Cusp Local Law and Complex Hermitian Universality joint work with - - PowerPoint PPT Presentation

Cusp Local Law and Complex Hermitian Universality

joint work with Johannes Alt, László Erdős and Torben Krüger

Dominik Schröder† Feburary 19, 2019

Workshop on Statistical Mechanics, Les Diablerets †Partially supported by ERC Advanced Grant No. 338804

Correlated Wigner Random Matrices

A random matrix H = H∗ =     h11 . . . h1N . . . ... . . . hN1 . . . hNN     ∈ CN×N is called a correlated Wigner random matrix if it has the following properties:

• Bounded expectation A .

.= E H: A ≤ C,

• Decaying correlations Cov(hij, hkl) = E wijwkl, where W .

.= H − A,

• Flat covariance operator S[R] .

.= E WRW, i.e.

c R ≤ S[R] ≤ C R , R .

.= 1

N Tr R for R = R∗ ≥ 0. For R

aa (matrix of all zeros, except for a 1 in the a a -th entry), aa bb

E wbawab E wab

2

Var hab

aa

1 N

Correlated Wigner Random Matrices

A random matrix H = H∗ =     h11 . . . h1N . . . ... . . . hN1 . . . hNN     ∈ CN×N is called a correlated Wigner random matrix if it has the following properties:

• Bounded expectation A .

.= E H: A ≤ C,

• Decaying correlations Cov(hij, hkl) = E wijwkl, where W .

.= H − A,

• Flat covariance operator S[R] .

.= E WRW, i.e.

c R ≤ S[R] ≤ C R , R .

.= 1

N Tr R for R = R∗ ≥ 0. For R = ∆aa (matrix of all zeros, except for a 1 in the (a, a)-th entry), (S[∆aa])bb = E wbawab = E |wab|2 = Var hab, ∆aa = 1 N .

Optimal Local Law

The resolvent G = G(z) .

.= (H − z)−1 is well approximated by the solution M = M(z) to the

matrix Dyson equation (MDE) 1 = (A − S[M] − z)M, ℑM = M − M∗ 2i > 0, ℑz > 0. Theorem ((Erdős, Krüger, S. 2017), (Alt, Erdős, Krüger, S. 2018), (Erdős, Krüger, S. 2018)) Let H be a real symmetric or complex Hermitian correlated Wigner matrix such that Cov(hab, hcd) (1 + |(a, b) − (c, d)|)−12−ǫ and M is bounded. Then the resolvent G(z) at z = E + iη satisfies u, (G(z) − M(z))v Nǫ u v

• ρ(E)

Nη + 1 Nη

• ,

X[G(z) − M(z)] Nǫ X 1 Nη (1) for all η ≫ ηf(E) above the fluctuation scale ηf and E in the edge and bulk regime. If H is a Wigner-type matrix, then (1) holds true also in the cusp regime. Previous results for less general ensembles by (Adhikari, Ajanki, Che, He, Knowles, Lee, O’Rourke, Schlein, Schnelli, Stetler, Rosenthal, Tau, Vu, Yau, Yin, …)

Derivation of the MDE

Derivation of the MDE

1 = HG − zG Gaussian integration by parts E xf x E x E f x Var x E f x E Hf H A E f H E EW

Wf W

A .

.

E H W .

.

H A Application to HG with G G H H z

1:

E HG A E G E EW

WG

A E G E EWGWG A E G E G G

WG

H W z

1

G H W z

1 WG

GWG Assuming G E G, thus 1 A G z G This motivates studying the solution M M z to the MDE 1 A M z M

Derivation of the MDE

1 = HG − zG Gaussian integration by parts E xf(x) = (E x)(E f(x)) + (Var x)(E f ′(x)) E Hf(H) = A E f(H) + E E W∂

Wf(W),

A .

.= E H,

W .

.= H − A.

