Random matrices, operators and analytic functions Benedek Valk o - PowerPoint PPT Presentation

Random matrices, operators and analytic functions Benedek Valk´ o (University of Wisconsin – Madison) joint with B. Vir´ ag (Toronto)

Eugene Wigner, 1950s: • energy levels of heavy nuclei ց • the spectrum of a complicated self-adjoint operator

Eugene Wigner, 1950s: • energy levels of heavy nuclei ց • the spectrum of a complicated self-adjoint operator ց • the eigenvalues of a large random symmetric matrix

Eugene Wigner, 1950s: • energy levels of heavy nuclei ց • the spectrum of a complicated self-adjoint operator ց • the eigenvalues of a large random symmetric matrix In this talk: random matrices � (random) operators

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 How about other observables related to the matrix?

A classical example: Gaussian Unitary Ensemble M = A + A ∗ 2 , A is n × n with iid standard complex normal entries √ π e −| x | 2 on C A i , j has density 1

A classical example: Gaussian Unitary Ensemble M = A + A ∗ 2 , A is n × n with iid standard complex normal entries √ π e −| x | 2 on C A i , j has density 1 35 30 25 20 15 10 5 Global picture: Wigner semicircle law � 60 � 40 � 20 0 20 40 60

A classical example: Gaussian Unitary Ensemble M = A + A ∗ 2 , A is n × n with iid standard complex normal entries √ π e −| x | 2 on C A i , j has density 1 35 30 25 20 15 10 5 Global picture: Wigner semicircle law � 60 � 40 � 20 0 20 40 60 Local picture: point process limit in the bulk and near the edge b H L n - a L Bulk: Dyson-Gaudin-Mehta, edge: Tracy-Widom

A classical example: Gaussian Unitary Ensemble M = A + A ∗ 2 , A is n × n with iid standard complex normal entries √ π e −| x | 2 on C A i , j has density 1 35 30 25 20 15 10 5 Global picture: Wigner semicircle law � 60 � 40 � 20 0 20 40 60 Local picture: point process limit in the bulk and near the edge b H L n - a L Bulk: Dyson-Gaudin-Mehta, edge: Tracy-Widom The limit processes are characterized by their joint intensity functions. Roughly: how likely that we see random points near certain values.

Another classical example: Circular Unitary Ensemble Eigenvalues of a uniform n × n unitary matrix: { e i θ ( n ) j , θ ( n ) ∈ ( − π, π ] } j

Another classical example: Circular Unitary Ensemble Eigenvalues of a uniform n × n unitary matrix: { e i θ ( n ) j , θ ( n ) ∈ ( − π, π ] } j Global picture: Uniform law

Another classical example: Circular Unitary Ensemble Eigenvalues of a uniform n × n unitary matrix: { e i θ ( n ) j , θ ( n ) ∈ ( − π, π ] } j Global picture: Uniform law Local picture: point process limit near any given point E.g. { n θ ( n ) , 1 ≤ j ≤ n } converges to a point process on R . j

Another classical example: Circular Unitary Ensemble Eigenvalues of a uniform n × n unitary matrix: { e i θ ( n ) j , θ ( n ) ∈ ( − π, π ] } j Global picture: Uniform law Local picture: point process limit near any given point E.g. { n θ ( n ) , 1 ≤ j ≤ n } converges to a point process on R . j Limit is the same as the bulk limit of GUE

Point process limit b H L n - a L Finite n : spectrum of a random Hermitian/unitary matrix zeros of the characteristic polynomial

Point process limit b H L n - a L Finite n : spectrum of a random Hermitian/unitary matrix zeros of the characteristic polynomial Limit point process: spectrum of ??, zeros of ??

Quick detour to number theory ∞ 1 � Riemann zeta function: ζ ( s ) = n s , for Re s > 1. n =1 (Analytic continuation to C \ { 1 } )

Quick 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 .

Quick 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 . Hilbert-P´ olya conjecture: the Riemann hypotheses is true because { s ∈ R : ζ ( 1 2 + i s ) = 0 } = spectrum of an unbounded self-adjoint operator

Quick 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 . Hilbert-P´ olya conjecture: the Riemann hypotheses is true because { s ∈ R : ζ ( 1 2 + i s ) = 0 } = spectrum of an unbounded self-adjoint operator Dyson-Montgomery conjecture: After some scaling: { s ∈ R : ζ ( 1 2 + i s ) = 0 } ∼ bulk limit process of GUE

Quick 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 . Hilbert-P´ olya conjecture: the Riemann hypotheses is true because { s ∈ R : ζ ( 1 2 + i s ) = 0 } = spectrum of an unbounded self-adjoint operator Dyson-Montgomery conjecture: After some scaling: { s ∈ R : ζ ( 1 2 + i s ) = 0 } ∼ bulk limit process of GUE Is there a natural operator for the bulk limit of GUE?

Back to random matrices Joint eigenvalue density for GUE and CUE: n 1 1 e − 1 2 λ 2 � � � | e − i θ j − e i θ k | 2 . | λ j − λ i | 2 i , Z ′ Z n n i < j ≤ n i =1 j < k ≤ n

Back to random matrices Joint eigenvalue density for GUE and CUE: n 1 1 e − 1 2 λ 2 � � � | e − i θ j − e i θ k | 2 . | λ j − λ i | 2 i , Z ′ Z n n i < j ≤ n i =1 j < k ≤ n 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, β > 0

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, β > 0 Examples: ◮ Hermite or Gaussian: normal density ◮ circular: uniform on the unit circle ◮ Laguerre or Wishart: gamma density ◮ Jacobi or MANOVA: beta density β = 1 , 2 , 4: classical random matrix models

Scaling limits - global picture Circular β -ensemble � uniform law 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.

Point process 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 (later shown to be the same)

Point process 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 (later shown to be the same) Instead of joint intensities, the limit processes are described via their counting functions using coupled systems of stochastic differential equations. λ → sign( λ ) · (# of points in [0 , λ ])

Point process 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 (later shown to be the same) Instead of joint intensities, the limit processes are described via their counting functions using coupled systems of stochastic differential equations. λ → sign( λ ) · (# of points in [0 , λ ]) Universality: Erd˝ os-Yau-Bourgade, Krishnapur-Rider-Vir´ ag, Rider-Waters

Random operators

Random operators Dumitriu-Edelman ’02: tridiagonal representation for Gaussian and Laguerre β -ensembles Edelman-Sutton ’06: random tridiagonal matrices � random differential operators Conj.: Limit processes ∼ spectra of random differential operators

Random operators Dumitriu-Edelman ’02: tridiagonal representation for Gaussian and Laguerre β -ensembles Edelman-Sutton ’06: random tridiagonal matrices � random differential operators Conj.: Limit processes ∼ spectra of random differential operators Soft edge: Ram´ ırez-Rider-Vir´ ag ’06 (Gaussian, Laguerre) A β = − d 2 2 dx 2 + x + √ β dB Hard edge: Ram´ ırez-Rider ’08 (Laguerre) � � √ 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, dB : white noise domain: [0 , ∞ ) → R , L 2 and boundary conditions

The Sine β operator - operator in the bulk