Random matrices, differential operators and carousels matyd/BMC/slides/Valko - Random... · PDF file Random matrices, di erential operators and carousels Benedek Valko (University

Random matrices, differential operators and carousels

Benedek Valkó (University of Wisconsin – Madison)

joint with B. Virág (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

1 ns , 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

1 ns , 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

1 ns , 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-Pólya 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.

Hilbert-Pólya 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.

Hilbert-Pólya 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?

Disclaimer: A positive answer would not get us closer to any of the conjectures or 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 given by the bulk limit of GUE?

Disclaimer: A positive answer would not get us closer to any of the conjectures or 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 given by the bulk limit of GUE?

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

Starting point for deriving the Sine2 process:

Joint eigenvalue density of GUE:

1

Zn

∏ i

Starting point for deriving the Sine2 process:

Joint eigenvalue density of GUE:

1

Zn

∏ i

β-ensemble: finite point process with joint density

1

Zn,f ,β

∏ i

Scaling limits - global picture

Hermite β-ensemble semicircle law Laguerre 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-Raḿırez-Virág (Hermite, Laguerre) Airyβ process

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

Bulk: Killip-Stoiciu, V.-Virág (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])

Local limits

Soft edge: Rider-Raḿırez-Virág (Hermite, Lague