Random matrices, differential operators andcarousels

Benedek Valko(University of Wisconsin – Madison)

joint with B. Virag (Toronto)

March 24, 2016

Basic question of RMT:

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

-60 -40 -20 0 20 40 60

5

10

15

20

25

30

35

Ln

b HLn - aL

global local

In this talk: local picture (point process limits)

Basic question of RMT:

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

-60 -40 -20 0 20 40 60

5

10

15

20

25

30

35

Ln

b HLn - aL

global local

In this talk: local picture (point process limits)

Basic question of RMT:

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

-60 -40 -20 0 20 40 60

5

10

15

20

25

30

35

Ln

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.

Global picture: Wigner semicircle law -60 -40 -20 0 20 40 60

5

10

15

20

25

30

35

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

A classical example: Gaussian Unitary Ensemble

M = A+A∗√2

, A is n × n with iid complex std normal.

Global picture: Wigner semicircle law

-60 -40 -20 0 20 40 60

5

10

15

20

25

30

35

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

A classical example: Gaussian Unitary Ensemble

M = A+A∗√2

, A is n × n with iid complex std normal.

Global picture: Wigner semicircle law -60 -40 -20 0 20 40 60

5

10

15

20

25

30

35

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

A classical example: Gaussian Unitary Ensemble

M = A+A∗√2

, A is n × n with iid complex std normal.

Global picture: Wigner semicircle law -60 -40 -20 0 20 40 60

5

10

15

20

25

30

35

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

A classical example: Gaussian Unitary Ensemble

M = A+A∗√2

, A is n × n with iid complex std normal.

Global picture: Wigner semicircle law -60 -40 -20 0 20 40 60

5

10

15

20

25

30

35

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

Ln

b HLn - aL

Finite n: spectrum of a random Hermitian matrix

Limit point process: spectrum of ??

Point process limit

Ln

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

1ns , for Re s > 1.

(Analytic continuation to C \ {1})

Riemann hypothesis: the non-trivial zeros are on the line Re s = 12 .

Dyson-Montgomery conjecture:

After some scaling:

non-trivial zeros of ζ(1

2+ i s) ∼ bulk limit process of GUE

(Sine2 process)

I Strong numerical evidence: Odlyzko

I Certain weaker versions are proved(Montgomery, Rudnick-Sarnak)

Detour to number theory

Riemann zeta function: ζ(s) =∞∑n=1

1ns , for Re s > 1.

(Analytic continuation to C \ {1})

Riemann hypothesis: the non-trivial zeros are on the line Re s = 12 .

Dyson-Montgomery conjecture:

After some scaling:

non-trivial zeros of ζ(1

2+ i s) ∼ bulk limit process of GUE

(Sine2 process)

I Strong numerical evidence: Odlyzko

I Certain weaker versions are proved(Montgomery, Rudnick-Sarnak)

Detour to number theory

Riemann zeta function: ζ(s) =∞∑n=1

1ns , for Re s > 1.

(Analytic continuation to C \ {1})

Riemann hypothesis: the non-trivial zeros are on the line Re s = 12 .

Dyson-Montgomery conjecture:

After some scaling:

non-trivial zeros of ζ(1

2+ i s) ∼ bulk limit process of GUE

(Sine2 process)

I Strong numerical evidence: Odlyzko

I Certain weaker versions are proved(Montgomery, Rudnick-Sarnak)

Hilbert-Polya 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 operatorwith the appropriate spectrum.

Hilbert-Polya 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 operatorwith the appropriate spectrum.

Hilbert-Polya 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 operatorwith the appropriate spectrum.

Natural question:

Is there a self-adjoint differential operator with a spectrum givenby the bulk limit of GUE?

Disclaimer: A positive answer would not get us closer to any of the conjecturesor the Riemann hypothesis (unfortunately...)