Application to HG with G G H H z

1:

E HG A E G E EW

WG

A E G E EWGWG A E G E G G

WG

H W z

1

G H W z

1 WG

GWG Assuming G E G, thus 1 A G z G This motivates studying the solution M M z to the MDE 1 A M z M

Derivation of the MDE

1 = HG − zG Gaussian integration by parts E xf(x) = (E x)(E f(x)) + (Var x)(E f ′(x)) E Hf(H) = A E f(H) + E E W∂

Wf(W),

A .

.= E H,

W .

.= H − A.

Application to HG with G = G(H) = (H − z)−1: E HG = A E G + E E W∂

WG = A E G − E

E WG WG = A E G − E S[G]G ∂

WG = lim ǫ→0

(H + ǫ W − z)−1 − G ǫ = − lim

ǫ→0

(H + ǫ W − z)−1ǫ WG ǫ = −G WG Assuming G E G, thus 1 A G z G This motivates studying the solution M M z to the MDE 1 A M z M

Derivation of the MDE

1 = HG − zG Gaussian integration by parts E xf(x) = (E x)(E f(x)) + (Var x)(E f ′(x)) E Hf(H) = A E f(H) + E E W∂

Wf(W),

A .

.= E H,

W .

.= H − A.

Application to HG with G = G(H) = (H − z)−1: E HG = A E G + E E W∂

WG = A E G − E

E WG WG = A E G − E S[G]G ∂

WG = lim ǫ→0

(H + ǫ W − z)−1 − G ǫ = − lim

ǫ→0

(H + ǫ W − z)−1ǫ WG ǫ = −G WG Assuming G ≈ E G, thus 1 ≈ (A − S[G] − z)G. This motivates studying the solution M M z to the MDE 1 A M z M

Derivation of the MDE

1 = HG − zG Gaussian integration by parts E xf(x) = (E x)(E f(x)) + (Var x)(E f ′(x)) E Hf(H) = A E f(H) + E E W∂

Wf(W),

A .

.= E H,

W .

.= H − A.

Application to HG with G = G(H) = (H − z)−1: E HG = A E G + E E W∂

WG = A E G − E

E WG WG = A E G − E S[G]G ∂

WG = lim ǫ→0

(H + ǫ W − z)−1 − G ǫ = − lim

ǫ→0

(H + ǫ W − z)−1ǫ WG ǫ = −G WG Assuming G ≈ E G, thus 1 ≈ (A − S[G] − z)G. This motivates studying the solution M = M(z) to the MDE 1 = (A − S[M] − z)M.

Density of States = Empirical Distribution of Eigenvalues

−4 −2 2 4 0.1 0.2

Density of States = Empirical Distribution of Eigenvalues

−4 −2 2 4 0.1 0.2 Density of states ρ can be found by solving the matrix Dyson equation −M(z)−1 = z − A + S[M(z)], ℑM = M − M∗ 2i > 0, for ℑz > 0, to obtain the Stieltjes transform N−1 Tr M(z) and by Stieltjes inversion ρ(E) .

.= lim

ηց0

1 πN

• i

ℑMii(E + iη).

Density of States = Empirical Distribution of Eigenvalues

−4 −2 2 4 0.1 0.2 Complete classification of singularities of DOS ρ achieved in (Alt, Erdős, Krüger 2018)

• Only square-root edges and cubic root cusps
• Cusps are not as ubiquitous as edges but arise naturally when two support intervals

merge

Global and Mesoscopic Scales

Define the fluctuation scale ηf ηf(E)

−ηf(E)

ρ(E + y) dy = 1 N . Typical scaling ηf =        N−1 bulk N−3/4 cusp N−2/3 edge Global law: G M 1 for z 1 Local law (on mesoscopic scales): G M 1 for z

f

z Optimal local law: G M N N for

f

Global and Mesoscopic Scales

Define the fluctuation scale ηf ηf(E)

−ηf(E)

ρ(E + y) dy = 1 N . Typical scaling ηf =        N−1 bulk N−3/4 cusp N−2/3 edge Global law: |G − M| ≪ 1 for η = ℑz = O(1) Local law (on mesoscopic scales): |G − M| ≪ 1 for η = ℑz ≫ ηf (ℜz) Optimal local law: |G − M| Nǫ Nη for η ≫ ηf

Proof of the Optimal Local Law

Optimal Local Law

Central idea: The resolvent G(z) = (H − z)−1 almost fulfils the MDE, i.e. −1 = (z − A + S[M])M, −1 = (z − A + S[G])G + D, D .

