Local inhomogeneous circular law Johannes Alt Institute of Science - - PowerPoint PPT Presentation

Local inhomogeneous circular law

Johannes Alt

Institute of Science and Technology Austria

December 4, 2017 Mathematical Physics Seminar, Geneva University Joint work with László Erdős and Torben Krüger.

Partially funded by ERC Advanced Grant RANMAT No. 338804.

Table of Contents

1

Introduction

2

Local inhomogeneous circular law

3

Ideas of the proof

Johannes Alt (IST Austria) Local inhomogeneous circular law December 4, 2017

Introduction

Random matrix: matrix-valued random variable. Introduced by Wishart [1928] and Wigner [1955]. Applications/Motivations: Mathematical statistics, Quantum physics, Wireless communication, Neural networks, Number theory, Free probability

Some important questions in random matrix theory

Do the eigenvalues exhibit any deterministic behaviour? Are there any universal characteristics in this deterministic behaviour? How universal are their fluctuations?

Johannes Alt (IST Austria) Local inhomogeneous circular law December 4, 2017

Empirical spectral measure and density of states

Empirical spectral measure of an n × n random matrix X: µX = 1 n

n

• i=1

δzi where z1, . . . , zn ∈ C are the eigenvalues of X. Goal: Study eigenvalue density µX for large n. Typical situation: Empirical spectral measure becomes deterministic! There is a (deterministic) measure µ (self-consistent density of states) such that

• C

fdµX = 1 n

n

• i=1

f(zi) ≈

• C

fdµ for all f with compact support.

Johannes Alt (IST Austria) Local inhomogeneous circular law December 4, 2017

Spectra of Hermitian and non-Hermitian random matrices

Hermitian X = X∗ i.i.d. entries Semicircular law: µX ≈ 1 2π

• (4 − x2)+ dx

Non-Hermitian X i.i.d. entries Circular law: µX ≈ 1 π1D(0,1)d2z

−2 −1 1 2 0.1 0.2 0.3 −1 −0.5 0.5 1 −1 −0.5 0.5 1

In this talk: X = (xij)n

i,j=1 with xij centered, independent

but non-identically distributed

Johannes Alt (IST Austria) Local inhomogeneous circular law December 4, 2017

Example with non-identical variances

Random matrix X = (xij)n

i,j=1 with independent entries

E xij = 0, sij .

.= E |xij|2,

S .

.= (sij)n i,j=1. −1 1 −1 −0.5 0.5 1

(a) Eigenvalue locations

S = 1 1 0.2

(b) Variance profile S such that ρ(S) .

.= max|spec(S)| = 1.

Figure: Averaging 100 matrices of size n = 1000 with centered complex Gaussian entries and the variance profile S.

Johannes Alt (IST Austria) Local inhomogeneous circular law December 4, 2017

Example with non-identical variances

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6

|z|

Figure: Eigenvalue histogram and self-consistent density of states. Averaging 100 matrices of size n = 1000 with centered complex Gaussian entries and the variance profile S.

Johannes Alt (IST Austria) Local inhomogeneous circular law December 4, 2017

Behaviour of eigenvalues on different scales

Different scales (X normalized such that X = O(1)): Global law µX(U) → µ(U) diam(U) = O(1) Local law µX(Un) → µ(Un) diam(Un) = O(n−a), a ∈ (0, 1/2) Local scales: rescale and shift test function: fw,a(z) .

.= n2af(na(z − w))

for z, w ∈ C and a > 0. Local law: For a ∈ (0, 1/2), we have 1 n

n

• i=1

fw,a(zi) ≈

• C

fw,a(z)dµ(z).