Borodin-Olshanski, Maples-Najnudel-Nikeghbali:

‘operator-like object’ with generalized eigenvalues distributed as Sine2

Natural question:

Is there a self-adjoint differential operator with a spectrum givenby the bulk limit of GUE?

Disclaimer: A positive answer would not get us closer to any of the conjecturesor the Riemann hypothesis (unfortunately...)

Borodin-Olshanski, Maples-Najnudel-Nikeghbali:

‘operator-like object’ with generalized eigenvalues distributed as Sine2

Natural question:

Is there a self-adjoint differential operator with a spectrum givenby the bulk limit of GUE?

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

Joint eigenvalue density of GUE:

1

Zn

∏i<j≤n

|λj − λi |2n∏

i=1

e−12λ2i

Many of the classical random matrix ensembles have jointeigenvalue densities of the form

1

Zn,f ,β

∏i<j≤n

|λj − λi |βn∏

i=1

f (λi )

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

Starting point for deriving the Sine2 process:

Joint eigenvalue density of GUE:

1

Zn

∏i<j≤n

|λj − λi |2n∏

i=1

e−12λ2i

Many of the classical random matrix ensembles have jointeigenvalue densities of the form

1

Zn,f ,β

∏i<j≤n

|λj − λi |βn∏

i=1

f (λi )

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

Starting point for deriving the Sine2 process:

Joint eigenvalue density of GUE:

1

Zn

∏i<j≤n

|λj − λi |2n∏

i=1

e−12λ2i

Many of the classical random matrix ensembles have jointeigenvalue densities of the form

1

Zn,f ,β

∏i<j≤n

|λj − λi |βn∏

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 |βn∏

i=1

f (λi )

f (·): reference density

Examples:

I Hermite or Gaussian: normal density

I Laguerre or Wishart: gamma density

I Jacobi or MANOVA: beta density

I circular: uniform on the unit circle

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

Scaling limits - global picture

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

-2 2 1 2 3 4

↑ ↑ ↗ ↑ ↑ ↑soft edge bulk s. e. hard edge bulk s. e.

Local limits

Soft edge: Rider-Ramırez-Virag (Hermite, Laguerre)Airyβ process

Hard edge: Rider-Ramırez (Laguerre)Besselβ,a processes

Bulk: Killip-Stoiciu, V.-Virag (circular, Hermite)CβE and Sineβ processes

Instead of joint intensities, the limit processes are described viatheir counting functions using coupled systems of SDEs.

sign(λ) · (# of points in [0, λ])

Local limits

Soft edge: Rider-Ramırez-Virag (Hermite, Laguerre)Airyβ process

Hard edge: Rider-Ramırez (Laguerre)Besselβ,a processes

Bulk: Killip-Stoiciu, V.-Virag (circular, Hermite)CβE and Sineβ processes

Instead of joint intensities, the limit processes are described viatheir 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√βdB

dB: white noise

Hard edge: Besselβ,a is the spectrum of

Bβ,a = −e(a+1)x+ 2√βB(x) d

dx

{e−ax− 2√

βB(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?

Operators at the edge

Soft edge: Airyβ is the spectrum of

Aβ = − d2

dx2+ x +

2√βdB

dB: white noise

Hard edge: Besselβ,a is the spectrum of

Bβ,a = −e(a+1)x+ 2√βB(x) d

dx

{e−ax− 2√

βB(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?

Operators at the edge

Soft edge: Airyβ is the spectrum of

Aβ = − d2

dx2+ x +

2√βdB

dB: white noise

Hard edge: Besselβ,a is the spectrum of

Bβ,a = −e(a+1)x+ 2√βB(x) d

dx

{e−ax− 2√

βB(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-Virag):There is a self-adjoint differential operator (Dirac-operator)

f → 2R−1t

[0 −11 0

]f ′(t), f : [0, 1)→ R2.

where Rt is a random 2× 2 positive definite matrix valued functionso 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.

