Summary 3/3

• Wigner-Levy and free Levy ensembles

• spectral denisty can be calculated in both cases

• CLT for randomly rotated Wigner-Levy give free Levy

• Levy spacing (Wigner surmise for small λ and Poissonian for large

(Bouchaud-Cizeau); λ largest eigenvalue (Soshnikov))

• Random Levy Matrices Revisited ZB, J.Jurkiewicz, M.A. Nowak, G. Papp, I.

Zahed, cond-mat/0602087

• Applying Free Random Variables . . . ZB, A. Jarosz, J.Jurkiewicz, M.A. Nowak,

G. Papp, I. Zahed, physics/0603024

• Challenge: finite size effects for Wigner/Wishart ensembles with

pdf(x) ∼ x−1−µ for µ > 2

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Summary 3/3

• Wigner-Levy and free Levy ensembles

• spectral denisty can be calculated in both cases

• CLT for randomly rotated Wigner-Levy give free Levy

• Levy spacing: Wigner surmise for small λ and Poissonian for large

(Bouchaud-Cizeau); largest eigenvalue (Soshnikov);

• Random Levy Matrices Revisited ZB, J.Jurkiewicz, M.A. Nowak, G. Papp, I.

Zahed, cond-mat/0602087

• Applying Free Random Variables . . . ZB, A. Jarosz, J.Jurkiewicz, M.A. Nowak,

G. Papp, I. Zahed, physics/0603024

• Challenge: finite size effects for Wigner/Wishart ensembles with

pdf(x) ∼ x−1−µ for µ > 2

SEA06 MIT

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