General beta ensembles For any β > 0, introduce the law P n , � on n real points with density: n e � β 4 n � 2 Y k ⇥ Y | λ j � λ k | � / k =1 j < k 2 0 1 3 n λ 2 X X k = exp 4 � β @ n 4 � log | λ j � λ k | 5 . A k =1 j < k For β = 1 , 2 , 4 these are the eigenvalue densities for G { O,U,S } E. More broadly P � is referred to as the “beta-Hermite” ensemble. Interpreted as a 1-d caricature of a Coulomb gas, which happens to be solvable at three special values of the “charge”. Is there a one-parameter family of Tracy-Widom laws? Brian Rider (Temple University) Operator limits of random matrices 8 / 20
Stochastic Airy Operator Theorem (Ram´ ırez, R., Vir´ ag) For x 7! b ( x ) a standard Brownian motion, and any β > 0 define H � = � d 2 2 p β b 0 ( x ) . dx 2 + x + Let Λ 0 Λ 1 · · · denote the eigenvalues of H � acting on L 2 [0 , 1 ) with Dirichlet conditions at the origin. Then, with λ 1 > λ 2 > · · · the ordered points under P n , � it holds that n o n o n 2 / 3 (2 � λ ` ) ` =1 , k ) Λ ` ` =0 , k � 1 for any fixed k as n ! 1 . As b 0 ( x ) is a random distribution (Brownian motion is almost everywhere non-di ff erentiable), some work is required to make sense of H � Brian Rider (Temple University) Operator limits of random matrices 9 / 20
General beta Tracy-Widom The limiting largest point of the Hermite β -ensemble then converges to the ( negative) ground state eigenvalue of H � . In particular, ⇢Z 1 Z 1 2 � ( f 0 ( x )) 2 + xf 2 ( x ) f 2 ( x ) db ( x ) ⇥ ⇤ � TW � = inf dx + p β f 2 L 0 0 for Z 1 Z 1 ⇢ � ( f 0 ( x )) 2 + xf 2 ( x ) f 2 ( x ) dx = 1 , ⇥ ⇤ L = f : f (0) = 0 , dx < 1 . 0 0 Brian Rider (Temple University) Operator limits of random matrices 10 / 20
General beta Tracy-Widom The limiting largest point of the Hermite β -ensemble then converges to the ( negative) ground state eigenvalue of H � . In particular, ⇢Z 1 Z 1 2 � ( f 0 ( x )) 2 + xf 2 ( x ) f 2 ( x ) db ( x ) ⇥ ⇤ � TW � = inf dx + p β f 2 L 0 0 for Z 1 Z 1 ⇢ � ( f 0 ( x )) 2 + xf 2 ( x ) f 2 ( x ) dx = 1 , ⇥ ⇤ L = f : f (0) = 0 , dx < 1 . 0 0 Form is densely defined, and tempting to get a lower bound via Z 1 Z 1 Z 1 Z 1 � � � � f 2 db ( f 0 ) 2 ( x ) dx + c 0 b 2 ( x ) f 2 ( x ) dx , � � f 0 ( x ) f ( x ) b ( x ) dx � = 2 � c � � � � � � 0 0 0 0 but the law of the iterated log shows you have to be a bit more clever (even for large beta). Brian Rider (Temple University) Operator limits of random matrices 10 / 20
Where does this come from? For all β > 0 there is a simple tridiagonal matrix model for P � . Theorem (Dumitriu-Edelman) Let g 1 , g 2 , . . . , g n be independent N (0 , 2) and χ � n , χ � ( n � 1) , . . . , χ � be independent “chi” variables of the indicated parameter. Then the joint distribution of eigenvalues of the random Jacobi matrix 2 3 g 1 χ ( n � 1) � g 2 χ ( n � 1) � χ ( n � 2) � 6 7 1 6 7 ... ... ... H n , � = p n β 6 7 6 7 6 7 χ 2 � g n � 1 χ � 4 5 χ � g n is given by P n , � . (A χ r has density / x r � 1 e � x 2 / 2 , otherwise referred to as a certain Γ variable) , Brian Rider (Temple University) Operator limits of random matrices 11 / 20
Tridiagonals for the classical ensembles Any Hermitian matrix can be brought into tridiagonal form (while keeping the eigenvalues fixed) by a suitable sequence of Householder transformations. Brian Rider (Temple University) Operator limits of random matrices 12 / 20
Tridiagonals for the classical ensembles Any Hermitian matrix can be brought into tridiagonal form (while keeping the eigenvalues fixed) by a suitable sequence of Householder transformations. With M = M n = [ m ij ] 1 i , j n , m ij = m ji write m † � m ii M = m M n � 1 and build a ( n � 1) ⇥ ( n � 1) unitary U = [ u 1 . . . u n � 1 ] with m † u 1 = k m k . Then 1 2 ( k m k , 0 · · · 0) † 3 m ii 0 † � 0 † � 1 = 5 , M U † 4 0 0 U U † M n � 1 U ( k m k , 0 · · · 0) repeat. Brian Rider (Temple University) Operator limits of random matrices 12 / 20