The Sineβ operator

Thm (V-Virag):There is a self-adjoint differential operator (Dirac-operator)

f → 2R−1t

[0 −11 0

]f ′(t), f : [0, 1)→ R2.

where Rt is a random 2× 2 positive definite matrix valued functionso 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.

The Sineβ operator

Thm (V-Virag):There is a self-adjoint differential operator (Dirac-operator)

f → 2R−1t

[0 −11 0

]f ′(t), f : [0, 1)→ R2.

where Rt is a random 2× 2 positive definite matrix valued functionso 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

η0

γ(t)

η∞ zλ(t)

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

N(λ): # of times zλ hits η∞. This is the counting function of thepoint process.

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 itcontinuously around γ(t) with rate λ. (This is just an ODE!)

N(λ): # of times zλ hits η∞. This is the counting function of thepoint process.

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 itcontinuously around γ(t) with rate λ. (This is just an ODE!)

N(λ): # of times zλ hits η∞. This is the counting function of thepoint process.

A geometric description of SineβV.-Virag (’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 → − 4β log(1− t))

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[

0 −11 0

]f ′(t), U =

1√yt

[1 −xt0 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)e iθ(t,λ)

The spectrum can be identified from θ(·, ·) which basically evolvesaccording to a carousel.

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[

0 −11 0

]f ′(t), U =

1√yt

[1 −xt0 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)e iθ(t,λ)

The spectrum can be identified from θ(·, ·) which basically evolvesaccording to a carousel.

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[

0 −11 0

]f ′(t), U =

1√yt

[1 −xt0 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)e iθ(t,λ)

The spectrum can be identified from θ(·, ·) which basically evolvesaccording to a carousel.

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[

0 −11 0

]f ′(t), U =

1√yt

[1 −xt0 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)e iθ(t,λ)

The spectrum can be identified from θ(·, ·) which basically evolvesaccording to a carousel.

Carousel ∼ Dirac operator

(η0, η∞, γ) τ : f → 2R−1t

[0 −11 0

]f ′(t),

Under mild conditions: τ−1 is a Hilbert-Schmidt integral operatorwith kernel

K(x , y) =(u0u

T1 1(x < y) + u1u

T0 1(x ≥ y)

)R(y)

u0, u1: boundary conditions in τ

Nice property: if the path γ lives on [0,T ) then the operator canbe approximated using the path restricted to [0,T − ε).

Carousel ∼ Dirac operator

(η0, η∞, γ) τ : f → 2R−1t

[0 −11 0

]f ′(t),

Under mild conditions: τ−1 is a Hilbert-Schmidt integral operatorwith kernel

K(x , y) =(u0u

T1 1(x < y) + u1u

T0 1(x ≥ y)

)R(y)

u0, u1: boundary conditions in τ

Nice property: if the path γ lives on [0,T ) then the operator canbe approximated using the path restricted to [0,T − ε).

Carousel ∼ Dirac operator

(η0, η∞, γ) τ : f → 2R−1t

[0 −11 0

]f ′(t),

Under mild conditions: τ−1 is a Hilbert-Schmidt integral operatorwith kernel

K(x , y) =(u0u

T1 1(x < y) + u1u

T0 1(x ≥ y)

)R(y)

u0, u1: boundary conditions in τ

Nice property: if the path γ lives on [0,T ) then the operator canbe approximated using the path restricted to [0,T − ε).

Carousel ∼ Dirac operator

τ : f → 2(UTU)−1[

0 −11 0

]f ′(t), U =

1√yt

[1 −xt0 yy

]

Brownian carousel representation of Sineβ

⇓random differential operator for Sineβ

xt + iyt : time-changed hyperbolic Brownian motion

Carousel ∼ Dirac operator

τ : f → 2(UTU)−1[

0 −11 0

]f ′(t), U =

1√yt

[1 −xt0 yy

]

Brownian carousel representation of Sineβ

⇓random differential operator for Sineβ

xt + iyt : time-changed hyperbolic Brownian motion

Additional results

I CβEd= Sineβ

Using the hyperbolic carousel ∼ Dirac operator connection

(Nakano: independent proof)

