Triangular matrices and biorthogonal ensembles

/26 Triangular matrices and biorthogonal ensembles Dimitris Cheliotis Department of Mathematics University of Athens UK Easter Probability Meeting April 8, 206

2/26 Special densities on R n Example. n n GUE matrix: Hermitian with elements N R (0, ) N C (0, ) N R (0, )...... N C (0, ) N C (0, ) N R (0, ) Eigenvalue density C n r<s n n (x s x r ) 2 e x2 j /2 dx j j=

3/26 Example 2. LUE (Laguerre Unitary Ensemble) matrix: XX with X C n m, n m N C (0, ) N C (0, ) N C (0, ) N C (0, ) N C (0, ) N C (0, ) X :=...... N C (0, ) N C (0, ) N C (0, ) Eigenvalue density for XX C m,n r<s n (x s x r ) 2 n j= x m n j e x j xj >0 dx j

4/26 Example 3. JUE (Jacobi Unitary Ensemble) matrix: XX XX + YY X C n m, Y C n m 2, n m, m 2 with elements i.i.d. N C (0, ). Eigenvalue density for XX (XX + YY ) C n,m,m 2 r<s n (x s x r ) 2 n j= x m n j ( x j ) m2 n xj (0,) dx j

5/26 Example 4. Beenakker-Rajaei (Phys. Rev. Letters (993)) Transmission eigenvalues for a quantum conductor with n channels. T i (0, ), i n. Conductance G = G 0 n i= Density of λ i := T i T i, i =, 2,..., n. r<s n (x s x r ){f (x s ) f (x r )} T i n e Vn(x j ) xj (0,) dx j j= f (x) := {sinh ( x)} 2. Gives correct Var(G/G 0 ).

6/26 Fix θ > 0 C n i<j n {(x j x i )(x θ j x θ i )} n w(x i ) dx i i=

6/26 Fix θ > 0 C n i<j n {(x j x i )(x θ j x θ i )} n w(x i ) dx i i= A. Borodin (999). Special choices for w. Fix α, β >. () w(x) = x α ( x) β x (0,) Biorthogonal Jacobi ensemble. (2) w(x) = x α e x x>0 Biorthogonal Laguerre ensemble. (3) w(x) = x α e x2 Biorthogonal Hermite ensemble. Described limit point process near 0.

7/26 Recall r<s n (x s x r ) = det j (x j,k n k ) Biorthogonal ensembles Point process on R with n points and measure n det (ξ j(x k )) det (η j(x k )) dµ(x k ) j,k n j,k n k= GUE, LUE, JUE: Muttalib ensemble: ξ j (x) = η j (x) = x j ξ j (x) = x j, η j (x) = x (j )θ

8/26 Singular values A an n m complex matrix. Eigenvalues of AA : λ i (AA ), i =, 2,..., n. Singular values of A: s i (A) := λ i (AA ), i =, 2,..., n. s (A) s 2 (A) s n (A) We can write: A = UDV U C n n, V C m m unitary. D = diag(s (A), s 2 (A),..., s n m (A)) C n m

Singular values of a complex matrix 9/26 X = (X i,j ) i,j n with X i,j i.i.d. E(X, ) = 0, E( X, 2 ) =. Eigenvalues of XX : λ (n) λ (n) 2 λ (n) n > 0. Empirical spectral distribution of rescaled eigenvalues. L n := n n k= δ n λ(n) i Theorem (Marchenko-Pastur, 967) L n 4 x 0 x 4 dx 2π x If further X i,j N C (0, ). Density of (λ (n), λ(n) 2,, λ(n) n ): e n k= (k!)2 n k= x k i<j n (x i x j ) 2 x >x 2 > >x n>0

0/26 Triangular matrices X, 0 0 X 2, X 2,2 0 X (n) :=...... X n, X n,2 X n,n E(X, ) = 0, E( X, 2 ) =.

