Moments of Random Matrices and Hypergeometric Orthogonal Polynomials - PowerPoint PPT Presentation

Moments of Random Matrices and Hypergeometric Orthogonal Polynomials Francesco Mezzadri Integrability and Randonmness in Mathematical Physics and Geometry CIRM, Luminy, 8-12 April 2019 Collaborators: Fabio D Cunden (UCD), Neil O’Connell (UCD) and Nick Simm (Sussex)

Outline Moments of Random Matrices Moments & Hypergeometric OP’s Wronskians & Hypergeometric OP’s Moments for β = 1 and β = 4 Conclusions

Moments of Random Matrices ◮ j,p.d.f. of the eigenvalues at the classical RMT Ensembles n 1 � � | x k − x j | β dx 1 · · · dx n w β ( x j ) χ I ( x j ) C n ,β j =1 1 ≤ j < k ≤ n β = 1 , 2 , 4, I = R , I = R + and I = [0 , 1]

Moments of Random Matrices ◮ j,p.d.f. of the eigenvalues at the classical RMT Ensembles n 1 � � | x k − x j | β dx 1 · · · dx n w β ( x j ) χ I ( x j ) C n ,β j =1 1 ≤ j < k ≤ n β = 1 , 2 , 4, I = R , I = R + and I = [0 , 1] ◮ The weights are e − ( β/ 2) x 2  Hermite   x ( β/ 2)( m − n +1) − 1 e − ( β/ 2) x w β ( x ) = Laguerre β 2 ( m 1 − n +1) − 1 x β  2 ( m 2 − n +1) − 1 (1 − x ) Jacobi 

Moments of Random Matrices ◮ GUE ensemble n | x k − x j | 2 � − x 2 � � � P n ( x 1 , . . . , x n ) = C n exp j j =1 1 ≤ j < k ≤ n

Moments of Random Matrices ◮ GUE ensemble n | x k − x j | 2 � − x 2 � � � P n ( x 1 , . . . , x n ) = C n exp j j =1 1 ≤ j < k ≤ n ◮ The k-point correlation function k × k [ K n ( x i , x j )] k R k ( x 1 , . . . , x k ) = det i , j =1

Moments of Random Matrices ◮ GUE ensemble n | x k − x j | 2 � − x 2 � � � P n ( x 1 , . . . , x n ) = C n exp j j =1 1 ≤ j < k ≤ n ◮ The k-point correlation function k × k [ K n ( x i , x j )] k R k ( x 1 , . . . , x k ) = det i , j =1 ◮ The kernel is expressed in terms of Hermite polymonials, n − 1 H k ( x ) H k ( y ) K n ( x , y ) = e − ( x 2 + y 2 ) / 2 � √ π 2 k k ! k =0 H k ( x ) = ( − 1) k e x 2 d k dx k e − x 2 � ∞ H k ( x ) H j ( x ) e − x 2 dx = √ π 2 k k ! δ jk −∞

Moments of Random Matrices ◮ We define the eigenvalue density ρ ( β ) n ( x )   n ρ ( β ) � n ( x ) = R 1 ( x ) = E δ ( x − x j )   j =1

Moments of Random Matrices ◮ We define the eigenvalue density ρ ( β ) n ( x )   n ρ ( β ) � n ( x ) = R 1 ( x ) = E δ ( x − x j )   j =1 ◮ The main object we study � x k ρ ( β ) E Tr X k n = n ( x ) dx I

Moments of Random Matrices ◮ We define the eigenvalue density ρ ( β ) n ( x )   n ρ ( β ) � n ( x ) = R 1 ( x ) = E δ ( x − x j )   j =1 ◮ The main object we study � x k ρ ( β ) E Tr X k n = n ( x ) dx I ◮ Applications:

Moments of Random Matrices ◮ We define the eigenvalue density ρ ( β ) n ( x )   n ρ ( β ) � n ( x ) = R 1 ( x ) = E δ ( x − x j )   j =1 ◮ The main object we study � x k ρ ( β ) E Tr X k n = n ( x ) dx I ◮ Applications: ◮ Quantum Transport: Conductance, Shot noise, Wigner time-delay

