Random Matrices and Zeros of Polynomials Guilherme Silva Joint - PowerPoint PPT Presentation

Random Matrices and Zeros of Polynomials Guilherme Silva Joint work with Pavel Bleher (IUPUI) [Memoirs of the AMS, to appear] Guilherme Silva RMT and zeros of pols

Our interest today M random matrix size N × N Guilherme Silva RMT and zeros of pols

Our interest today M random matrix size N × N q N ( z ) = det( zI − M ) Guilherme Silva RMT and zeros of pols

Our interest today M random matrix size N × N q N ( z ) = det( zI − M ) P N ( z ) = E (det( zI − M )) Guilherme Silva RMT and zeros of pols

Our interest today M random matrix size N × N q N ( z ) = det( zI − M ) P N ( z ) = E (det( zI − M )) µ ( q N ) = 1 � δ w N q N ( w )=0 Guilherme Silva RMT and zeros of pols

Our interest today M random matrix size N × N q N ( z ) = det( zI − M ) P N ( z ) = E (det( zI − M )) µ ( q N ) = 1 µ ( P N ) = 1 � � δ w δ w N N q N ( w )=0 P N ( w )=0 Guilherme Silva RMT and zeros of pols

Our interest today M random matrix size N × N q N ( z ) = det( zI − M ) P N ( z ) = E (det( zI − M )) µ ( q N ) = 1 µ ( P N ) = 1 � � δ w δ w N N q N ( w )=0 P N ( w )=0 ?? Guilherme Silva RMT and zeros of pols

Eigenvalues for N = 50 , 60 , . . . , 1250 - 2 - 1 0 1 2 Guilherme Silva RMT and zeros of pols

Eigenvalues for N = 50 , 60 , . . . , 1250 - 2 - 1 0 1 2 Guilherme Silva RMT and zeros of pols

The general framework - potential theory on C Guilherme Silva RMT and zeros of pols

The general framework - potential theory on C Basic ingredients: ◮ D ⊂ C closed Guilherme Silva RMT and zeros of pols

The general framework - potential theory on C Basic ingredients: ◮ D ⊂ C closed ◮ Q : D → R “sufficiently nice” Guilherme Silva RMT and zeros of pols

The general framework - potential theory on C Basic ingredients: ◮ D ⊂ C closed ◮ Q : D → R “sufficiently nice” The equilibrium measure µ Q = µ Q,D of D is the probability measure that minimizes �� 1 � log | s − z | dµ ( s ) dµ ( z ) + Q ( z ) dµ ( z ) over all probability measures µ with supp µ ⊂ D . Guilherme Silva RMT and zeros of pols

The general framework - potential theory on C Basic ingredients: ◮ D ⊂ C closed ◮ Q : D → R “sufficiently nice” The equilibrium measure µ Q = µ Q,D of D is the probability measure that minimizes �� 1 � log | s − z | dµ ( s ) dµ ( z ) + Q ( z ) dµ ( z ) over all probability measures µ with supp µ ⊂ D . ◮ For D and Q nice enough, µ Q uniquely exists Guilherme Silva RMT and zeros of pols

The general framework - potential theory on C Basic ingredients: ◮ D ⊂ C closed ◮ Q : D → R “sufficiently nice” The equilibrium measure µ Q = µ Q,D of D is the probability measure that minimizes �� 1 � log | s − z | dµ ( s ) dµ ( z ) + Q ( z ) dµ ( z ) over all probability measures µ with supp µ ⊂ D . ◮ For D and Q nice enough, µ Q uniquely exists ◮ If D is unbounded, we have to impose sufficient growth for Q Guilherme Silva RMT and zeros of pols

The general framework - potential theory on C Basic ingredients: ◮ D ⊂ C closed ◮ Q : D → R “sufficiently nice” The equilibrium measure µ Q = µ Q,D of D is the probability measure that minimizes �� 1 � log | s − z | dµ ( s ) dµ ( z ) + Q ( z ) dµ ( z ) over all probability measures µ with supp µ ⊂ D . ◮ For D and Q nice enough, µ Q uniquely exists ◮ If D is unbounded, we have to impose sufficient growth for Q ◮ Finding supp µ Q is challenging Guilherme Silva RMT and zeros of pols

Unitary ensembles Guilherme Silva RMT and zeros of pols

Unitary ensembles ◮ Unitary ensembles: space H N of N × N hermitian matrices equipped with probability distribution 1 e − N Tr V ( M ) dM, (1) Z N where V is a real polynomial of even degree and dM is the Lebesgue measure on H N ≃ R N 2 . Guilherme Silva RMT and zeros of pols

Unitary ensembles ◮ Unitary ensembles: space H N of N × N hermitian matrices equipped with probability distribution 1 e − N Tr V ( M ) dM, (1) Z N where V is a real polynomial of even degree and dM is the Lebesgue measure on H N ≃ R N 2 . • Unitary because (1) is invariant under unitary conjugation M �→ UMU ∗ Guilherme Silva RMT and zeros of pols

Unitary ensembles ◮ Unitary ensembles: space H N of N × N hermitian matrices equipped with probability distribution 1 e − N Tr V ( M ) dM, (1) Z N where V is a real polynomial of even degree and dM is the Lebesgue measure on H N ≃ R N 2 . • Unitary because (1) is invariant under unitary conjugation M �→ UMU ∗ • When V ( x ) = x 2 / 2 , entries are independent Gaussian random variables (GUE) Guilherme Silva RMT and zeros of pols