0/26 Triangular matrices X, 0 0 X 2, X 2,2 0 X (n) :=...... X n, X n,2 X n,n E(X, ) = 0, E( X, 2 ) =. Eigenvalues of X (n)x (n) : λ (n) λ (n) 2 λ (n) n 0. Empirical spectral distribution of rescaled eigenvalues. L n := n n k= δ n λ(n) i

/26 Theorem (Dykema-Haagerup, 2004) With probability, x k dµ 0 (x) = µ 0 density with support [0, e]. L n µ 0 kk (k + )!, k 0

/26 Theorem (Dykema-Haagerup, 2004) With probability, x k dµ 0 (x) = µ 0 density with support [0, e]. L n µ 0 kk (k + )!, k 0 New proof with method of moments. Key: n n k Tr{(XX ) k } n Counting rooted alternating plane trees. k k (k + )! Theorem (C. 204). If E( X, 4 ) < then λ(n) n e.

2/26 Singular values of a triangular complex Gaussian matrix X i,j { 0 if i < j n N C (0, ) if n i j Eigenvalues of XX : λ > λ 2 > > λ n > 0

2/26 Singular values of a triangular complex Gaussian matrix X i,j { 0 if i < j n N C (0, ) if n i j Eigenvalues of XX : λ > λ 2 > > λ n > 0 Theorem (C. 204) Density of (λ, λ 2,..., λ n ): n e k= x k n k= k! i<j n (x i x j )(log x i log x j ) x >x 2 > >x n>0

2/26 Singular values of a triangular complex Gaussian matrix X i,j { 0 if i < j n N C (0, ) if n i j Eigenvalues of XX : λ > λ 2 > > λ n > 0 Theorem (C. 204) Density of (λ, λ 2,..., λ n ): n e k= x k n k= k! Biorthogonal ensemble: i<j n (x i x j )(log x i log x j ) x >x 2 > >x n>0 ξ j (x) = x j, η j (x) = (log x) j, dµ(x) = Ce x x>0 dx.

3/26 Connection with biorthogonal Laguerre ensemble (α = 0): n C n (θ) {(x j x i )(xj θ xi θ )} xi 0 e x i xi >0 dx i i<j n i=

3/26 Connection with biorthogonal Laguerre ensemble (α = 0): C n (θ) i<j n {(x j x i )(x θ j x θ i )} x θ y θ lim θ 0 θ n xi 0 e x i xi >0 dx i i= = log x log y

4/26 A special distribution on triangular matrices Parameters: θ 0, r > 0 The T (θ, r) distribution. X, 0 0 N C (0, ) X 2,2 0 X =...... N C (0, ) N C (0, ) X n,n c k := (k )θ + r Arithmetic progression in k =, 2,... X k,k : with density in C πγ(c k ) e z 2 z 2(c k ) X k,k = e iφ k χ 2ck / 2, φ k U(0, 2π)

5/26 Eigenvalue realization of the biorthogonal Laguerre ensemble Let X T (θ, r). Eigenvalues of XX : λ > λ 2 > > λ n > 0.

5/26 Eigenvalue realization of the biorthogonal Laguerre ensemble Let X T (θ, r). Eigenvalues of XX : λ > λ 2 > > λ n > 0. Theorem (C. 204) Density of (λ, λ 2,..., λ n ). C n,θ,r e n k= x k n j= x r j i<j n Biorthogonal Laguerre ensemble: α = r. (x i x j )(x θ i x θ j ) x >x 2 > >x n>0

6/26 Corollary L n := n µ θ has same moments as xx n k= δ n λ(n) i µ θ x = DT (ν θ, ) element in a -noncommutative probability space ν θ =uniform measure on {z C : z θ}

7/26 Further developments ) Eigenvalue realization of the biorthogonal Jacobi ensemble Weight x α ( x) β x (0,) instead of e x x α x>0.

7/26 Further developments ) Eigenvalue realization of the biorthogonal Jacobi ensemble Weight x α ( x) β x (0,) instead of e x x α x>0. Let X T (θ, r), Y T (θ, s). Eigenvalues of XX XX +YY : > λ > λ 2 > > λ n > 0. Theorem 2 (Forrester, Wang. 205) Density of (λ, λ 2,..., λ n ). C n,θ,r,s n j= x r j ( x j ) s i<j n (x i x j )(x θ i x θ j ) x >x 2 > >x n>0 Biorthogonal Jacobi ensemble: α = r, β = s.