I Dirac operator description for deterministic unitary matrices

I Random Dirac-operator description for other classical models

driving paths: ‘affine’ hyperbolic Brownian motions

I Soft edge limit: representation as a canonical system[0 −11 0

]f ′(t) = λRt f (t)

(rank(Rt) = 1)

I Bulk convergence via the operators

Additional results

I CβEd= Sineβ

Using the hyperbolic carousel ∼ Dirac operator connection

(Nakano: independent proof)

I Dirac operator description for deterministic unitary matrices

I Random Dirac-operator description for other classical models

driving paths: ‘affine’ hyperbolic Brownian motions

I Soft edge limit: representation as a canonical system[0 −11 0

]f ′(t) = λRt f (t)

(rank(Rt) = 1)

I Bulk convergence via the operators

Additional results

I CβEd= Sineβ

Using the hyperbolic carousel ∼ Dirac operator connection

(Nakano: independent proof)

I Dirac operator description for deterministic unitary matrices

I Random Dirac-operator description for other classical models

driving paths: ‘affine’ hyperbolic Brownian motions

I Soft edge limit: representation as a canonical system[0 −11 0

]f ′(t) = λRt f (t)

(rank(Rt) = 1)

I Bulk convergence via the operators

Additional results

I CβEd= Sineβ

Using the hyperbolic carousel ∼ Dirac operator connection

(Nakano: independent proof)

I Dirac operator description for deterministic unitary matrices

I Random Dirac-operator description for other classical models

driving paths: ‘affine’ hyperbolic Brownian motions

I Soft edge limit: representation as a canonical system[0 −11 0

]f ′(t) = λRt f (t)

(rank(Rt) = 1)

I Bulk convergence via the operators

Additional results

I CβEd= Sineβ

Using the hyperbolic carousel ∼ Dirac operator connection

(Nakano: independent proof)

I Dirac operator description for deterministic unitary matrices

I Random Dirac-operator description for other classical models

driving paths: ‘affine’ hyperbolic Brownian motions

I Soft edge limit: representation as a canonical system[0 −11 0

]f ′(t) = λRt f (t)

(rank(Rt) = 1)

I Bulk convergence via the operators

CβEd= Sineβ

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

1

Zn,β

∏i<j≤n

|e iλi − e iλj |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 hyperboliccarousel driven by a time-reversed version of the Sineβ carousel.

Reversing time in the carousel one can show that CβEd= Sineβ

CβEd= Sineβ

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

1

Zn,β

∏i<j≤n

|e iλi − e iλj |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 hyperboliccarousel driven by a time-reversed version of the Sineβ carousel.

Reversing time in the carousel one can show that CβEd= Sineβ

CβEd= Sineβ

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

1

Zn,β

∏i<j≤n

|e iλi − e iλj |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 hyperboliccarousel driven by a time-reversed version of the Sineβ carousel.

Reversing time in the carousel one can show that CβEd= Sineβ

CβEd= Sineβ

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

1

Zn,β

∏i<j≤n

|e iλi − e iλj |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 hyperboliccarousel driven by a time-reversed version of the Sineβ carousel.

Reversing time in the carousel one can show that CβEd= Sineβ

CβEd= Sineβ

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

1

Zn,β

∏i<j≤n

|e iλi − e iλj |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 hyperboliccarousel driven by a time-reversed version of the Sineβ carousel.

Reversing time in the carousel one can show that CβEd= Sineβ

CβEd= Sineβ

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

1

Zn,β

∏i<j≤n

|e iλi − e iλj |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 hyperboliccarousel driven by a time-reversed version of the Sineβ carousel.

