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Introduction Gaussian Unitary Ensemble (GUE) Gap Probabilities and Distribution of Largest Eigenvalue Matrix Ensembles with Generalized Cauchy Weights Summer School 2008, Disentis Gap Probabilities for Random Matrix Ensembles Felix Rubin July 21, 2008 Felix Rubin Gap Probabilities for Random Matrix Ensembles

Introduction Gaussian Unitary Ensemble (GUE) Gap Probabilities and Distribution of Largest Eigenvalue Matrix Ensembles with Generalized Cauchy Weights Outline 1 Introduction 2 Gaussian Unitary Ensemble (GUE) 3 Gap Probabilities and Distribution of Largest Eigenvalue Scaling Results and Painlev´ e 4 Matrix Ensembles with Generalized Cauchy Weights Results Further Steps Felix Rubin Gap Probabilities for Random Matrix Ensembles

Introduction Gaussian Unitary Ensemble (GUE) Gap Probabilities and Distribution of Largest Eigenvalue Matrix Ensembles with Generalized Cauchy Weights Introduction Main Question: Given a large matrix with random entries, what can be said about the distribution of its eigenvalues? In particular: What can be said about the distribution of the largest eigenvalue? Started with Physicists in the 50’s. Model to understand statistical behavior of slow neutron resonances (Wigner). 70’s: Applications to number theory (Montgomery). Felix Rubin Gap Probabilities for Random Matrix Ensembles

Introduction Gaussian Unitary Ensemble (GUE) Gap Probabilities and Distribution of Largest Eigenvalue Matrix Ensembles with Generalized Cauchy Weights Gaussian Unitary Ensemble (GUE) I Definition A random N × N Hermitian matrix belongs to the GUE , if the diagonal elements x jj and the upper triangular elements x jk = u jk + iv jk ( j < k ) are independently chosen with normal densities of the form: 1 ∼ N (0 , 1 √ π e − x 2 2) (diagonal) , jj 2 ∼ N (0 , 1 4) + i N (0 , 1 π e − 2( u 2 jk + v 2 jk ) 4) (upper triangular) Joint p.d.f: N 1 π e − 2 | x jk | 2 = 1 2 √ π e − x 2 � � exp {− Tr ( X 2 ) } . p ( X ) = jj Z N j =1 1 ≤ j < k ≤ N Felix Rubin Gap Probabilities for Random Matrix Ensembles

Introduction Gaussian Unitary Ensemble (GUE) Gap Probabilities and Distribution of Largest Eigenvalue Matrix Ensembles with Generalized Cauchy Weights Gaussian Unitary Ensemble (GUE) II Eigenvalue distribution? (Eigenvalues: ( x 1 , . . . , x N ) ⊂ R N ) Apply basis transformation and integrate out elements independent of the eigenvalues: Eigenvalue measure on R N : If x 1 < . . . < x N ,   N u N ( x 1 , . . . , x N ) = 1 | x j − x k | 2 exp   � � x 2  − j Z N  1 ≤ j < k ≤ N j =1 = 1 � 2 � det( p j − 1 ( x i ) e ( − x 2 i ) / 2 ) 1 ≤ i , j ≤ N Z N = det( K N ( x i , x j )) N i , j =1 , Felix Rubin Gap Probabilities for Random Matrix Ensembles

Introduction Gaussian Unitary Ensemble (GUE) Gap Probabilities and Distribution of Largest Eigenvalue Matrix Ensembles with Generalized Cauchy Weights Gaussian Unitary Ensemble (GUE) III where N − 1 j ( y ) e − x 2+ y 2 � p H j ( x ) p H K N ( x , y ) = 2 j =0 = const · e − ( x 2 + y 2 ) / 2 p H N ( x ) p H N − 1 ( y ) − p H N ( y ) p H N − 1 ( x ) . x − y p H is the i -th normalized Hermite polynomial of degree i . i The eigenvalue distribution can be viewed as a point process on R via the application ( x 1 , . . . , x N ) �→ � N i =1 δ x i . Point processes with a measure of this determinantal form are called determinantal point processes . Felix Rubin Gap Probabilities for Random Matrix Ensembles

Introduction Gaussian Unitary Ensemble (GUE) Gap Probabilities and Distribution of Largest Eigenvalue Matrix Ensembles with Generalized Cauchy Weights Gaussian Unitary Ensemble (GUE) IV Definition We define the n-th correlation function ρ n by: ρ n ( x 1 , . . . , x n ) = det( K N ( x i , x j )) n i , j =1 , for n ≤ N . The correlation function can be viewed as a particle density. Namely, if [ x i , x i + ∆ x i ], 1 ≤ i ≤ n , are all disjoint, ρ n ( x 1 , . . . , x n ) = P [ there is exactly one particle in [ x i , x i + ∆ x i ] , 1 ≤ i ≤ n ] lim ∆ x 1 . . . ∆ x n ∆ x i → 0 Felix Rubin Gap Probabilities for Random Matrix Ensembles

Introduction Gaussian Unitary Ensemble (GUE) Scaling Results and Painlev´ e Gap Probabilities and Distribution of Largest Eigenvalue Matrix Ensembles with Generalized Cauchy Weights Gap Probabilities and Distribution of Largest Eigenvalue I Question: P [ there is no eigenvalue in ( a , b ) = 0] =? , a < b ∈ R . Lemma Let φ be a bounded and measurable function with bounded support B. Then ∞ n 1 � � � � E [ (1 + φ ( x j ))] = φ ( x j ) ρ n ( x 1 , . . . , x n ) dx 1 . . . dx n . n ! R n n =0 j j =1 Thus, ∞ ( − 1) n � � ( t , ∞ ) n ρ n ( x 1 , . . . , x n ) d n x . P [ x max ≤ t ] = n ! n =0 Felix Rubin Gap Probabilities for Random Matrix Ensembles

