Random matrices, differential operators and carousels Benedek Valk - - PowerPoint PPT Presentation

Random matrices, differential operators and carousels

Benedek Valk´

• (University of Wisconsin – Madison)

joint with B. Vir´ ag (Toronto) March 24, 2016

Basic question of RMT:

What can we say about the spectrum of a large random matrix?

Basic question of RMT:

What can we say about the spectrum of a large random matrix?

60 40 20 20 40 60 5 10 15 20 25 30 35

b HLn - aL

global local

Basic question of RMT:

What can we say about the spectrum of a large random matrix?

60 40 20 20 40 60 5 10 15 20 25 30 35

b HLn - aL

global local In this talk: local picture (point process limits)

A classical example: Gaussian Unitary Ensemble

M = A+A∗

√ 2 ,

A is n × n with iid complex std normal.

A classical example: Gaussian Unitary Ensemble

M = A+A∗

√ 2 ,

A is n × n with iid complex std normal. Global picture: Wigner semicircle law

A classical example: Gaussian Unitary Ensemble

M = A+A∗

√ 2 ,

A is n × n with iid complex std normal. Global picture: Wigner semicircle law

A classical example: Gaussian Unitary Ensemble

M = A+A∗

√ 2 ,

A is n × n with iid complex std normal. Global picture: Wigner semicircle law

Local picture: point process limit in the bulk and near the edge (Bulk: Dyson-Gaudin-Mehta, edge: Tracy-Widom)

The limit processes are characterized by their joint intensity functions.

A classical example: Gaussian Unitary Ensemble

M = A+A∗

√ 2 ,

A is n × n with iid complex std normal. Global picture: Wigner semicircle law

Local picture: point process limit in the bulk and near the edge (Bulk: Dyson-Gaudin-Mehta, edge: Tracy-Widom)

The limit processes are characterized by their joint intensity functions. Roughly: what is the probability of finding points near x1, . . . , xn

Point process limit

b HLn - aL

Finite n: spectrum of a random Hermitian matrix

Point process limit

b HLn - aL

Finite n: spectrum of a random Hermitian matrix Limit point process: spectrum of ??

Detour to number theory

Riemann zeta function: ζ(s) =

∞

• n=1

1 ns , for Re s > 1.

(Analytic continuation to C \ {1})

Riemann hypothesis: the non-trivial zeros are on the line Re s = 1

2.

Detour to number theory

Riemann zeta function: ζ(s) =

∞

• n=1

1 ns , for Re s > 1.

(Analytic continuation to C \ {1})

Riemann hypothesis: the non-trivial zeros are on the line Re s = 1

2.

Dyson-Montgomery conjecture: After some scaling: non-trivial zeros of ζ(1 2 + i s) ∼ bulk limit process of GUE

(Sine2 process)

Detour to number theory

Riemann zeta function: ζ(s) =

∞

• n=1

1 ns , for Re s > 1.

(Analytic continuation to C \ {1})

Riemann hypothesis: the non-trivial zeros are on the line Re s = 1

2.

Dyson-Montgomery conjecture: After some scaling: non-trivial zeros of ζ(1 2 + i s) ∼ bulk limit process of GUE

(Sine2 process)

◮ Strong numerical evidence: Odlyzko ◮ Certain weaker versions are proved

(Montgomery, Rudnick-Sarnak)

Hilbert-P´

• lya conjecture: the Riemann hypotheses is true because

non-trivial zeros of ζ(1 2 + i s) = ev’s of an unbounded self-adjoint operator

Hilbert-P´

• lya conjecture: the Riemann hypotheses is true because

non-trivial zeros of ζ(1 2 + i s) = ev’s of an unbounded self-adjoint operator A famous attempt to make this approach rigorous: de Branges

Hilbert-P´

• lya conjecture: the Riemann hypotheses is true because

non-trivial zeros of ζ(1 2 + i s) = ev’s of an unbounded self-adjoint operator A famous attempt to make this approach rigorous: de Branges

(based on the theory of Hilbert spaces of entire functions)

This approach would produce a self-adjoint differential operator with the appropriate spectrum.

Natural question: Is there a self-adjoint differential operator with a spectrum given by the bulk limit of GUE?