Tridiagonals for the classical ensembles Any Hermitian matrix can be brought into tridiagonal form (while keeping the eigenvalues fixed) by a suitable sequence of Householder transformations. With M = M n = [ m ij ] 1 i , j n , m ij = m ji write m † � m ii M = m M n � 1 and build a ( n � 1) ⇥ ( n � 1) unitary U = [ u 1 . . . u n � 1 ] with m † u 1 = k m k . Then 1 2 ( k m k , 0 · · · 0) † 3 m ii 0 † � 0 † � 1 = 5 , M U † 4 0 0 U U † M n � 1 U ( k m k , 0 · · · 0) repeat. Exercise: Convince yourself that when you carry out the above for GOE or GUE you get the advertised β = 1 or β = 2 tridiagonal. Note: (i) Gaussian vectors are rotation invariant, (ii) the squared norm of a d -dim Gaussian vector is a χ 2 d . Brian Rider (Temple University) Operator limits of random matrices 12 / 20
Reverse engineering the Jacobian Instructive to view the Dumitriu-Edelman matrix model as placing a measure down on random tridiagonals. Brian Rider (Temple University) Operator limits of random matrices 13 / 20
Reverse engineering the Jacobian Instructive to view the Dumitriu-Edelman matrix model as placing a measure down on random tridiagonals. With T ( A , B ) = tridiag( B , A , B ) for B = ( B 1 , . . . , B n � 1 ) 2 R + n � 1 and A = ( A 1 , . . . , A n ) 2 R n their result reads: Brian Rider (Temple University) Operator limits of random matrices 13 / 20
Reverse engineering the Jacobian Instructive to view the Dumitriu-Edelman matrix model as placing a measure down on random tridiagonals. With T ( A , B ) = tridiag( B , A , B ) for B = ( B 1 , . . . , B n � 1 ) 2 R + n � 1 and A = ( A 1 , . . . , A n ) 2 R n their result reads: Distribute ( A , B ) according to the density ⇣ ⌘ n � 1 n � 1 4 tr � n β T 2 ( a , b ) i +2 P n � 1 / e � n β 4 ( P n i =1 a 2 i =1 b 2 b � ( n � i ) b � ( n � i ) i ) Y Y = e i i i =1 i =1 then the eigenvalues of T ( A , B ) have density n e � β 4 n � 2 Y k ⇥ Y | λ j � λ k | � . / k =1 j < k Brian Rider (Temple University) Operator limits of random matrices 13 / 20
Reverse engineering the Jacobian Instructive to view the Dumitriu-Edelman matrix model as placing a measure down on random tridiagonals. With T ( A , B ) = tridiag( B , A , B ) for B = ( B 1 , . . . , B n � 1 ) 2 R + n � 1 and A = ( A 1 , . . . , A n ) 2 R n their result reads: Distribute ( A , B ) according to the density ⇣ ⌘ n � 1 n � 1 4 tr � n β T 2 ( a , b ) i +2 P n � 1 / e � n β 4 ( P n i =1 a 2 i =1 b 2 b � ( n � i ) b � ( n � i ) i ) Y Y = e i i i =1 i =1 then the eigenvalues of T ( A , B ) have density n e � β 4 n � 2 Y k ⇥ Y | λ j � λ k | � . / k =1 j < k The map needed is to go from tridiagonal ( a , b )-coordinates to eigenvalue and eigenvector (really norming constant ) ( λ , q )-coordinates. Brian Rider (Temple University) Operator limits of random matrices 13 / 20
Stochastic Airy heuristics Edelman-Sutton had conjectured the Stochastic Airy limit via the natural continuum limit of the tridiagonals. That is, they suggested that n 2 / 3 (2 I � H n , � ) � d 2 p β b 0 ( x ) 2 dx 2 + x + as operators. (Scaling H n , � itself like λ max in Tracy-Widom.) Brian Rider (Temple University) Operator limits of random matrices 14 / 20
Stochastic Airy heuristics Edelman-Sutton had conjectured the Stochastic Airy limit via the natural continuum limit of the tridiagonals. That is, they suggested that n 2 / 3 (2 I � H n , � ) � d 2 p β b 0 ( x ) 2 dx 2 + x + as operators. (Scaling H n , � itself like λ max in Tracy-Widom.) The only thing really moving in H n , � is those o ff diagonal χ s. 1 Excerise: Make precise the statement that, for fixed k and n ! 1 , p β n χ β ( n � k ) ' 1 � k 2 n + g for g a Gaussian. This give the leading order n 2 / 3 (2 I � H n , � ) = n 2 / 3 tridiag( � 1 , 2 , � 1) + · · · which has the clear interpretation as � d 2 dx 2 , discretized on scale ( ∆ x ) = n � 1 / 3 . Brian Rider (Temple University) Operator limits of random matrices 14 / 20