8/26 2) Large deviations with density (λ (n), λ(n) 2,..., λ(n) n ). e n n i= V (x i ) Z n n i= x b j x i x j g(x i ) g(x j ) i<j V, g appropriate (g C ([0, )), g > 0,...). Theorem 3: (R. Butez, 206) The sequence of measures n n i= δ λ (n) i satisfies a large deviations principle with speed n 2 and certain good rate function I. I has a unique minimizer.

9/26 Proof of Theorem Step : Density of XX. Write uniquely XX = TT X = TV V = diag(eiθ, e iθ 2,..., e iθn ) T : Lower triangular with t j,j > 0 X g (T, V ) T h TT Density of TT at an a C n n positive definite.

20/26 Jacobians: t = h (a) X Jg(t, v) = g (T, V ) T h TT n n t j,j Jh(t) = 2 n j= j= t 2(n j)+ j,j f (a) = f h(t ) (a) = f T (h (a)) Jh (a) = f T (t) Jh(t) n f T,V (t, v) = f g (X )(t, v) = f X (g(t, v)) Jg(t, v) = f X (t) t j,j, f T (t) = (2π) n f X (t) n j= t j,j j=

2/26 Collecting everything: f X (x) = C n,b,θ e tr(xx ) f (a) =(2π) n f X (t) n j= t j,j Jh(t) = C n,θ,b e tr(a) ( n j= t 2 j,j n x k,k 2(c k ) k= ) cn ( n j= ) (+θ) t 2(n j) j,j = C n,θ,b e tr(a) {det(a)} cn {det(a ) det(a 2 ) det(a n )} θ+ a k := (a i,j ) i,j k, the k principal minor of a.

22/26 Step 2 : Eigenvalues of XX. Density: { C n i x j ) i<j n(x 2} f XX (HD x H )dh x >x 2 >...>x n>0 U(n) D x := diag(x, x 2,..., x n ) dh : normalized Haar measure on U(n)

22/26 Step 2 : Eigenvalues of XX. Density: { C n i x j ) i<j n(x 2} f XX (HD x H )dh x >x 2 >...>x n>0 U(n) D x := diag(x, x 2,..., x n ) dh : normalized Haar measure on U(n) a := HD x H Eigenvalues of the minors a n, a n 2..., a 2, a of a. λ (n ) R n, λ (n 2) R n 2,..., λ () R

23/26 a : x x 2 x 3 x 4 x 5 a 4 : λ (4) λ (4) 2 λ (4) 3 λ (4) 4 a 3 : λ (3) λ (3) 2 λ (3) 3 a 2 : λ (2) λ (2) 2 a : λ ()

23/26 a : x x 2 x 3 x 4 x 5 a 4 : λ (4) λ (4) 2 λ (4) 3 λ (4) 4 a 3 : λ (3) λ (3) 2 λ (3) 3 a 2 : λ (2) λ (2) 2 a : λ () G(x) := {(x (n ), x (n 2),..., x () ) : x x (n ) x (n 2) x () } subset of R n R n 2 R 2 R. Λ := (λ (n ), λ (n 2),..., λ () ) G(x)

a : x x 2 x 3 x 4 x 5 a 4 : λ (4) λ (4) 2 λ (4) 3 λ (4) 4 a 3 : λ (3) λ (3) 2 λ (3) 3 a 2 : λ (2) λ (2) 2 a : λ () G(x) := {(x (n ), x (n 2),..., x () ) : x x (n ) x (n 2) x () } subset of R n R n 2 R 2 R. Λ := (λ (n ), λ (n 2),..., λ () ) G(x) Y. Baryshnikov (200): Λ is uniformly distributed on G(x) 23/26

24/26 U(n) f XX (HD x H )dh = C n,θ,b e n j= x ( n ) j cn x j The last line equals Vol(G(x)) G(x) { n j= j= j k= x (j) k n } θ+ j j= k= ( n ) θ(n ) i<j n θ n(n )/2 x (x i θ xj θ j ) j= i<j n (x i x j ) and c n θ(n ) = r. dx (j) k

25/26 Collecting everything: { C n i x j ) i<j n(x 2} f XX (HD x H )dh x >x 2 >...>x n>0 U(n) = C { nc n,θ,b θ n(n )/2 (x i x j ) 2} e n j= x j i<j n ( n j= x j ) r i<j n (x θ i x θ j ) i<j n (x i x j ) x >x 2 >...>x n>0

Thank you 26/26