Natural question: Is there a self-adjoint differential operator with a spectrum given by the bulk limit of GUE?

Disclaimer: A positive answer would not get us closer to any of the conjectures

• r the Riemann hypothesis (unfortunately...)

Natural question: Is there a self-adjoint differential operator with a spectrum given by the bulk limit of GUE?

Disclaimer: A positive answer would not get us closer to any of the conjectures

• r the Riemann hypothesis (unfortunately...)

Borodin-Olshanski, Maples-Najnudel-Nikeghbali: ‘operator-like object’ with generalized eigenvalues distributed as Sine2

Starting point for deriving the Sine2 process:

Starting point for deriving the Sine2 process: Joint eigenvalue density of GUE: 1 Zn

• i<j≤n

|λj − λi|2

n

• i=1

e− 1

2 λ2 i

Starting point for deriving the Sine2 process: Joint eigenvalue density of GUE: 1 Zn

• i<j≤n

|λj − λi|2

n

• i=1

e− 1

2 λ2 i

Many of the classical random matrix ensembles have joint eigenvalue densities of the form 1 Zn,f ,β

• i<j≤n

|λj − λi|β

n

• i=1

f (λi) with β = 1, 2 or 4 and f a specific reference density.

β-ensemble: finite point process with joint density 1 Zn,f ,β

• i<j≤n

|λj − λi|β

n

• i=1

f (λi) f (·): reference density Examples:

◮ Hermite or Gaussian: normal density ◮ Laguerre or Wishart: gamma density ◮ Jacobi or MANOVA: beta density ◮ circular: uniform on the unit circle

β = 1, 2, 4: classical random matrix models

Scaling limits - global picture

Hermite β-ensemble semicircle law Laguerre β-ensemble Marchenko-Pastur law

2 2 1 2 3 4

↑ ↑ ր ↑ ↑ ↑

soft edge bulk

• s. e.

hard edge bulk

• s. e.

Local limits

Soft edge: Rider-Ram´ ırez-Vir´ ag (Hermite, Laguerre)

Airyβ process

Hard edge: Rider-Ram´ ırez (Laguerre)

Besselβ,a processes

Bulk: Killip-Stoiciu, V.-Vir´ ag (circular, Hermite)

CβE and Sineβ processes

Local limits

Soft edge: Rider-Ram´ ırez-Vir´ ag (Hermite, Laguerre)

Airyβ process

Hard edge: Rider-Ram´ ırez (Laguerre)

Besselβ,a processes

Bulk: Killip-Stoiciu, V.-Vir´ ag (circular, Hermite)

CβE and Sineβ processes

Instead of joint intensities, the limit processes are described via their counting functions using coupled systems of SDEs. sign(λ) · (# of points in [0, λ])

Operators at the edge

Soft edge: Airyβ is the spectrum of Aβ = − d2 dx2 + x + 2 √β dB

dB: white noise

Operators at the edge

Soft edge: Airyβ is the spectrum of Aβ = − d2 dx2 + x + 2 √β dB

dB: white noise

Hard edge: Besselβ,a is the spectrum of Bβ,a = −e(a+1)x+

2 √β B(x) d

dx

• e−ax−

2 √β B(x) d

dx

• B: standard Brownian motion

Random second order self-adjoint differential operators on [0, ∞).

Edelman-Sutton: non-rigorous versions of these operators

Operators at the edge

Soft edge: Airyβ is the spectrum of Aβ = − d2 dx2 + x + 2 √β dB

dB: white noise

Hard edge: Besselβ,a is the spectrum of Bβ,a = −e(a+1)x+

2 √β B(x) d

dx

• e−ax−

2 √β B(x) d

dx

• B: standard Brownian motion

Random second order self-adjoint differential operators on [0, ∞).

Edelman-Sutton: non-rigorous versions of these operators What about the bulk? Is there an operator for CβE or Sineβ?

The Sineβ operator

Thm (V-Vir´ ag): There is a self-adjoint differential operator (Dirac-operator) f → 2R−1

t

−1 1

• f ′(t),

f : [0, 1) → R2. where Rt is a random 2 × 2 positive definite matrix valued function so that the spectrum is the Sineβ process.