Stochastic Airy heuristics Edelman-Sutton had conjectured the Stochastic Airy limit via the natural continuum limit of the tridiagonals. That is, they suggested that n 2 / 3 (2 I � H n , � ) � d 2 p β b 0 ( x ) 2 dx 2 + x + as operators. (Scaling H n , � itself like λ max in Tracy-Widom.) The only thing really moving in H n , � is those o ff diagonal χ s. 1 Excerise: Make precise the statement that, for fixed k and n ! 1 , p β n χ β ( n � k ) ' 1 � k 2 n + g for g a Gaussian. This give the leading order n 2 / 3 (2 I � H n , � ) = n 2 / 3 tridiag( � 1 , 2 , � 1) + · · · which has the clear interpretation as � d 2 dx 2 , discretized on scale ( ∆ x ) = n � 1 / 3 . Excerise: Convince yourself that the natural continuum interpretation of n 2 / 3 (tridiag(1 , 0 , 1) � H n , β ) as n ! 1 is ⌦ ( x + 2 p β b 0 ( x )). Brian Rider (Temple University) Operator limits of random matrices 14 / 20
The Riccati substitution Consider τ = � d 2 dx 2 + q ( x ) for a nice (deterministic, smooth) potential q and its Dirichlet eigenvalue problem on [0 , L < 1 ] τψ ( x ) = λψ ( x ) , ψ (0) = ψ ( L ) = 0 . Brian Rider (Temple University) Operator limits of random matrices 15 / 20
The Riccati substitution Consider τ = � d 2 dx 2 + q ( x ) for a nice (deterministic, smooth) potential q and its Dirichlet eigenvalue problem on [0 , L < 1 ] τψ ( x ) = λψ ( x ) , ψ (0) = ψ ( L ) = 0 . Sturm’s Oscillation theorem tells you: Consider the corresponding solution ψ = ψ ( x , λ ) for fixed λ to the initial value problem with ψ (0 , λ ) = 0 and ψ 0 (0 , λ ) = 1. Then it holds that n o n o # eigenvalues λ = # zeros of x 7! ψ ( x , λ ) in [0 , L ] . Brian Rider (Temple University) Operator limits of random matrices 15 / 20
The Riccati substitution Consider τ = � d 2 dx 2 + q ( x ) for a nice (deterministic, smooth) potential q and its Dirichlet eigenvalue problem on [0 , L < 1 ] τψ ( x ) = λψ ( x ) , ψ (0) = ψ ( L ) = 0 . Sturm’s Oscillation theorem tells you: Consider the corresponding solution ψ = ψ ( x , λ ) for fixed λ to the initial value problem with ψ (0 , λ ) = 0 and ψ 0 (0 , λ ) = 1. Then it holds that n o n o # eigenvalues λ = # zeros of x 7! ψ ( x , λ ) in [0 , L ] . The Riccati substitution takes the equation satisfied by p ( x ) = 0 ( x , � ) ( x , � ) : p 0 ( x ) = q ( x ) � λ � p 2 ( x ) . This starts at p (0) = + 1 , hits �1 when ψ hits zero, immediately “reappearing” at + 1 . Brian Rider (Temple University) Operator limits of random matrices 15 / 20
The Riccati di ff usion p β b 0 ( x ): 2 What this means for q ( x ) = x + Theorem Consider the solution p t = p � t to the Itˆ o equation p β db t + ( λ + t � p 2 dp t = 2 t ) dt , started from + 1 at time zero, and restarted there after any explosion to �1 . Then P ( TW � λ ) = P (+ 1 , 0) ( p � never explodes ) , with the distribution of the k th largest point being given by the probability of at most k explosions. Note: Can absorb the spectral parameter λ into a starting time, or, replace the probabilities on the right with P (+ 1 , � ) for p = p 0 . 2 Exercise: Show that p t � p β b t solves an ODE with random coe ffi cients - convince yourself that the process really can be started from 1 Brian Rider (Temple University) Operator limits of random matrices 16 / 20
Application: Tracy-Widom( β ) tails Combining the defining variational principle Z 1 Z 1 2 x ) 2 + xf 2 ( f 0 f 2 ⇥ ⇤ p β � TW � = inf dx + x db x x f 2 L 0 0 with the Riccati di ff usion description p β db t + ( t � p 2 P ( TW � λ ) = P (+ 1 , � ) ( p never explodes) , dp t = 2 t ) dt we can prove: Theorem (Ram´ ırez, R., Vir´ ag) For all β > 0 it holds 3 P ( TW � > a ) = e � 2 2 (1+ o (1)) 3 � a and P ( TW � < � a ) = e � β 24 a 3 (1+ o (1)) as a ! 1 . Brian Rider (Temple University) Operator limits of random matrices 17 / 20