Johannes Alt (IST Austria) Local inhomogeneous circular law December 4, 2017

Related previous results for non-Hermitian case

Global laws (a = 0)

Ginibre [1965]: i.i.d. entries with complex Gaussian distribution Girko [1984]: independent entries with variance profile Bai [1997]: i.i.d. entries with bounded density Tao, Vu [2010]: i.i.d. entries with second moments Cook [2016], Cook, Hachem, Najim, Renfrew [2016]: independent entries with irreducible variance profile; control on smallest singular value

Local laws (a ∈ (0, 1/2))

Bourgade, Yau, Yin [2014]; Yin [2014] Bulk and edge local law on optimal scale for independent entries with identical variances E|xij|2 = n−1

Local statistics (a = 1/2)

Ginibre [1965]: k-point correlation function for complex Gaussians → expected universal statistics Tao, Vu [2014]: i.i.d. entries with four matching moments

Johannes Alt (IST Austria) Local inhomogeneous circular law December 4, 2017

Local inhomogeneous circular law

Assumptions

The random matrix X = (xij)n

i,j=1 has independent centered entries.

We assume There are some constants s∗, s∗ > 0 such that s∗ n ≤ E|xij|2 ≤ s∗ n for all i, j. Uniformly bounded moments E|xij √n|m ≤ cm for all m ∈ N and i, j. The distribution of xij √n has an Lp density for some p > 1.

Johannes Alt (IST Austria) Local inhomogeneous circular law December 4, 2017

Local inhomogeneous circular law

We recall the matrix of variances S .

.= (sij)n i,j=1,

sij = E|xij|2. For z ∈ C and η > 0, let (v1, v2) ∈ R2n

+ be the unique solution of

1 v1 = η + Sv2 + |z|2 η + Stv1 , 1 v2 = η + Stv1 + |z|2 η + Sv2 .

Self-consistent density of states of X

σ(z) .

.=

   − 1 2π ∞ ∆zv1(η; z)dη, if |z|2 < ρ(S), 0, if |z|2 ≥ ρ(S). (ρ(S) .

.= max|Spec(S)| spectral radius of S, v1 average of v1)

Johannes Alt (IST Austria) Local inhomogeneous circular law December 4, 2017

Local inhomogeneous circular law

With R .

.=

• ρ(S), we recall

σ(z) .

.=

   − 1 2π ∞ ∆zv1(η; z)dη, if |z| < R, 0, if |z| ≥ R.

Proposition (Properties of σ) (A., Erdős, Krüger; 2016)

(i) there are c2 > c1 > 0 such that for all z ∈ D(0, R) c1 ≤ σ(z) ≤ c2. (ii) infinitely often differentiable in D(0, R). (iii) supp σ = D(0, R) Cook et al. [2016]: σ is a continuous, rotationally symmetric probability density with supp σ ⊆ D(0, R).

Johannes Alt (IST Austria) Local inhomogeneous circular law December 4, 2017

Local inhomogeneous circular law

Local scales: fw,a(z) .

.= n2af(na(z − w))

Theorem (A., Erdős, Krüger; 2016)

(i) For any ε, D > 0, a ∈ (0, 1/2), |w| < R .

.=

• ρ(S) and f ∈ C2

0(C),

we have P

• 1

n

n

• i=1

fw,a(zi) −

• C

fw,a(z)σ(z)d2z

• ≤ ∆f1

n1−2a−ε

• ≥ 1 − C

nD (ii) For any D > 0 and ε > 0, we have P

• Spec(X) ⊆ D(0, R + ε)
• ≥ 1 − C

nD

Johannes Alt (IST Austria) Local inhomogeneous circular law December 4, 2017

Ideas of the proof

Girko’s Hermitization trick

Idea: Transform problem to question for a Hermitian random matrix Instead of X study the family of Hermitian random 2n × 2n matrices Hz .

.=

• X − z1

X∗ − ¯ z1

• for z ∈ C.

Relation between X and Hz: log|det(X − z1)| = 1 2 log|det Hz|. This connects the quantity we want to study with Hz since 1 n

n

• i=1

fw,a(zi) = 1 2πn

• C

∆fw,a(z) log|det(X − z1)|d2z.