.= WG + S[G]G

G − M = (1 − MS[·]M)−1[MD] + . . . , stability operator B .

.= 1 − MS[·]M

Stability analysis (Alt, Erdős, Krüger 2018) The operator B has a smallest eigenvalue and eigenmatrix B V V with spectral projection P V V, On the complement Q .

.

1 P, we have the bound BQ

1

C, Scaling 1 bulk, edge,

2

cusp. High probability error bound In all spectral regimes, and also for correlated matrices for general X (Erdős, Krüger, S. 2017) XD N X N For indep. matrices improved cusp bound (Erdős, Krüger, S. 2018) VMD N

2

N

Optimal Local Law

Central idea: The resolvent G(z) = (H − z)−1 almost fulfils the MDE, i.e. −1 = (z − A + S[M])M, −1 = (z − A + S[G])G + D, D .

.= WG + S[G]G

G − M = (1 − MS[·]M)−1[MD] + . . . , stability operator B .

.= 1 − MS[·]M

Stability analysis (Alt, Erdős, Krüger 2018)

• The operator B has a smallest eigenvalue

β and eigenmatrix B[V] = βV with spectral projection P[V] = V,

• On the complement Q .

.= 1 − P, we have

the bound

• (BQ)−1

≤ C,

• Scaling |β| ∼

       1 bulk, ρ edge, ρ2 cusp. High probability error bound In all spectral regimes, and also for correlated matrices for general X (Erdős, Krüger, S. 2017) XD N X N For indep. matrices improved cusp bound (Erdős, Krüger, S. 2018) VMD N

2

N

Optimal Local Law

Central idea: The resolvent G(z) = (H − z)−1 almost fulfils the MDE, i.e. −1 = (z − A + S[M])M, −1 = (z − A + S[G])G + D, D .

.= WG + S[G]G

G − M = (1 − MS[·]M)−1[MD] + . . . , stability operator B .

.= 1 − MS[·]M

Stability analysis (Alt, Erdős, Krüger 2018)

• The operator B has a smallest eigenvalue

β and eigenmatrix B[V] = βV with spectral projection P[V] = V,

• On the complement Q .

.= 1 − P, we have

the bound

• (BQ)−1

≤ C,

• Scaling |β| ∼

       1 bulk, ρ edge, ρ2 cusp. High probability error bound

• In all spectral regimes, and also for

correlated matrices for general X (Erdős, Krüger, S. 2017) |XD| Nǫ X ρ Nη ,

• For indep. matrices improved cusp

bound (Erdős, Krüger, S. 2018) |VMD| Nǫ ρ2 Nη .

Cumulant Expansion

Classical cumulant expansion E xf(x) =

• k

κk+1(x) k! E f (k)(x).

• First application to RMT in (Khorunzhy, Khoruzhenko, Pastur 1996), revived in (He,

Knowles 2017).

• An iterated and multivariate version (Erdős, Krüger, S. 2017) allows to compute

E |XD|p , automatically exploiting the cancellation in D to p-th order.