Unitary ensembles ◮ Unitary ensembles: space H N of N × N hermitian matrices equipped with probability distribution 1 e − N Tr V ( M ) dM, (1) Z N where V is a real polynomial of even degree and dM is the Lebesgue measure on H N ≃ R N 2 . • Unitary because (1) is invariant under unitary conjugation M �→ UMU ∗ • When V ( x ) = x 2 / 2 , entries are independent Gaussian random variables (GUE) • The factor N makes sure that eigenvalues remain bounded Guilherme Silva RMT and zeros of pols

Unitary ensembles - techniques ◮ We can see the diagonalization   λ 1 0 · · · 0 0 λ 2 · · · 0    U ∗ M = U  . . .  ... . . .   . . .  0 0 · · · λ N as a change of variables Guilherme Silva RMT and zeros of pols

Unitary ensembles - techniques ◮ We can see the diagonalization   λ 1 0 · · · 0 0 λ 2 · · · 0    U ∗ M = U  . . .  ... . . .   . . .  0 0 · · · λ N as a change of variables ◮ Computing the Jacobian of the change of variables we get that 1 e − N Tr V ( M ) dM = Z N 1 � ( λ j − λ k ) 2 � e − NV ( λ j ) dλ 1 . . . dλ N dU Z N j j<k where dU is the Haar measure on U N Guilherme Silva RMT and zeros of pols

Unitary ensembles - techniques ◮ We can see the diagonalization   λ 1 0 · · · 0 0 λ 2 · · · 0    U ∗ M = U  . . .  ... . . .   . . .  0 0 · · · λ N as a change of variables ◮ Computing the Jacobian of the change of variables we get that 1 e − N Tr V ( M ) dM = Z N 1 � ( λ j − λ k ) 2 � e − NV ( λ j ) dλ 1 . . . dλ N dU Z N j j<k where dU is the Haar measure on U N ◮ Consequences: • Eigenvalues and eigenvectors are independent Guilherme Silva RMT and zeros of pols

Unitary ensembles - techniques ◮ We can see the diagonalization   λ 1 0 · · · 0 0 λ 2 · · · 0    U ∗ M = U  . . .  ... . . .   . . .  0 0 · · · λ N as a change of variables ◮ Computing the Jacobian of the change of variables we get that 1 e − N Tr V ( M ) dM = Z N 1 � ( λ j − λ k ) 2 � e − NV ( λ j ) dλ 1 . . . dλ N dU Z N j j<k where dU is the Haar measure on U N ◮ Consequences: • Eigenvalues and eigenvectors are independent • Eigenvectors are uniformly distributed on U ( N ) Guilherme Silva RMT and zeros of pols

Unitary ensembles - techniques ◮ We can see the diagonalization   λ 1 0 · · · 0 0 λ 2 · · · 0    U ∗ M = U  . . .  ... . . .   . . .  0 0 · · · λ N as a change of variables ◮ Computing the Jacobian of the change of variables we get that 1 e − N Tr V ( M ) dM = Z N 1 � ( λ j − λ k ) 2 � e − NV ( λ j ) dλ 1 . . . dλ N dU Z N j j<k where dU is the Haar measure on U N ◮ Consequences: • Eigenvalues and eigenvectors are independent • Eigenvectors are uniformly distributed on U ( N ) • Eigenvalues exhibit local repulsion Guilherme Silva RMT and zeros of pols

Unitary ensembles - global behavior of eigenvalues We can rewrite 1 1 e − NV ( λ j ) = e − N 2 H ( λ 1 ,...,λ N ) � ( λ j − λ k ) 2 � Z N Z N j<k j where 1 | λ j − λ k | + 1 1 � � H ( λ 1 , . . . , λ N ) = log V ( λ j ) N 2 N j � = k j Guilherme Silva RMT and zeros of pols

Unitary ensembles - global behavior of eigenvalues We can rewrite 1 1 e − NV ( λ j ) = e − N 2 H ( λ 1 ,...,λ N ) � ( λ j − λ k ) 2 � Z N Z N j<k j where 1 | λ j − λ k | + 1 1 � � H ( λ 1 , . . . , λ N ) = log V ( λ j ) N 2 N j � = k j �� 1 � = log | x − y | dµ N ( x ) dµ N ( y ) + V dµ N , x � = y µ N := µ ( q N ) = 1 � δ λ k N k Guilherme Silva RMT and zeros of pols

Unitary ensembles - global behavior of eigenvalues We can rewrite 1 1 e − NV ( λ j ) = e − N 2 H ( λ 1 ,...,λ N ) � ( λ j − λ k ) 2 � Z N Z N j<k j where 1 | λ j − λ k | + 1 1 � � H ( λ 1 , . . . , λ N ) = log V ( λ j ) N 2 N j � = k j �� 1 � = log | x − y | dµ N ( x ) dµ N ( y ) + V dµ N , x � = y µ N := µ ( q N ) = 1 � δ λ k N k ◮ Thus the most likely eigenvalue configurations µ ( q N ) ’s should be close to µ V ! Guilherme Silva RMT and zeros of pols

Unitary ensembles and orthogonal polynomials After some massage, we get that 1 e − NV ( λ j ) = det( K N ( λ k , λ j )) 1 ≤ k,j ≤ n � ( λ j − λ k ) 2 � Z N j j<k Guilherme Silva RMT and zeros of pols