Johannes Alt (IST Austria) Local inhomogeneous circular law December 4, 2017

Girko’s Hermitization trick

log|det Hz| = −2n ∞ Im mz(iη)dη, where mz is the Stieltjes transform of the empirical spectral measure of Hz mz(ζ) =

• R

1 x − ζ dµHz(x) = 1 2n

2n

• i=1

1 λi(z) − ζ , with ζ ∈ C, Im ζ > 0 and λi(z) eigenvalues of Hz. Starting formula 1 n

n

• i=1

fw,a(zi) −

• C

fw,a(z)σ(z)d2z = − 1 2π

• C

∆fw,a(z) ∞

• Im mz(iη) − v1(η; z)
• dη d2z.

Goal (Local law)

|mz(iη) − iv1(η; z)| 1 nη (with high prob.)

Johannes Alt (IST Austria) Local inhomogeneous circular law December 4, 2017

Local laws for Hermitian random matrices

For ζ ∈ C, Im ζ > 0, we have G(ζ) .

.= (Hz − ζ)−1,

mz(ζ) = 1 2n Tr G(ζ). The local law follows from G(ζ) ≈ M z(ζ), Im ζ ≥ n−1+γ. Here, M = M z is the unique solution of Matrix Dyson equation −M(ζ)−1 = ζ − A + S[M(ζ)] with positive definite imaginary part Im M = 1

2i(M − M ∗) and

A .

.= EH,

S[R] .

.= E [(H − A)R(H − A)] .

Existence and uniqueness: Helton, Rashidi Far, Speicher [2007]. Relationship between v1 and M: v1(η; z) = 1 2n Tr Im M z(iη).

Johannes Alt (IST Austria) Local inhomogeneous circular law December 4, 2017

Heuristic derivation of the Matrix Dyson equation

If Y is a centered real Gaussian random variable and f a differentiable function then E [Y f(Y )] = E

• Y 2

E

• f′(Y )
• .

(1) With the notations A .

.= EH,

Y .

.= H − A,

S[R] .

.= E [Y RY ] ,

D .

.= (−Y − S[G])G

the definition of the resolvent G can be rewritten as −1 = (ζ − H)G = (ζ − A − Y )G = (ζ − A + S[G])G + D. For real Gaussian H, using a multidimensional analogue of (1), we obtain E[Y G] = −E[S[G]G] ⇒ ED = 0. Therefore, if G can be approximated by a deterministic matrix M then it is plausible that −1 = (ζ − A + S[M]) M.

Johannes Alt (IST Austria) Local inhomogeneous circular law December 4, 2017

General strategy for proving G ≈ M

The resolvent G satisfies −1 = (ζ − A + S[G])G + D, D = (A − H − S[G])G. This is a perturbed version of the Matrix Dyson equation (MDE) −M(ζ)−1 = ζ − A + S[M(ζ)]. How to prove that G − M is small? Two steps:

1 D is small. 2 G − M ≤ CD.

For

1 , we can use estimates by Ajanki, Erdős, Krüger [2016]. They also

show

2 but under the condition

E |x, (H − A)y|2 ≥ c nx2

2y2 2,

for all x, y ∈ C2n. (2) On the global scale, the general MDE analysis also yields G − M ≤ CηD even without (2). However, getting the optimal η-dependence in the local law in [AEK2016] makes crucial use of (2) which is violated in our case.

Johannes Alt (IST Austria) Local inhomogeneous circular law December 4, 2017

Stability operator for the vector equation

In our setup, we can reduce MDE at ζ = iη to vector equation for v .

.= −i diag M = (v1, v2).

In fact, 1 v1 = η + Sv2 + |z|2 η + Stv1 , 1 v2 = η + Stv1 + |z|2 η + Sv2 . Introducing g = diag G, we obtain the stability equation L (g − iv) = r, r = O(D) with the linear stability operator Ly .

.= y + v2(Soy) − |z|2

v2 (η + Sdv)2 (Sdy), y ∈ C2n, So = S St

• ,

Sd = St S

• .

→ We need to invert L to control g − iv. The bound L−1 ≤ Cη is easy from MDE, but useless if η ≪ 1.

