Unitary Matrix Models: universality conjecture in the bulk and - PowerPoint PPT Presentation

Unitary Matrix Models Poplavskyi M. Random matrices Unitary Matrix Models: universality conjecture in the bulk and orthogonal and on the edge of the spectrum. polynomials Overview of previous results Global M. Poplavskyi regime Bulk universality Department of Statistical Methods in Mathematical Physics Edge B.Verkin Institute for Low Temperature Physics and Engineering of the NASU universality VI School on Mathematical Physics, September 6, 2011

Some types of Random Matrix Ensembles Unitary Matrix Models Matrix Ensembles with independent entries Poplavskyi M. Wigner matrices Random n matrices � � F ( n 1 / 2 dM i , j ) F 0 ( n 1 / 2 dM i , i ) P n ( dM ) = and orthogonal 1 ≤ i < j ≤ n i = 1 polynomials Overview of Marchenko-Pastur ensemble previous results � F ( n 1 / 2 dA i , j ) , M n = n − 1 AA ∗ P n ( dA ) = Global 1 ≤ n , j ≤ m regime Hermitian and Unitary Matrix Ensembles Bulk universality Hermitian and Real Symmetric Matrix Models Edge universality � − β n � P n ,β ( d β M ) = Z − 1 n ,β exp 2 Tr V ( M ) d β M . Unitary Matrix Models � U + U ∗ � �� p n ( U ) d µ n ( U ) = Z − 1 exp − nTrV d µ n ( U ) . n 2

Joint Eigenvalue Distribution Unitary � � n Matrix e i λ ( n ) Models Let Let be an eigenvalues of matrix U . j Poplavskyi M. j = 1 n � � � V ( cos λ j ) − n � p n ( λ 1 , . . . , λ n ) = 1 2 � � Random � e i λ j − e i λ k e . � j = 1 matrices Z n and 1 ≤ j < k ≤ n orthogonal polynomials � p ( n ) Overview of ( λ 1 , . . . , λ l ) = p n ( λ 1 , . . . , λ l , λ l + 1 , . . . , λ n ) d λ l + 1 . . . d λ n . previous l results OPUC with a varying weight and determinant formulas Global regime � � e ik λ � n � e i λ � � e i λ � Bulk k = 0 → P ( n ) P ( n ) P ( n ) ( e i λ ) e − nV ( cos λ ) d λ = δ k , l universality : k k l Edge universality � e i λ , e i µ � n − 1 � e i λ � � P ( n ) P ( n ) ( e i µ ) e − nV ( cos λ ) / 2 e − nV ( cos µ ) / 2 K n = j j j = 0 � � e i λ j , e i λ k �� ( λ 1 , . . . , λ l ) = ( n − l )! l � � p ( n ) det � K n � l n ! j , k = 1

Global and local regimes Unitary Matrix Models � � Poplavskyi λ ( n ) Global regime: N n (∆) = n − 1 ♯ M. ∈ ∆ , l = 1 , . . . , n , ∆ ∈ [ − π, π ) j Random � � matrices p ( n ) ? and N n (∆) = ( λ ) d λ → N (∆) = ρ ( λ ) d λ, n → ∞ . orthogonal 1 ∆ ∆ polynomials Overview of previous results Local regime: Global � � → − regime − → ξ [ c V δ n ] − l p ( n ) ? → det { K ( ξ j , ξ k ) } l Λ 0 + j , k = 1 . Bulk l c V δ n universality Edge � universality ρ ( λ ) d λ ∼ 1 δ n is a typical distance between eigenvalues ⇒ n . | λ − λ 0 |≤ δ n Bulk universality: ρ ( λ 0 ) � = 0 ⇒ δ n = n − 1 . Edge universality: ρ ( λ ) ∼ | λ − λ 0 | 1 / 2 ⇒ δ n = n − 2 / 3 .

Global and local regimes Unitary Matrix Models � � Poplavskyi λ ( n ) Global regime: N n (∆) = n − 1 ♯ M. ∈ ∆ , l = 1 , . . . , n , ∆ ∈ [ − π, π ) j Random � � matrices p ( n ) ? and N n (∆) = ( λ ) d λ → N (∆) = ρ ( λ ) d λ, n → ∞ . orthogonal 1 ∆ ∆ polynomials Overview of previous results Local regime: Global � � → − regime − → ξ [ c V δ n ] − l p ( n ) ? → det { K ( ξ j , ξ k ) } l Λ 0 + j , k = 1 . Bulk l c V δ n universality Edge � universality ρ ( λ ) d λ ∼ 1 δ n is a typical distance between eigenvalues ⇒ n . | λ − λ 0 |≤ δ n Bulk universality: ρ ( λ 0 ) � = 0 ⇒ δ n = n − 1 . Edge universality: ρ ( λ ) ∼ | λ − λ 0 | 1 / 2 ⇒ δ n = n − 2 / 3 .