Introduction Gaussian Unitary Ensemble (GUE) Scaling Results and Painlev´ e Gap Probabilities and Distribution of Largest Eigenvalue Matrix Ensembles with Generalized Cauchy Weights Gap Probabilities and Distribution of Largest Eigenvalue II The correlation kernel K N ( x , y ) can be viewed as the kernel of an integral operator K on L 2 ( R ): If f ∈ L 2 ( R ), � Kf ( x ) = K N ( x , y ) f ( y ) dy . R One can define the Fredholm determinant of the operator K as: ∞ ( − 1) n � � det( K N ( x i , x j )) n i , j =1 d n x . det( Id − K ) = 1 + n ! n =1 If ρ n ( x 1 , . . . , x n ) = det( K N ( x i , x j )) 1 ≤ i , j ≤ n , we thus have: P [ x max ≤ t ] = det( Id − K ) | L 2 ( t , ∞ ) . Felix Rubin Gap Probabilities for Random Matrix Ensembles

Introduction Gaussian Unitary Ensemble (GUE) Scaling Results and Painlev´ e Gap Probabilities and Distribution of Largest Eigenvalue Matrix Ensembles with Generalized Cauchy Weights Scaling Results and Painlev´ e I If one scales around the largest eigenvalue, say x max ( N ), of the GUE, one obtains for N → ∞ : √ � � s √ P x max ( N ) ≤ 2 N + − → F TW ( s ) = det( Id − K Airy ) | L 2 ( s , ∞ ) 2 N 1 / 6 � ∞ � � ( x − s ) q 2 ( x ) dx − F TW ( s ) = exp , s e-II equation q ′′ = sq + 2 q 3 with q being the solution of a Painlev´ boundary condition q ( s ) ∼ Ai ( s ) for s → ∞ . Felix Rubin Gap Probabilities for Random Matrix Ensembles

Introduction Gaussian Unitary Ensemble (GUE) Results Gap Probabilities and Distribution of Largest Eigenvalue Further Steps Matrix Ensembles with Generalized Cauchy Weights Matrix Ensembles with Generalized Cauchy Weights I (Joint work with Joseph Najnudel and Ashkan Nikeghbali) Consider the Unitary group U ( N ) with the Haar measure µ N . The eigenvalue distribution function here is: N | e i θ j − e i θ k | 2 � � const · d θ j , 1 ≤ j < k ≤ N j =1 where e i θ j , j = 1 , . . . , N , are the eigenvalues of U ∈ U ( N ) with θ j ∈ [ − π, π ]. Felix Rubin Gap Probabilities for Random Matrix Ensembles

Introduction Gaussian Unitary Ensemble (GUE) Results Gap Probabilities and Distribution of Largest Eigenvalue Further Steps Matrix Ensembles with Generalized Cauchy Weights Matrix Ensembles with Generalized Cauchy Weights II Generalize this eigenvalue distribution: Introduce a complex parameter s , ℜ s ≥ − 1 2 , and write: N � | e i θ j − e i θ k | 2 � const · w U ( θ j ) d θ j , 1 ≤ j < k ≤ N j =1 where w U ( θ j ) = (1 + e i θ j ) s (1 + e − i θ j ) s . U ( N ) is linked to H ( N ) (Hermitian matrices) via the Cayley transform X ∈ H ( N ) �→ i − X i + X ∈ U ( N ) . Felix Rubin Gap Probabilities for Random Matrix Ensembles

Introduction Gaussian Unitary Ensemble (GUE) Results Gap Probabilities and Distribution of Largest Eigenvalue Further Steps Matrix Ensembles with Generalized Cauchy Weights Matrix Ensembles with Generalized Cauchy Weights III The Cayley transform sends the generalized Haar measure to the following Cauchy type measure on H ( N ): N � ( x j − x k ) 2 � const · w H ( x j ) dx j , 1 ≤ j < k ≤ N j =1 where w H ( x j ) = (1 + ix j ) − s − N (1 − ix j ) − s − N . The correlation kernel for this eigenvalue process is: K N ( x , y ) = φ ( x ) ψ ( y ) − φ ( y ) ψ ( x ) , x − y � � with φ ( x ) = Cw H ( x ) p N ( x ), and ψ ( x ) = Cw H ( x ) p N − 1 ( x ). (Borodin, Olshanski, 2001). Felix Rubin Gap Probabilities for Random Matrix Ensembles

Introduction Gaussian Unitary Ensemble (GUE) Results Gap Probabilities and Distribution of Largest Eigenvalue Further Steps Matrix Ensembles with Generalized Cauchy Weights Results: An ODE related to the Painlev´ e-VI equation I Consider dt log det( Id − K N ) | L 2 ( t , ∞ ) = d d dt log P [ no eigenvalue inside ( t , ∞ )] . It is known that this is equal to R ( t , t ), where R ( x , y ) . = K N (1 − K N ) − 1 is the resolvent kernel of K N . Using a general method given by Tracy, Widom (1994), we prove a differential equation for the above quantity. All one needs to find are the following recurrence equations for φ and ψ : m ( x ) φ ′ ( x ) = A ( x ) φ ( x ) + B ( x ) ψ ( x ) m ( x ) ψ ′ ( x ) = − C ( x ) φ ( x ) − A ( x ) ψ ( x ) , where A , B and m are polynomials in x . Felix Rubin Gap Probabilities for Random Matrix Ensembles