Johannes Alt (IST Austria) Local inhomogeneous circular law December 4, 2017

Symmetrization

Identify bad directions and use spectral decomposition → need a symmetric operator → Key representation: L = V −1(1 − T F )V , where V invertible, T , F self-adjoint with T 2 = 1. If sij ≥ s∗/n then V , V −1 ∼ 1, F 2 ≤ 1 − cη and hence L−12 ≤ Cη−1. Moreover, for some ε > 0, we have spectral gaps Spec(F ) ⊂ {−F 2} ∪ [−F 2 + ε, F 2 − ε] ∪ {F 2}. Therefore, the blow-up of L−1 can potentially be caused by the two extremal eigendirections f + and f − of F , which satisfy F f ± = ±F 2f ± .

Johannes Alt (IST Austria) Local inhomogeneous circular law December 4, 2017

Stability of Dyson equation

f + eigendirection Positive lower bound on v1, v2 ⇒ T f +2 ≤ 1 − δ and hence (1 − T F )f +2 ≥ δf +2 (3) → f + is harmless. f − eigendirection In contrast to (3), (1 − T F )f − = O(η).

Contraction-Inversion Lemma

|f −, p| ≤ cηp ⇒ (1 − T F )−1p ≤ Cp. This assumption is satisfied for r = O(D) since v1 = v2, g1 = g2. Hence, inverting L in L(g − iv) = r yields g − iv ≤ CD.

Johannes Alt (IST Austria) Local inhomogeneous circular law December 4, 2017

Bounding derivatives of v

Set ρ(S) = 1. Set v .

.= (v1, v2), τ . .= |z|2, ∂αv . .= ∂α1 η ∂α2 τ v.

Claim

sup{∂αv: η > 0, τ ≤ 1 − ε} 1. For the first derivatives, we have L(∂ηv) = −v2 + τv2 (η + Sdv)2 , L(∂τv) = − v2 η + Sdv In general, L(∂αv) = rα, where rα is determined by derivatives of lower

• rder. The Contraction - Inversion lemma implies ∂αv rα 1

provided f −, V rα = O(η). Setting e− .

.= (1, −1) and a . .= e−

• v(η + Sov), we find

a − f − = O(η) and using L∗(V a) = ηe− we have that a, V rα = V a, L(∂αv) = η∂αe−, v = η∂α (v1 − v2) = 0.

Johannes Alt (IST Austria) Local inhomogeneous circular law December 4, 2017

Proof of c1 ≤ σ(z) ≤ c2 for |z| ≤ 1

We are now allowed to interchange derivatives and limits in η and τ freely. Therefore, it is easy to obtain σ(z) = 1 π∂τ (τu(η = 0, τ)) |τ=|z|2, where u(η) .

.=

v(η) η + Sdv(η), v .

.= (v1, v2).

We compute σ(z) = − 2 πSov0, ∂τv0 = lim

η↓0

2 π √ v˜ v, 1 √u (1 − T F )−1 √u √ v˜ v

• ,

where v0 = v(η = 0) and ˜ v = (v2, v1). Now, we need to focus on τ = |z|2 ≈ 1. We find v1 ∼ v2 ∼ (1 − τ)1/2 and (1 − T F )−1 blows up like (1 − τ)−1 on √ uv˜

• v. Hence, upper and lower bounds on σ follow.

Johannes Alt (IST Austria) Local inhomogeneous circular law December 4, 2017

Conclusion

Local inhomogeneous circular law: Convergence of the bulk eigenvalue density for random matrices with independent, centered entries (optimal scale, optimal error bounds) Density of states: rotationally invariant, supported on a disk with jump on the boundary, smooth Detailed stability analysis of Dyson equation with additional instability Work in Progress Local law close to the edge: |z0|2 ≈ ρ(S) Local law if the entries of X are correlated Dream Universality of k-point correlation functions

Johannes Alt (IST Austria) Local inhomogeneous circular law December 4, 2017