The Sineβ operator

Thm (V-Vir´ ag): There is a self-adjoint differential operator (Dirac-operator) f → 2R−1

t

−1 1

• f ′(t),

f : [0, 1) → R2. where Rt is a random 2 × 2 positive definite matrix valued function so that the spectrum is the Sineβ process.

Rt is given a simple function of a hyperbolic Brownian motion.

The Sineβ operator

Thm (V-Vir´ ag): There is a self-adjoint differential operator (Dirac-operator) f → 2R−1

t

−1 1

• f ′(t),

f : [0, 1) → R2. where Rt is a random 2 × 2 positive definite matrix valued function so that the spectrum is the Sineβ process.

Rt is given a simple function of a hyperbolic Brownian motion. This is a first order differential operator.

Digression: the hyperbolic plane H

Disk model Halfplane model

A geometric description of Sineβ

Hyperbolic carousel: (η0, η∞, γ) point process η0, η∞: points on the boundary of the hyperbolic plane H γ : [0, 1) → H: a path in the hyperbolic plane

η γ η∞

λ

A geometric description of Sineβ

Hyperbolic carousel: (η0, η∞, γ) point process η0, η∞: points on the boundary of the hyperbolic plane H γ : [0, 1) → H: a path in the hyperbolic plane

η0 γ(t) η∞ zλ(t)

For each λ ∈ R we start a point zλ from η0 and rotate it continuously around γ(t) with rate λ. (This is just an ODE!)

A geometric description of Sineβ

Hyperbolic carousel: (η0, η∞, γ) point process η0, η∞: points on the boundary of the hyperbolic plane H γ : [0, 1) → H: a path in the hyperbolic plane

η0 γ(t) η∞ zλ(t)

For each λ ∈ R we start a point zλ from η0 and rotate it continuously around γ(t) with rate λ. (This is just an ODE!) N(λ): # of times zλ hits η∞. This is the counting function of the point process.

A geometric description of Sineβ

V.-Vir´ ag (’07): if γ is a time changed hyperbolic Brownian motion, η∞ is its limit point and η0 is a fixed boundary point then (η0, η∞, γ) Sineβ

(β only appears in the time change: t → − 4

β log(1 − t))

Carousel ∼ Dirac operator

Carousel ∼ Dirac operator

Suppose that γ(t) = xt + iyt in the half-plane coordinates.

From the carousel (η0, η∞, γ) we can build the self-adjoint Dirac operator

τ : f → 2(UTU)−1 −1 1

• f ′(t),

U = 1 √yt 1 −xt yy

• (η0, η∞ boundary conditions)

Carousel ∼ Dirac operator

Suppose that γ(t) = xt + iyt in the half-plane coordinates.

From the carousel (η0, η∞, γ) we can build the self-adjoint Dirac operator

τ : f → 2(UTU)−1 −1 1

• f ′(t),

U = 1 √yt 1 −xt yy

• (η0, η∞ boundary conditions)

point process produced by (η0, η∞, γ)= spectrum of τ

Carousel ∼ Dirac operator

Suppose that γ(t) = xt + iyt in the half-plane coordinates.

From the carousel (η0, η∞, γ) we can build the self-adjoint Dirac operator

τ : f → 2(UTU)−1 −1 1

• f ′(t),

U = 1 √yt 1 −xt yy

• (η0, η∞ boundary conditions)

point process produced by (η0, η∞, γ)= spectrum of τ Main idea of the proof: Sturm-Liouville oscillation theory τv(t, λ) = λv(t, λ), v1(t, λ) + iv2(t, λ) = r(t, λ)eiθ(t,λ) The spectrum can be identified from θ(·, ·) which basically evolves according to a carousel.