Reversing time in the carousel one can show that CβEd= Sineβ

Dirac operators for unitary matrices

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

Apply G-S to e,Ve, . . . ,V n−1e Szego recursion for OPUC[Φk+1(z)Φ∗k+1(z)

]=

[1 −αk

−αk 1

] [z 00 1

] [Φk(z)Φ∗k(z)

],

[Φ∗0(z)Φ0(z)

]=

[11

]αk : Verblunsky coefficients, |αk | ≤ 1

z is an e.v. ⇔[z 00 1

] [Φn−1(z)Φ∗n−1(z)

]‖[αn−1

1

]

Dirac operators for unitary matrices

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

Apply G-S to e,Ve, . . . ,V n−1e Szego recursion for OPUC

[Φk+1(z)Φ∗k+1(z)

]=

[1 −αk

−αk 1

] [z 00 1

] [Φk(z)Φ∗k(z)

],

[Φ∗0(z)Φ0(z)

]=

[11

]αk : Verblunsky coefficients, |αk | ≤ 1

z is an e.v. ⇔[z 00 1

] [Φn−1(z)Φ∗n−1(z)

]‖[αn−1

1

]

Dirac operators for unitary matrices

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

Apply G-S to e,Ve, . . . ,V n−1e Szego recursion for OPUC[Φk+1(z)Φ∗k+1(z)

]=

[1 −αk

−αk 1

] [z 00 1

] [Φk(z)Φ∗k(z)

],

[Φ∗0(z)Φ0(z)

]=

[11

]αk : Verblunsky coefficients, |αk | ≤ 1

z is an e.v. ⇔[z 00 1

] [Φn−1(z)Φ∗n−1(z)

]‖[αn−1

1

]

Dirac operators for unitary matrices

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

Apply G-S to e,Ve, . . . ,V n−1e Szego recursion for OPUC[Φk+1(z)Φ∗k+1(z)

]=

[1 −αk

−αk 1

] [z 00 1

] [Φk(z)Φ∗k(z)

],

[Φ∗0(z)Φ0(z)

]=

[11

]αk : Verblunsky coefficients, |αk | ≤ 1

z is an e.v. ⇔[z 00 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 00 1

] [Φk(z)Φ∗k(z)

],

[Φ∗0(z)Φ0(z)

]=

[11

]

We can introduce a transformed version of

[Φk(z)Φ∗k(z)

]satisfying

gk+1 = M−1k

[e i

µ2n 0

0 e−iµ2n

]Mkgk , z = e i

µn

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.

Dirac operators for unitary matrices

[Φk+1(z)Φ∗k+1(z)

]=

[1 −αk

−αk 1

] [z 00 1

] [Φk(z)Φ∗k(z)

],

[Φ∗0(z)Φ0(z)

]=

[11

]

We can introduce a transformed version of

[Φk(z)Φ∗k(z)

]satisfying

gk+1 = M−1k

[e i

µ2n 0

0 e−iµ2n

]Mkgk , z = e i

µn

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.

Dirac operators for unitary matrices

[Φk+1(z)Φ∗k+1(z)

]=

[1 −αk

−αk 1

] [z 00 1

] [Φk(z)Φ∗k(z)

],

[Φ∗0(z)Φ0(z)

]=

[11

]

We can introduce a transformed version of

[Φk(z)Φ∗k(z)

]satisfying

gk+1 = M−1k

[e i

µ2n 0

0 e−iµ2n

]Mkgk , z = e i

µn

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 |βn∏

i=1

f (λi )

Dumitriu-Edelman: tridiagonal matrix models for Hermite andLaguerre β-ensembles

Killip-Nenciu: models for the circular β-ensembles, using theSzego-recursion and random Verblunsky coefficients

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

More β-ensembles

1

Zn,f ,β

∏i<j≤n

|λj − λi |βn∏

i=1

f (λi )

Dumitriu-Edelman: tridiagonal matrix models for Hermite andLaguerre β-ensembles

Killip-Nenciu: models for the circular β-ensembles, using theSzego-recursion and random Verblunsky coefficients

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

More β-ensembles

1

Zn,f ,β

∏i<j≤n

|λj − λi |βn∏

i=1

f (λi )

Dumitriu-Edelman: tridiagonal matrix models for Hermite andLaguerre β-ensembles

Killip-Nenciu: models for the circular β-ensembles, using theSzego-recursion and random Verblunsky coefficients

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

Operator convergence

One can find the discrete versions of the limit operators in thefinite tridiagonal models.