Graphical Iterated Cumulant Expansion

N2 E |diag(x)D|2 = Val

• S

x2 1

• + Val
• T

x x

• + Val

       

S S x 1 1 x

        + Val        

T S x 1 x 1

        + Val        

S T x 1 1 x

        + Val        

T T x 1 1 x

        + . . . For example:

S x2 1

E

a b

x2

asabGabGba a b

x2

asabGabGba

N

1 a b

Gab

2

GG G gain of N per off-diag. G

Graphical Iterated Cumulant Expansion

N2 E |diag(x)D|2 = Val

• S

x2 1

• + Val
• T

x x

• + Val

       

S S x 1 1 x

        + Val        

T S x 1 x 1

        + Val        

S T x 1 1 x

        + Val        

T T x 1 1 x

        + . . . For example: Val

• S

x2 1

• = E
• a,b

x2

asabG∗ abGba a b

x2

asabGabGba

N

1 a b

Gab

2

GG G gain of N per off-diag. G

Graphical Iterated Cumulant Expansion

N2 E |diag(x)D|2 = Val

• S

x2 1

• + Val
• T

x x

• + Val

       

S S x 1 1 x

        + Val        

T S x 1 x 1

        + Val        

S T x 1 1 x

        + Val        

T T x 1 1 x

        + . . . For example: Val

• S

x2 1

• = E
• a,b

x2

asabG∗ abGba

• a,b

x2

asabG∗ abGba

• N−1

a,b

|Gab|2 = GG∗ = ℑG η ρ η ⇒ gain of ρ Nη per off-diag. G

Cusp Fluctuation Averaging

The following σ-cell subgraphs

S x 1 , S 1 x

,

S x 1

are key for the cusp-specific cancellation.

S x 1 = P S P x 1

+

Q S

+

P S Q

For x VM cusp cancellation

P S P x 1

N where 0 at the cusp and 1 at the edge and in the bulk. An additional cumulant expansion gives

B

N complete the proof by counting the number of

• cells.

Cusp Fluctuation Averaging

The following σ-cell subgraphs

S x 1 , S 1 x

,

S x 1

are key for the cusp-specific cancellation.

S x 1 = P S P x 1

+

Q S

+

P S Q

For x = diag(VM) cusp cancellation

• P

S P x 1

• ∼ ρ + σ

N , where σ = 0 at the cusp and σ ∼ 1 at the edge and in the bulk. An additional cumulant expansion gives

B

N complete the proof by counting the number of

• cells.

Cusp Fluctuation Averaging

The following σ-cell subgraphs

S x 1 , S 1 x

,

S x 1

are key for the cusp-specific cancellation.

S x 1 = P S P x 1

+

Q S

+

P S Q

For x = diag(VM) cusp cancellation

• P

S P x 1

• ∼ ρ + σ

N , where σ = 0 at the cusp and σ ∼ 1 at the edge and in the bulk. An additional cumulant expansion gives Val     

B

     ρ Nη Val       complete the proof by counting the number of

• cells.

Cusp Fluctuation Averaging

The following σ-cell subgraphs

S x 1 , S 1 x

,

S x 1

are key for the cusp-specific cancellation.

S x 1 = P S P x 1

+

Q S

+

P S Q

For x = diag(VM) cusp cancellation

• P

S P x 1

• ∼ ρ + σ

N , where σ = 0 at the cusp and σ ∼ 1 at the edge and in the bulk. An additional cumulant expansion gives Val     

B

     ρ Nη Val       ⇒ complete the proof by counting the number of σ-cells.

Universality for RM with Gaussian Component via Contour Integration

Step 3: Universality via Contour Integration

The Brezin-Hikami formula expresses the kernel K(u, v) of H + √ tU as N (2πi)2t

• Υ

dz

• Γ

dw exp(N[f(w, v) − f(z, u)]) w − z , f(z, u) .

.= (z − u)2

2t + 1 N

• i

log(z − λi) in terms of the eigenvalues λi of H. Υ Γ Band rigidity (Alt, Erdős, Krüger, S. 2018) ensures that the number of eigenvalues is fixed in each support interval, as long as they are at least N−3/4+ǫ-separated.