References Unitary Matrix Models Poplavskyi M. Random L. Pastur, M. Shcherbina ’97, ’07 - proved bulk and edge universality for matrices HMM. and orthogonal polynomials A. Kolyandr ’97 - studied the global regime for UMM. Overview of K. Johansson ’98 - studied the question about length of longest previous results increasing subsequence. Global regime P . Deift and colaborators ’99,’99- proved uniform assymptotics for OPRL Bulk with a varying weight. universality M.J. Cantero, L. Moral, L. Velasquez ’03 - obtained the five term Edge universality recurrence relation for OPUC. K. T.-R. McLaughlin ’06- proved assymptotics for OPUC ( ρ ( λ ) > 0).

Global regime Unitary Matrix The joint eigenvalue distribution can be rewritten in terms of Hamiltonian Models p n (Λ) = 1 Z n e − nH (Λ) with Poplavskyi M. Random � � n matrices � � V ( cos λ j ) − 2 � � � e i λ j − e i λ k and H (Λ) = log � . orthogonal n polynomials j = 1 1 ≤ j < k ≤ n Overview of previous Consider the linear functional results � π � π � � e i λ − e i µ � Global � � regime E [ m ] = V ( cos λ ) m ( d λ ) − log � m ( d λ ) m ( d µ ) , Bulk − π − π universality in the class of unit measures on the interval [ − π, π ] . Edge universality Theorem Let potential V ( cos λ ) be a C 2 [ − π, π ] , then there exists a unique minimizer of the functional,called an equilibrium measure. This measure has a density ρ ( λ ) and NCM measure of eigenvalues converges in probability to the equilibrium measure.

Bulk universality Unitary Matrix Models Poplavskyi M. Theorem Random Let potential V ( cos λ ) be a C 2 [ − π, π ] function and there exists some matrices and subinterval ( a , b ) ⊂ supp ρ ( λ ) such that orthogonal V ′′′ ( λ ) ≤ C 1 , ρ ( λ ) ≥ C 2 , λ ∈ ( a , b ) . Then the universality conjecture polynomials sup Overview of λ ∈ ( a , b ) previous is true for every λ 0 ∈ ( a , b ) with kernel K ( x , y ) = sin π ( x − y ) results and Global π ( x − y ) regime c V = ρ ( λ 0 ) .The limit is uniform for any − → ξ in a compact subset of R l . Bulk universality Basic ideas of the proof Edge universality Prove the uniform convergence of ρ n ( λ ) to ρ ( λ ) . Derive the integro-differential equation for the K n . Find the class of functions in which this equation has a unique solution.

Bulk universality Unitary Matrix Models Poplavskyi M. Theorem Random Let potential V ( cos λ ) be a C 2 [ − π, π ] function and there exists some matrices and subinterval ( a , b ) ⊂ supp ρ ( λ ) such that orthogonal V ′′′ ( λ ) ≤ C 1 , ρ ( λ ) ≥ C 2 , λ ∈ ( a , b ) . Then the universality conjecture polynomials sup Overview of λ ∈ ( a , b ) previous is true for every λ 0 ∈ ( a , b ) with kernel K ( x , y ) = sin π ( x − y ) results and Global π ( x − y ) regime c V = ρ ( λ 0 ) .The limit is uniform for any − → ξ in a compact subset of R l . Bulk universality Basic ideas of the proof Edge universality Prove the uniform convergence of ρ n ( λ ) to ρ ( λ ) . Derive the integro-differential equation for the K n . Find the class of functions in which this equation has a unique solution.

Bulk universality Unitary Matrix Models Poplavskyi M. Theorem Random Let potential V ( cos λ ) be a C 2 [ − π, π ] function and there exists some matrices and subinterval ( a , b ) ⊂ supp ρ ( λ ) such that orthogonal V ′′′ ( λ ) ≤ C 1 , ρ ( λ ) ≥ C 2 , λ ∈ ( a , b ) . Then the universality conjecture polynomials sup Overview of λ ∈ ( a , b ) previous is true for every λ 0 ∈ ( a , b ) with kernel K ( x , y ) = sin π ( x − y ) results and Global π ( x − y ) regime c V = ρ ( λ 0 ) .The limit is uniform for any − → ξ in a compact subset of R l . Bulk universality Basic ideas of the proof Edge universality Prove the uniform convergence of ρ n ( λ ) to ρ ( λ ) . Derive the integro-differential equation for the K n . Find the class of functions in which this equation has a unique solution.

Bulk universality Unitary Matrix Models Poplavskyi M. Theorem Random Let potential V ( cos λ ) be a C 2 [ − π, π ] function and there exists some matrices and subinterval ( a , b ) ⊂ supp ρ ( λ ) such that orthogonal V ′′′ ( λ ) ≤ C 1 , ρ ( λ ) ≥ C 2 , λ ∈ ( a , b ) . Then the universality conjecture polynomials sup Overview of λ ∈ ( a , b ) previous is true for every λ 0 ∈ ( a , b ) with kernel K ( x , y ) = sin π ( x − y ) results and Global π ( x − y ) regime c V = ρ ( λ 0 ) .The limit is uniform for any − → ξ in a compact subset of R l . Bulk universality Basic ideas of the proof Edge universality Prove the uniform convergence of ρ n ( λ ) to ρ ( λ ) . Derive the integro-differential equation for the K n . Find the class of functions in which this equation has a unique solution.