Soft edge: in the appropriate scaling the tridiagonal matrix can bewritten as a sum of a discrete Laplacian, a discrete white noisepotential and a potential approximating the function x

Hard edge: the inverse of the tridiagonal matrix (as a product oftwo bidiagonal matrices) can be written as an integral operatorapproximating the inverse of the Bβ,a operator

What about the bulk?

Operator convergence

One can find the discrete versions of the limit operators in thefinite tridiagonal models.

Soft edge: in the appropriate scaling the tridiagonal matrix can bewritten as a sum of a discrete Laplacian, a discrete white noisepotential and a potential approximating the function x

Hard edge: the inverse of the tridiagonal matrix (as a product oftwo bidiagonal matrices) can be written as an integral operatorapproximating the inverse of the Bβ,a operator

What about the bulk?

Operator convergence

One can find the discrete versions of the limit operators in thefinite tridiagonal models.

Soft edge: in the appropriate scaling the tridiagonal matrix can bewritten as a sum of a discrete Laplacian, a discrete white noisepotential and a potential approximating the function x

Hard edge: the inverse of the tridiagonal matrix (as a product oftwo bidiagonal matrices) can be written as an integral operatorapproximating 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

The previous methods required the derivation of a one-parameterfamily of SDE system.

Here we need to understand the limit of the integral kernel: asingle SDE.

Operator level bulk limit

discrete model

↓discrete ‘differential operator’

↓discrete integral operator

↓limiting integral operator

The previous methods required the derivation of a one-parameterfamily of SDE system.

Here we need to understand the limit of the integral kernel: asingle SDE.

Dirac operators for other models

I finite circular β-ensemble and circular Jacobi ensembles

I limits of circular Jacobi ensembles

I hard edge limits

I certain one dimensional random Schrodinger operators

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

I finite circular β-ensemble and circular Jacobi ensembles: γ isa random walk in H

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

Dirac operators for other models

I finite circular β-ensemble and circular Jacobi ensembles

I limits of circular Jacobi ensembles

I hard edge limits

I certain one dimensional random Schrodinger operators

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

I finite circular β-ensemble and circular Jacobi ensembles: γ isa random walk in H

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

Dirac operators for other models

I finite circular β-ensemble and circular Jacobi ensembles

I limits of circular Jacobi ensembles

I hard edge limits

I certain one dimensional random Schrodinger operators

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

I finite circular β-ensemble and circular Jacobi ensembles: γ isa random walk in H

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

Dirac operators from tridiagonal matrices?

The eigenvalue equation is a three-term recursion

Mu = λu a`u`−1 + b`u` + a`u`+1 = λu`

This can be reformulated with transfer matrices:

T`

[u`−1u`

]−[

u`u`+1

]=

[0 0λ 0

] [u`u`+1

],

[u0u1

]=

[01

].

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

Dirac operators from tridiagonal matrices?

The eigenvalue equation is a three-term recursion

Mu = λu a`u`−1 + b`u` + a`u`+1 = λu`

This can be reformulated with transfer matrices:

T`

[u`−1u`

]−[

u`u`+1

]=

[0 0λ 0

] [u`u`+1

],

[u0u1

]=

[01

].

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

Dirac operators from tridiagonal matrices?

The eigenvalue equation is a three-term recursion

Mu = λu a`u`−1 + b`u` + a`u`+1 = λu`

This can be reformulated with transfer matrices:

T`

[u`−1u`

]−[

u`u`+1

]=

[0 0λ 0

] [u`u`+1

],

[u0u1

]=

[01

].

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

THANK YOU!