Carousel ∼ Dirac operator

(η0, η∞, γ)

• τ : f → 2R−1

t

−1 1

• f ′(t),

Carousel ∼ Dirac operator

(η0, η∞, γ)

• τ : f → 2R−1

t

−1 1

• f ′(t),

Under mild conditions: τ −1 is a Hilbert-Schmidt integral operator with kernel K(x, y) =

• u0uT

1 1(x < y) + u1uT 0 1(x ≥ y)

• R(y)

u0, u1: boundary conditions in τ

Carousel ∼ Dirac operator

(η0, η∞, γ)

• τ : f → 2R−1

t

−1 1

• f ′(t),

Under mild conditions: τ −1 is a Hilbert-Schmidt integral operator with kernel K(x, y) =

• u0uT

1 1(x < y) + u1uT 0 1(x ≥ y)

• R(y)

u0, u1: boundary conditions in τ Nice property: if the path γ lives on [0, T) then the operator can be approximated using the path restricted to [0, T − ε).

Carousel ∼ Dirac operator

τ : f → 2(UTU)−1 −1 1

• f ′(t),

U = 1 √yt 1 −xt yy

• Brownian carousel representation of Sineβ

⇓ random differential operator for Sineβ

Carousel ∼ Dirac operator

τ : f → 2(UTU)−1 −1 1

• f ′(t),

U = 1 √yt 1 −xt yy

• Brownian carousel representation of Sineβ

⇓ random differential operator for Sineβ xt + iyt: time-changed hyperbolic Brownian motion

Additional results

◮ CβE d

= Sineβ Using the hyperbolic carousel ∼ Dirac operator connection

(Nakano: independent proof)

Additional results

◮ CβE d

= Sineβ Using the hyperbolic carousel ∼ Dirac operator connection

(Nakano: independent proof)

◮ Dirac operator description for deterministic unitary matrices

Additional results

◮ CβE d

= Sineβ Using the hyperbolic carousel ∼ Dirac operator connection

(Nakano: independent proof)

◮ Dirac operator description for deterministic unitary matrices ◮ Random Dirac-operator description for other classical models

driving paths: ‘affine’ hyperbolic Brownian motions

Additional results

◮ CβE d

= Sineβ Using the hyperbolic carousel ∼ Dirac operator connection

(Nakano: independent proof)

◮ Dirac operator description for deterministic unitary matrices ◮ Random Dirac-operator description for other classical models

driving paths: ‘affine’ hyperbolic Brownian motions

◮ Soft edge limit: representation as a canonical system

−1 1

• f ′(t) = λRtf (t)

(rank(Rt) = 1)

Additional results

◮ CβE d

= Sineβ Using the hyperbolic carousel ∼ Dirac operator connection

(Nakano: independent proof)

◮ Dirac operator description for deterministic unitary matrices ◮ Random Dirac-operator description for other classical models

driving paths: ‘affine’ hyperbolic Brownian motions

◮ Soft edge limit: representation as a canonical system

−1 1

• f ′(t) = λRtf (t)

(rank(Rt) = 1)

◮ Bulk convergence via the operators

CβE

d

= Sineβ

Circular β-ensemble: n points on the unit circle with joint density 1 Zn,β

• i<j≤n

|eiλi − eiλj|β

CβE

d

= Sineβ

Circular β-ensemble: n points on the unit circle with joint density 1 Zn,β

• i<j≤n

|eiλi − eiλj|β Killip-Stoiciu: {nλi, 1 ≤ i ≤ n} converges to a point process CβE

CβE

d

= Sineβ

Circular β-ensemble: n points on the unit circle with joint density 1 Zn,β

• i<j≤n

|eiλi − eiλj|β Killip-Stoiciu: {nλi, 1 ≤ i ≤ n} converges to a point process CβE Description: coupled SDE system counting function

CβE

d

= Sineβ

Circular β-ensemble: n points on the unit circle with joint density 1 Zn,β

• i<j≤n

|eiλi − eiλj|β Killip-Stoiciu: {nλi, 1 ≤ i ≤ n} converges to a point process CβE Description: coupled SDE system counting function

Similar to the Sineβ description, but time is reversed, different end cond.

CβE

d

= Sineβ

Circular β-ensemble: n points on the unit circle with joint density 1 Zn,β

• i<j≤n

|eiλi − eiλj|β Killip-Stoiciu: {nλi, 1 ≤ i ≤ n} converges to a point process CβE Description: coupled SDE system counting function

Similar to the Sineβ description, but time is reversed, different end cond.

One can reformulate the Killip-Stoiciu description as a hyperbolic carousel driven by a time-reversed version of the Sineβ carousel.