Moments of Random Matrices ◮ We define the eigenvalue density ρ ( β ) n ( x )   n ρ ( β ) � n ( x ) = R 1 ( x ) = E δ ( x − x j )   j =1 ◮ The main object we study � x k ρ ( β ) E Tr X k n = n ( x ) dx I ◮ Applications: ◮ Quantum Transport: Conductance, Shot noise, Wigner time-delay ◮ Quantum Field Theory — maps enumerations

Moments of Random Matrices ◮ We define the eigenvalue density ρ ( β ) n ( x )   n ρ ( β ) � n ( x ) = R 1 ( x ) = E δ ( x − x j )   j =1 ◮ The main object we study � x k ρ ( β ) E Tr X k n = n ( x ) dx I ◮ Applications: ◮ Quantum Transport: Conductance, Shot noise, Wigner time-delay ◮ Quantum Field Theory — maps enumerations ◮ Others...

Moments of Random Matrices ◮ If X n is a GUE matrix then [ k / 2] ǫ g ( k ) � Q C k ( n ) = E Tr X 2 k = n k +1 n 2 g . n g =0 ǫ g ( k ) is the number of maps of genus g with k edges.

Moments of Random Matrices ◮ If X n is a GUE matrix then [ k / 2] ǫ g ( k ) � Q C k ( n ) = E Tr X 2 k = n k +1 n 2 g . n g =0 ǫ g ( k ) is the number of maps of genus g with k edges. ◮ Take X n in the LUE, i.e. w ( λ ) = λ n e − n λ .

Moments of Random Matrices ◮ If X n is a GUE matrix then [ k / 2] ǫ g ( k ) � Q C k ( n ) = E Tr X 2 k = n k +1 n 2 g . n g =0 ǫ g ( k ) is the number of maps of genus g with k edges. ◮ Take X n in the LUE, i.e. w ( λ ) = λ n e − n λ . ◮ Then τ = (1 / n ) Tr X − 1 is the Wigner delay time n

Moments of Random Matrices ◮ If X n is a GUE matrix then [ k / 2] ǫ g ( k ) � Q C k ( n ) = E Tr X 2 k = n k +1 n 2 g . n g =0 ǫ g ( k ) is the number of maps of genus g with k edges. ◮ Take X n in the LUE, i.e. w ( λ ) = λ n e − n λ . ◮ Then τ = (1 / n ) Tr X − 1 is the Wigner delay time n ◮ The CGF H n ( t ) satisfies Painlev´ e III (FM & Simm, 2013) n ) 2 − H ′ n ) 2 = 4 H n n ) 2 ( zH ′′ � ( H ′ � � 4 z ( H ′ − n − (4 z + ( b − 2 n ) 2 ) H ′ H ′ n + n 2 . � n − 2 n ( b − 2 n )

Moments of Random Matrices ◮ Take the cumulant expansion of τ , 1 c g ( ν ) � C ν = (2 n 2 ) ν − 1 n g g ≥ 0

Moments of Random Matrices ◮ Take the cumulant expansion of τ , 1 c g ( ν ) � C ν = (2 n 2 ) ν − 1 n g g ≥ 0 ◮ c 0 ( ν ) are integers for all ν (FM & Simm, 2013)

Moments of Random Matrices ◮ Take the cumulant expansion of τ , 1 c g ( ν ) � C ν = (2 n 2 ) ν − 1 n g g ≥ 0 ◮ c 0 ( ν ) are integers for all ν (FM & Simm, 2013) ◮ Take M ( β ) ( n ) = n k − 1 E Tr X − k k ≥ 0 , β = 1 , 2 n k

Moments of Random Matrices ◮ Take the cumulant expansion of τ , 1 c g ( ν ) � C ν = (2 n 2 ) ν − 1 n g g ≥ 0 ◮ c 0 ( ν ) are integers for all ν (FM & Simm, 2013) ◮ Take M ( β ) ( n ) = n k − 1 E Tr X − k k ≥ 0 , β = 1 , 2 n k ◮ Now take the asymptotics expansion of the moments ∞ M ( β ) � κ ( β ) g ( k ) n − g , ( n ) = β = 1 , 2 . k g =0

Moments of Random Matrices ◮ Conjecture: κ (2) ∈ N (Cunden, FM, Simm & Vivo 2016) g

Moments of Random Matrices ◮ Conjecture: κ (2) ∈ N (Cunden, FM, Simm & Vivo 2016) g ◮ Cumulant expansion of negative powers of matrices in the LUE 1 Tr X − µ 1 , . . . , Tr X − µ k � � � n − g c g ( µ 1 , . . . , µ k ) , C k = n n (2 N 2 ) k − 1 g ≥ 0 with ( µ 1 , . . . , µ k ) ∈ N k .

