Universality results for the Cauchy-Laguerre chain matrix model - PowerPoint PPT Presentation

Universality results for the Cauchy-Laguerre chain matrix model Thomas Bothner Centre de recherches math´ ematiques, Universit´ e de Montr´ eal September 21st, 2014 Cincinnati Symposium on Probability Theory and Applications, Cincinnati, OH, USA Thomas Bothner Universality results for the Cauchy-Laguerre chain matrix model

Setup and outline This talk discusses joint work (BB 14 [6]) with Marco Bertola on the Cauchy matrix chain, the space M p + ( n ) , p , n ∈ Z � 2 of p -tuples ( M 1 , . . . , M p ) of n × n positive - definite Hermitian matrices with joint probability density function e � tr P p j =1 U j ( M j ) d µ ( M 1 , . . . , M p ) ∝ j =1 det( M j + M j +1 ) n d M 1 · . . . · d M p . (1) Q p � 1 The density depends on p potentials U j : R + → R which we specify later on. Thomas Bothner Universality results for the Cauchy-Laguerre chain matrix model

Setup and outline This talk discusses joint work (BB 14 [6]) with Marco Bertola on the Cauchy matrix chain, the space M p + ( n ) , p , n ∈ Z � 2 of p -tuples ( M 1 , . . . , M p ) of n × n positive - definite Hermitian matrices with joint probability density function e � tr P p j =1 U j ( M j ) d µ ( M 1 , . . . , M p ) ∝ j =1 det( M j + M j +1 ) n d M 1 · . . . · d M p . (1) Q p � 1 The density depends on p potentials U j : R + → R which we specify later on. Several key features (“Integrability”) of the model allow us to Reduce (1) to a density function defined on the eigenvalues Thomas Bothner Universality results for the Cauchy-Laguerre chain matrix model

Setup and outline This talk discusses joint work (BB 14 [6]) with Marco Bertola on the Cauchy matrix chain, the space M p + ( n ) , p , n ∈ Z � 2 of p -tuples ( M 1 , . . . , M p ) of n × n positive - definite Hermitian matrices with joint probability density function e � tr P p j =1 U j ( M j ) d µ ( M 1 , . . . , M p ) ∝ j =1 det( M j + M j +1 ) n d M 1 · . . . · d M p . (1) Q p � 1 The density depends on p potentials U j : R + → R which we specify later on. Several key features (“Integrability”) of the model allow us to Reduce (1) to a density function defined on the eigenvalues Rewrite correlation functions in determinantal form and connect to orthogonal polynomials Thomas Bothner Universality results for the Cauchy-Laguerre chain matrix model

Setup and outline This talk discusses joint work (BB 14 [6]) with Marco Bertola on the Cauchy matrix chain, the space M p + ( n ) , p , n ∈ Z � 2 of p -tuples ( M 1 , . . . , M p ) of n × n positive - definite Hermitian matrices with joint probability density function e � tr P p j =1 U j ( M j ) d µ ( M 1 , . . . , M p ) ∝ j =1 det( M j + M j +1 ) n d M 1 · . . . · d M p . (1) Q p � 1 The density depends on p potentials U j : R + → R which we specify later on. Several key features (“Integrability”) of the model allow us to Reduce (1) to a density function defined on the eigenvalues Rewrite correlation functions in determinantal form and connect to orthogonal polynomials Express orthogonal polynomials in terms of a Riemann-Hilbert problem Thomas Bothner Universality results for the Cauchy-Laguerre chain matrix model

Setup and outline This talk discusses joint work (BB 14 [6]) with Marco Bertola on the Cauchy matrix chain, the space M p + ( n ) , p , n ∈ Z � 2 of p -tuples ( M 1 , . . . , M p ) of n × n positive - definite Hermitian matrices with joint probability density function e � tr P p j =1 U j ( M j ) d µ ( M 1 , . . . , M p ) ∝ j =1 det( M j + M j +1 ) n d M 1 · . . . · d M p . (1) Q p � 1 The density depends on p potentials U j : R + → R which we specify later on. Several key features (“Integrability”) of the model allow us to Reduce (1) to a density function defined on the eigenvalues Rewrite correlation functions in determinantal form and connect to orthogonal polynomials Express orthogonal polynomials in terms of a Riemann-Hilbert problem Derive strong asymptotics for the orthogonal polynomials and thus prove universality results for specific potentials Thomas Bothner Universality results for the Cauchy-Laguerre chain matrix model

Setup and outline This talk discusses joint work (BB 14 [6]) with Marco Bertola on the Cauchy matrix chain, the space M p + ( n ) , p , n ∈ Z � 2 of p -tuples ( M 1 , . . . , M p ) of n × n positive - definite Hermitian matrices with joint probability density function e � tr P p j =1 U j ( M j ) d µ ( M 1 , . . . , M p ) ∝ j =1 det( M j + M j +1 ) n d M 1 · . . . · d M p . (1) Q p � 1 The density depends on p potentials U j : R + → R which we specify later on. Several key features (“Integrability”) of the model allow us to Reduce (1) to a density function defined on the eigenvalues Rewrite correlation functions in determinantal form and connect to orthogonal polynomials Express orthogonal polynomials in terms of a Riemann-Hilbert problem Derive strong asymptotics for the orthogonal polynomials and thus prove universality results for specific potentials This four step program has been successfully completed for the Hermitian one-matrix model, i.e. p = 1: Thomas Bothner Universality results for the Cauchy-Laguerre chain matrix model

Hermitian one-matrix model Joint probability density on eigenvalues, for M ∈ M ( n ) , U : R → R , n ∆ ( X ) 2 e � P n 1 ) d n x = 1 j =1 U ( x j ) Y d µ ( M ) ∝ e � tr U ( M ) d M P ( { x j } n d x j Z n j =1 with Vandermonde ∆ ( X ) = Q j < k ( x j − x k ). (PR 60 [18], D 62 [10]) Thomas Bothner Universality results for the Cauchy-Laguerre chain matrix model