CβE

d

= Sineβ

Circular β-ensemble: n points on the unit circle with joint density 1 Zn,β

• i<j≤n

|eiλi − eiλj|β Killip-Stoiciu: {nλi, 1 ≤ i ≤ n} converges to a point process CβE Description: coupled SDE system counting function

Similar to the Sineβ description, but time is reversed, different end cond.

One can reformulate the Killip-Stoiciu description as a hyperbolic carousel driven by a time-reversed version of the Sineβ carousel. Reversing time in the carousel one can show that CβE d = Sineβ

Dirac operators for unitary matrices

V : n × n unitary matrix with distinct eigenvalues e: a cyclic unit vector

Dirac operators for unitary matrices

V : n × n unitary matrix with distinct eigenvalues e: a cyclic unit vector Apply G-S to e, Ve, . . . , V n−1e Szeg˝

• recursion for OPUC

Dirac operators for unitary matrices

V : n × n unitary matrix with distinct eigenvalues e: a cyclic unit vector Apply G-S to e, Ve, . . . , V n−1e Szeg˝

• recursion for OPUC
• Φk+1(z)

Φ∗

k+1(z)

• =
• 1

−¯ αk −αk 1 z 1 Φk(z) Φ∗

k(z)

• ,
• Φ∗

0(z)

Φ0(z)

• =
• 1

1

• αk: Verblunsky coefficients, |αk| ≤ 1

Dirac operators for unitary matrices

V : n × n unitary matrix with distinct eigenvalues e: a cyclic unit vector Apply G-S to e, Ve, . . . , V n−1e Szeg˝

• recursion for OPUC
• Φk+1(z)

Φ∗

k+1(z)

• =
• 1

−¯ αk −αk 1 z 1 Φk(z) Φ∗

k(z)

• ,
• Φ∗

0(z)

Φ0(z)

• =
• 1

1

• αk: Verblunsky coefficients, |αk| ≤ 1

z is an e.v. ⇔ z 1 Φn−1(z) Φ∗

n−1(z)

• ¯

αn−1 1

Dirac operators for unitary matrices

• Φk+1(z)

Φ∗

k+1(z)

• =
• 1

−¯ αk −αk 1 z 1 Φk(z) Φ∗

k(z)

• ,
• Φ∗

0(z)

Φ0(z)

• =
• 1

1

• We can introduce a transformed version of

Φk(z) Φ∗

k(z)

• satisfying

gk+1 = M−1

k

• ei µ

2n

e−i µ

2n

• Mkgk,

z = ei µ

n

Mk: product of 1 −¯ αj −αj 1

• matrices

Dirac operators for unitary matrices

• Φk+1(z)

Φ∗

k+1(z)

• =
• 1

−¯ αk −αk 1 z 1 Φk(z) Φ∗

k(z)

• ,
• Φ∗

0(z)

Φ0(z)

• =
• 1

1

• We can introduce a transformed version of

Φk(z) Φ∗

k(z)

• satisfying

gk+1 = M−1

k

• ei µ

2n

e−i µ

2n

• Mkgk,

z = ei µ

n

Mk: product of 1 −¯ αj −αj 1

• matrices

This gives an actual Dirac operator with piecewise continuous Rt.

Dirac operators for unitary matrices

• Φk+1(z)

Φ∗

k+1(z)

• =
• 1

−¯ αk −αk 1 z 1 Φk(z) Φ∗

k(z)

• ,
• Φ∗

0(z)

Φ0(z)

• =
• 1

1

• We can introduce a transformed version of

Φk(z) Φ∗

k(z)

• satisfying

gk+1 = M−1

k

• ei µ

2n

e−i µ

2n

• Mkgk,

z = ei µ

n

Mk: product of 1 −¯ αj −αj 1

• matrices

This gives an actual Dirac operator with piecewise continuous Rt. The path γ: a discrete walk in H.

