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

Random matrices, differential operators and carousels Benedek Valk´ o (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? 35 30 25 20 15 10 5 � 60 � 40 � 20 0 20 40 60 b H L n - a L global local

Basic question of RMT: What can we say about the spectrum of a large random matrix? 35 30 25 20 15 10 5 � 60 � 40 � 20 0 20 40 60 b H L n - a L 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. √ 35 30 25 20 15 10 Global picture: Wigner semicircle law 5 � 60 � 40 � 20 0 20 40 60

A classical example: Gaussian Unitary Ensemble M = A + A ∗ 2 , A is n × n with iid complex std normal. √ 35 30 25 20 15 10 Global picture: Wigner semicircle law 5 � 60 � 40 � 20 0 20 40 60 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. √ 35 30 25 20 15 10 Global picture: Wigner semicircle law 5 � 60 � 40 � 20 0 20 40 60 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 x 1 , . . . , x n

Point process limit b H L n - a L Finite n : spectrum of a random Hermitian matrix

Point process limit b H L n - a L Finite n : spectrum of a random Hermitian matrix Limit point process: spectrum of ??

Detour to number theory ∞ 1 Riemann zeta function: ζ ( s ) = � n s , for Re s > 1. n =1 (Analytic continuation to C \ { 1 } ) Riemann hypothesis: the non-trivial zeros are on the line Re s = 1 2 .

Detour to number theory ∞ 1 Riemann zeta function: ζ ( s ) = � n s , for Re s > 1. n =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 (Sine 2 process)

Detour to number theory ∞ 1 Riemann zeta function: ζ ( s ) = � n s , for Re s > 1. n =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 (Sine 2 process) ◮ Strong numerical evidence: Odlyzko ◮ Certain weaker versions are proved (Montgomery, Rudnick-Sarnak)

Hilbert-P´ olya 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´ olya 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´ olya 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 or 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 or the Riemann hypothesis (unfortunately...) Borodin-Olshanski, Maples-Najnudel-Nikeghbali: ‘operator-like object’ with generalized eigenvalues distributed as Sine 2

Starting point for deriving the Sine 2 process:

Starting point for deriving the Sine 2 process: Joint eigenvalue density of GUE: n 1 e − 1 2 λ 2 � | λ j − λ i | 2 � i Z n i < j ≤ n i =1

Starting point for deriving the Sine 2 process: Joint eigenvalue density of GUE: n 1 e − 1 2 λ 2 � | λ j − λ i | 2 � i Z n i < j ≤ n i =1 Many of the classical random matrix ensembles have joint eigenvalue densities of the form n 1 � � | λ j − λ i | β f ( λ i ) Z n , f ,β i < j ≤ n i =1 with β = 1 , 2 or 4 and f a specific reference density.

β -ensemble: finite point process with joint density n 1 � � | λ j − λ i | β f ( λ i ) Z n , f ,β i < j ≤ n i =1 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) Airy β process Hard edge: Rider-Ram´ ırez (Laguerre) Bessel β, a processes Bulk: Killip-Stoiciu, V.-Vir´ ag (circular, Hermite) C β E and Sine β processes

Local limits Soft edge: Rider-Ram´ ırez-Vir´ ag (Hermite, Laguerre) Airy β process Hard edge: Rider-Ram´ ırez (Laguerre) Bessel β, a processes Bulk: Killip-Stoiciu, V.-Vir´ ag (circular, Hermite) C β E and Sine β 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: Airy β is the spectrum of A β = − d 2 2 dx 2 + x + √ β dB dB : white noise

Operators at the edge Soft edge: Airy β is the spectrum of A β = − d 2 2 dx 2 + x + √ β dB dB : white noise Hard edge: Bessel β, a is the spectrum of � � √ β B ( x ) d √ β B ( x ) d 2 2 B β, a = − e ( a +1) x + e − ax − dx 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: Airy β is the spectrum of A β = − d 2 2 dx 2 + x + √ β dB dB : white noise Hard edge: Bessel β, a is the spectrum of � � √ β B ( x ) d √ β B ( x ) d 2 2 B β, a = − e ( a +1) x + e − ax − dx 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 C β E or Sine β ?

The Sine β operator Thm (V-Vir´ ag): There is a self-adjoint differential operator (Dirac-operator) � 0 � − 1 f → 2 R − 1 f : [0 , 1) → R 2 . f ′ ( t ) , t 1 0 where R t is a random 2 × 2 positive definite matrix valued function so that the spectrum is the Sine β process.

The Sine β operator Thm (V-Vir´ ag): There is a self-adjoint differential operator (Dirac-operator) � 0 � − 1 f → 2 R − 1 f : [0 , 1) → R 2 . f ′ ( t ) , t 1 0 where R t is a random 2 × 2 positive definite matrix valued function so that the spectrum is the Sine β process. R t is given a simple function of a hyperbolic Brownian motion.

The Sine β operator Thm (V-Vir´ ag): There is a self-adjoint differential operator (Dirac-operator) � 0 � − 1 f → 2 R − 1 f : [0 , 1) → R 2 . f ′ ( t ) , t 1 0 where R t is a random 2 × 2 positive definite matrix valued function so that the spectrum is the Sine β process. R t 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

η ∞ λ γ η A geometric description of Sine β Hyperbolic carousel: ( η 0 , η ∞ , γ ) � point process η 0 , η ∞ : points on the boundary of the hyperbolic plane H γ : [0 , 1) → H : a path in the hyperbolic plane

A geometric description of Sine β Hyperbolic carousel: ( η 0 , η ∞ , γ ) � point process η 0 , η ∞ : points on the boundary of the hyperbolic plane H γ : [0 , 1) → H : a path in the hyperbolic plane η ∞ z λ ( t ) γ ( t ) η 0 For each λ ∈ R we start a point z λ from η 0 and rotate it continuously around γ ( t ) with rate λ . (This is just an ODE!)