Hermitian one-matrix model Joint probability density on eigenvalues, for M ∈ M ( n ) , U : R → R , n ∆ ( X ) 2 e � P n 1 ) d n x = 1 j =1 U ( x j ) Y d µ ( M ) ∝ e � tr U ( M ) d M P ( { x j } n d x j Z n j =1 with Vandermonde ∆ ( X ) = Q j < k ( x j − x k ). (PR 60 [18], D 62 [10]) Determinantal reduction for the ` -point correlation function n ` ! Z ⇤ ` R ( ` ) ( { x j } ` R n � ` P ( { x j } n Y ⇥ 1 ) = 1 ) d x j = det K 11 ( x i , x j ) i , j =1 ( n − ` )! j = ` +1 Thomas Bothner Universality results for the Cauchy-Laguerre chain matrix model

Hermitian one-matrix model Joint probability density on eigenvalues, for M ∈ M ( n ) , U : R → R , n ∆ ( X ) 2 e � P n 1 ) d n x = 1 j =1 U ( x j ) Y d µ ( M ) ∝ e � tr U ( M ) d M P ( { x j } n d x j Z n j =1 with Vandermonde ∆ ( X ) = Q j < k ( x j − x k ). (PR 60 [18], D 62 [10]) Determinantal reduction for the ` -point correlation function n ` ! Z ⇤ ` R ( ` ) ( { x j } ` R n � ` P ( { x j } n Y ⇥ 1 ) = 1 ) d x j = det K 11 ( x i , x j ) i , j =1 ( n − ` )! j = ` +1 with correlation kernel n � 1 ⇡ k ( x ) ⇡ k ( y ) 1 K 11 ( x , y ) = e � 1 2 U ( x ) e � 1 X 2 U ( y ) h k k =0 and monic orthogonal polynomials { ⇡ k } k � 0 Z ⇡ n ( x ) ⇡ m ( x ) e � U ( x ) d x = h n � nm . (D 70 [11]) R Thomas Bothner Universality results for the Cauchy-Laguerre chain matrix model

Riemann-Hilbert characterization for { ⇡ k } k � 0 (FIK 91 [13]): Thomas Bothner Universality results for the Cauchy-Laguerre chain matrix model

Riemann-Hilbert characterization for { ⇡ k } k � 0 (FIK 91 [13]): Determine 2 × 2 function Γ ( z ) ≡ Γ ( z ; n ) such that Γ ( z ) analytic for z ∈ C \ R 1 Thomas Bothner Universality results for the Cauchy-Laguerre chain matrix model

Riemann-Hilbert characterization for { ⇡ k } k � 0 (FIK 91 [13]): Determine 2 × 2 function Γ ( z ) ≡ Γ ( z ; n ) such that Γ ( z ) analytic for z ∈ C \ R 1 Γ ( z ) admits boundary values Γ ± ( z ) for z ∈ R related via 2  � e � U ( z ) 1 Γ + ( z ) = Γ � ( z ) z ∈ R , 0 1 Thomas Bothner Universality results for the Cauchy-Laguerre chain matrix model

Riemann-Hilbert characterization for { ⇡ k } k � 0 (FIK 91 [13]): Determine 2 × 2 function Γ ( z ) ≡ Γ ( z ; n ) such that Γ ( z ) analytic for z ∈ C \ R 1 Γ ( z ) admits boundary values Γ ± ( z ) for z ∈ R related via 2  � e � U ( z ) 1 Γ + ( z ) = Γ � ( z ) z ∈ R , 0 1 As z → ∞ , 3 ⇣ ⇣ z � 1 ⌘⌘ z n � 3 , Γ ( z ) = I + O z → ∞ Thomas Bothner Universality results for the Cauchy-Laguerre chain matrix model

Riemann-Hilbert characterization for { ⇡ k } k � 0 (FIK 91 [13]): Determine 2 × 2 function Γ ( z ) ≡ Γ ( z ; n ) such that Γ ( z ) analytic for z ∈ C \ R 1 Γ ( z ) admits boundary values Γ ± ( z ) for z ∈ R related via 2  � e � U ( z ) 1 Γ + ( z ) = Γ � ( z ) z ∈ R , 0 1 As z → ∞ , 3 ⇣ ⇣ z � 1 ⌘⌘ z n � 3 , Γ ( z ) = I + O z → ∞ The RHP for Γ ( z ; n ) is uniquely solvable i ff ⇡ n ( z ) exists, Thomas Bothner Universality results for the Cauchy-Laguerre chain matrix model

Riemann-Hilbert characterization for { ⇡ k } k � 0 (FIK 91 [13]): Determine 2 × 2 function Γ ( z ) ≡ Γ ( z ; n ) such that Γ ( z ) analytic for z ∈ C \ R 1 Γ ( z ) admits boundary values Γ ± ( z ) for z ∈ R related via 2  � e � U ( z ) 1 Γ + ( z ) = Γ � ( z ) z ∈ R , 0 1 As z → ∞ , 3 ⇣ ⇣ z � 1 ⌘⌘ z n � 3 , Γ ( z ) = I + O z → ∞ The RHP for Γ ( z ; n ) is uniquely solvable i ff ⇡ n ( z ) exists, moreover  Γ � 1 ( x ; n ) Γ ( y ; n ) � 2 U ( y ) i K 11 ( x , y ) = e � 1 2 U ( x ) e � 1 2 ⇡ x − y 21 Thomas Bothner Universality results for the Cauchy-Laguerre chain matrix model