More β-ensembles

1 Zn,f ,β

• i<j≤n

|λj − λi|β

n

• i=1

f (λi)

More β-ensembles

1 Zn,f ,β

• i<j≤n

|λj − λi|β

n

• i=1

f (λi) Dumitriu-Edelman: tridiagonal matrix models for Hermite and Laguerre β-ensembles Killip-Nenciu: models for the circular β-ensembles, using the Szeg˝

• -recursion and random Verblunsky coefficients

More β-ensembles

1 Zn,f ,β

• i<j≤n

|λj − λi|β

n

• i=1

f (λi) Dumitriu-Edelman: tridiagonal matrix models for Hermite and Laguerre β-ensembles Killip-Nenciu: models for the circular β-ensembles, using the Szeg˝

• -recursion and random Verblunsky coefficients

Edelman-Sutton: the rescaled tridiagonal models can be viewed as discrete versions of random differential operators

Operator convergence

One can find the discrete versions of the limit operators in the finite tridiagonal models. Soft edge: in the appropriate scaling the tridiagonal matrix can be written as a sum of a discrete Laplacian, a discrete white noise potential and a potential approximating the function x

Operator convergence

One can find the discrete versions of the limit operators in the finite tridiagonal models. Soft edge: in the appropriate scaling the tridiagonal matrix can be written as a sum of a discrete Laplacian, a discrete white noise potential and a potential approximating the function x Hard edge: the inverse of the tridiagonal matrix (as a product of two bidiagonal matrices) can be written as an integral operator approximating the inverse of the Bβ,a operator

Operator convergence

One can find the discrete versions of the limit operators in the finite tridiagonal models. Soft edge: in the appropriate scaling the tridiagonal matrix can be written as a sum of a discrete Laplacian, a discrete white noise potential and a potential approximating the function x Hard edge: the inverse of the tridiagonal matrix (as a product of two bidiagonal matrices) can be written as an integral operator approximating the inverse of the Bβ,a operator What about the bulk?

Operator level bulk limit

discrete model ↓ discrete ‘differential operator’ ↓ discrete integral operator ↓ limiting integral operator

Operator level bulk limit

discrete model ↓ discrete ‘differential operator’ ↓ discrete integral operator ↓ limiting integral operator The previous methods required the derivation of a one-parameter family of SDE system. Here we need to understand the limit of the integral kernel: a single SDE.

Dirac operators for other models

◮ finite circular β-ensemble and circular Jacobi ensembles ◮ limits of circular Jacobi ensembles ◮ hard edge limits ◮ certain one dimensional random Schr¨

• dinger operators

Dirac operators for other models

◮ finite circular β-ensemble and circular Jacobi ensembles ◮ limits of circular Jacobi ensembles ◮ hard edge limits ◮ certain one dimensional random Schr¨

• dinger operators

In each case the path γ is a random walk or diffusion on H.

Dirac operators for other models

◮ finite circular β-ensemble and circular Jacobi ensembles ◮ limits of circular Jacobi ensembles ◮ hard edge limits ◮ certain one dimensional random Schr¨

• dinger operators

In each case the path γ is a random walk or diffusion on H.

◮ finite circular β-ensemble and circular Jacobi ensembles: γ is

a random walk in H

◮ Hard edge: γ is a real BM with drift embedded in H ◮ circular Jacobi: γ is a ‘hyperbolic BM with drift’

Dirac operators from tridiagonal matrices?

The eigenvalue equation is a three-term recursion Mu = λu

• aℓuℓ−1 + bℓuℓ + aℓuℓ+1 = λuℓ

Dirac operators from tridiagonal matrices?

The eigenvalue equation is a three-term recursion Mu = λu

• aℓuℓ−1 + bℓuℓ + aℓuℓ+1 = λuℓ

This can be reformulated with transfer matrices: Tℓ uℓ−1 uℓ

• −
• uℓ

uℓ+1

• =

λ uℓ uℓ+1

• ,

u0 u1

• =

1

• .

Dirac operators from tridiagonal matrices?

The eigenvalue equation is a three-term recursion Mu = λu

• aℓuℓ−1 + bℓuℓ + aℓuℓ+1 = λuℓ

This can be reformulated with transfer matrices: Tℓ uℓ−1 uℓ

• −
• uℓ

uℓ+1

• =

λ uℓ uℓ+1

• ,

u0 u1

• =

1

• .

After conjugation and some averaging, one can recover the eigenvalue equation of a Dirac operator.

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