Moments of Random Matrices ◮ Conjecture: κ (2) ∈ N (Cunden, FM, Simm & Vivo 2016) g ◮ Cumulant expansion of negative powers of matrices in the LUE 1 Tr X − µ 1 , . . . , Tr X − µ k � � � n − g c g ( µ 1 , . . . , µ k ) , C k = n n (2 N 2 ) k − 1 g ≥ 0 with ( µ 1 , . . . , µ k ) ∈ N k . ◮ The c g ( µ 1 , . . . , µ k ) are Hurwitz numbers (Cunden, Dahlqvist & O’Connell 2018)

Moments & Hypergeometric OP’s ◮ If X n is a GUE matrix then E Tr X 2 k is a polynomial in n n n = 14 n 5 + 70 n 3 + 21 n . E Tr X 8 ( ∗ )

Moments & Hypergeometric OP’s ◮ If X n is a GUE matrix then E Tr X 2 k is a polynomial in n n n = 14 n 5 + 70 n 3 + 21 n . E Tr X 8 ( ∗ ) ◮ Can we say something about it as a function of k ? ( k +2) Q C k +1 ( n ) = 2 n (2 k +1) Q C k ( n )+ k (2 k +1)(2 k − 1) Q C k − 1 ( n ) , (Harer and Zagier, 1986)

Moments & Hypergeometric OP’s ◮ If X n is a GUE matrix then E Tr X 2 k is a polynomial in n n n = 14 n 5 + 70 n 3 + 21 n . E Tr X 8 ( ∗ ) ◮ Can we say something about it as a function of k ? ( k +2) Q C k +1 ( n ) = 2 n (2 k +1) Q C k ( n )+ k (2 k +1)(2 k − 1) Q C k − 1 ( n ) , (Harer and Zagier, 1986) ◮ It turns out that 1 = 4 3 k 3 + 4 k 2 + 20 (2 k − 1)!! E Tr X 2 k 3 k + 4 . 4

Moments & Hypergeometric OP’s ◮ If X n is a GUE matrix then E Tr X 2 k is a polynomial in n n n = 14 n 5 + 70 n 3 + 21 n . E Tr X 8 ( ∗ ) ◮ Can we say something about it as a function of k ? ( k +2) Q C k +1 ( n ) = 2 n (2 k +1) Q C k ( n )+ k (2 k +1)(2 k − 1) Q C k − 1 ( n ) , (Harer and Zagier, 1986) ◮ It turns out that 1 = 4 3 k 3 + 4 k 2 + 20 (2 k − 1)!! E Tr X 2 k 3 k + 4 . 4 This is a Meixner polynomial!

Moments & Hypergeometric OP’s ◮ Meixner polynomials have the representation � − n , − x ; 1 − 1 � M n ( x ; γ, c ) = 2 F 1 γ c

Moments & Hypergeometric OP’s ◮ Meixner polynomials have the representation � − n , − x ; 1 − 1 � M n ( x ; γ, c ) = 2 F 1 γ c ◮ They obey the orthogonality relation ∞ ( γ ) x � x ! c x M n ( x ; γ, c ) M m ( x ; γ, c ) x =0 c − n n ! = ( γ ) x (1 − c ) γ δ mn , γ > 0 , 0 < c < 1

Moments & Hypergeometric OP’s ◮ Meixner polynomials have the representation � − n , − x ; 1 − 1 � M n ( x ; γ, c ) = 2 F 1 γ c ◮ They obey the orthogonality relation ∞ ( γ ) x � x ! c x M n ( x ; γ, c ) M m ( x ; γ, c ) x =0 c − n n ! = ( γ ) x (1 − c ) γ δ mn , γ > 0 , 0 < c < 1 ◮ They obey the recurrence relation ( c − 1) xM n ( x ; γ, c ) = c ( n + γ ) M n +1 ( x ; γ, c ) − [ n + ( n + γ ) c ] M n ( x ; γ, c ) + nM n − 1 ( x ; γ, c )

Moments & Hypergeometric OP’s ◮ Take X n in the LUE with parameter α = m − n and define Q C k ( m , n ) = E Tr X k n