Step 3: Universality via Contour Integration

The Brezin-Hikami formula expresses the kernel K(u, v) of H + √ tU as N (2πi)2t

• Υ

dz

• Γ

dw exp(N[f(w, v) − f(z, u)]) w − z , f(z, u) .

.= (z − u)2

2t + 1 N

• i

log(z − λi) in terms of the eigenvalues λi of H. − − − − + + + + We use a fourth order saddle-point analysis and the local law to obtain the Pearcey kernel for t ≫ N−1/2.

Step 3: Universality via Contour Integration

The Brezin-Hikami formula expresses the kernel K(u, v) of H + √ tU as N (2πi)2t

• Υ

dz

• Γ

dw exp(N[f(w, v) − f(z, u)]) w − z , f(z, u) .

.= (z − u)2

2t + 1 N

• i

log(z − λi) in terms of the eigenvalues λi of H. A precise quantitative understanding of the semicircular flow is essential.

Semicircular Flow

If M solves the MDE corresponding to H, then mfc

s (z) = M(z + smfc s (z)) ,

ℑmfc

s (z) > 0,

ℑz > 0 is the Stieltjes transform of the self-consistent density ρfc

s of H + √sU. The density ρfc s is the

free convolution ρfc

s = ρ ⊞ √sρsc

with a semicircular distribution of variance s. Scaling in the vicinity of a cusp (formed at time s ): Before the cusp: gap

s s

• f size

s s

s s 3 2 moving linearly

s

sm0 . Afuer the cusp: minimum

s of size fc s s

s s

1 2 moving linearly s s

s s mfc

s s

.

Semicircular Flow

If M solves the MDE corresponding to H, then mfc

s (z) = M(z + smfc s (z)) ,

ℑmfc

s (z) > 0,

ℑz > 0 is the Stieltjes transform of the self-consistent density ρfc

s of H + √sU. The density ρfc s is the

free convolution ρfc

s = ρ ⊞ √sρsc

with a semicircular distribution of variance s. Scaling in the vicinity of a cusp (formed at time s∗):

• Before the cusp: gap [e−

s , e+ s ] of size e+ s − e− s ∼ (s∗ − s)3/2 moving linearly

e±

s ≈ e± 0 − sm0(e± 0 ).

• Afuer the cusp: minimum ms of size ρfc

s (ms) ∼ (s − s∗)1/2 moving linearly

ms ≈ ms∗ − (s − s∗)mfc

s∗(ms∗).

Thank you very much for your attention!

References

References

• J. Alt, L. Erdős, T. Krüger, “The Dyson equation with linear self-energy: spectral bands,

edges and cusps”, preprint (2018), arXiv:1804.07752.

• J. Alt, L. Erdős, T. Krüger, D. S., “Correlated random matrices: band rigidity and edge

universality”, preprint (2018), arXiv:1804.07744.

• L. Erdős, T. Krüger, D. S., “Cusp universality for random matrices I: local law and the

complex Hermitian case”, preprint (2018), arXiv:1809.03971.

• L. Erdős, T. Krüger, D. S., “Random matrices with slow correlation decay”, accepted for

publication in Forum Math. Sigma (2017), arXiv:1705.10661.

• Y. He, A. Knowles, “Mesoscopic eigenvalue statistics of Wigner matrices”, Ann. Appl.
• Probab. 27, 1510–1550 (2017).
• A. M. Khorunzhy, B. A. Khoruzhenko, L. A. Pastur, “Asymptotic properties of large random

matrices with independent entries”, J. Math. Phys. 37, 5033–5060 (1996).

Harish-Chandra-Itzykson-Zuber integral

Let A, B be Hermitian N × N matrices with eigenvalues (λi) and (µi). Then the integral

• U
• ver the unitary group is given by
• U

exp(t Tr(AUBU∗)) dU = N−1

• i=1

i!

• det
• tλiµj
• i,j

t(N2−N)/2∆(λ)∆(µ) , where ∆(λ) .

.=

• i<j

(λj − λi) denotes the Vandermonde determinant.