Proof of left-tail upper bound Using that � TW � is the ground state eigenvalue of H � one has R ( f 0 2 x + xf 2 2 R f 2 ! x ) dx ] + x db x p β P ( TW � < � a ) = P ( Λ 0 ( H � ) > a ) P > a R f 2 x dx for any nice function f 6⌘ 0 vanishing at the origin. Brian Rider (Temple University) Operator limits of random matrices 18 / 20
Proof of left-tail upper bound Using that � TW � is the ground state eigenvalue of H � one has R ( f 0 2 x + xf 2 2 R f 2 ! x ) dx ] + x db x p β P ( TW � < � a ) = P ( Λ 0 ( H � ) > a ) P > a R f 2 x dx for any nice function f 6⌘ 0 vanishing at the origin. qR R f 2 Exercise: For deterministic f it holds x db x ⇠ f 4 x ⇥ g for g ⇠ N (0 , 1). Brian Rider (Temple University) Operator limits of random matrices 18 / 20
Proof of left-tail upper bound Using that � TW � is the ground state eigenvalue of H � one has R ( f 0 2 x + xf 2 2 R f 2 ! x ) dx ] + x db x p β P ( TW � < � a ) = P ( Λ 0 ( H � ) > a ) P > a R f 2 x dx for any nice function f 6⌘ 0 vanishing at the origin. qR R f 2 Exercise: For deterministic f it holds x db x ⇠ f 4 x ⇥ g for g ⇠ N (0 , 1). Choose f ( x ) = ( x p a ) ^ ( a � x ) + ^ ( a � x ) + p and collect: x dx ⇠ a 3 x dx ⇠ a 3 x dx ⇠ a 3 Z Z Z f 2 xf 2 f 4 a 2 , 6 , 3 , f 0 ( x ) 2 dx = O ( a ) to finish. R while Brian Rider (Temple University) Operator limits of random matrices 18 / 20
Proof of left-tail lower bound p β db t + ( t � p 2 We look at the event that the di ff usion dp t = 2 t ) dt , started from position + 1 at time � a never explodes (hits �1 ). Brian Rider (Temple University) Operator limits of random matrices 19 / 20
Proof of left-tail lower bound p β db t + ( t � p 2 We look at the event that the di ff usion dp t = 2 t ) dt , started from position + 1 at time � a never explodes (hits �1 ). Want to estimate the probability of a “likely path”. Intuitively, p wants to hang around the origin until it makes it into the safe parabola (where drift can be positive). Brian Rider (Temple University) Operator limits of random matrices 19 / 20
Proof of left-tail lower bound p β db t + ( t � p 2 We look at the event that the di ff usion dp t = 2 t ) dt , started from position + 1 at time � a never explodes (hits �1 ). Want to estimate the probability of a “likely path”. Intuitively, p wants to hang around the origin until it makes it into the safe parabola (where drift can be positive). With that P ( TW � < � a ) = P ( 1 , � a ) ( p never explodes ) � P (1 , � a ) ( p never explodes) � P (1 , � a ) ( p t 2 [0 , 2] for all t 2 [ � a , 0]) P 0 , 0 ( p never explodes) What we’ve bought: The second factor has no dependence on a ! 1 . Brian Rider (Temple University) Operator limits of random matrices 19 / 20
Left-tail lower bound con’t Cameron-Martin-Girsanov: Let P denote the measure induced on continuous paths by the solution of x t = p σ b t + R t · f ( x s ) ds . Over finite time windows this will be absolutely continuous to Brownian motion measure with dP R T R T � 1 S f ( b t ) db t � 1 S f 2 ( b t ) dt F [ S , T ] = e � σ 2 σ dBM � (assuming nice enough f , both processes started from the same place, etc.) Brian Rider (Temple University) Operator limits of random matrices 20 / 20
Left-tail lower bound con’t Cameron-Martin-Girsanov: Let P denote the measure induced on continuous paths by the solution of x t = p σ b t + R t · f ( x s ) ds . Over finite time windows this will be absolutely continuous to Brownian motion measure with dP R T R T � 1 S f ( b t ) db t � 1 S f 2 ( b t ) dt F [ S , T ] = e � σ 2 σ dBM � (assuming nice enough f , both processes started from the same place, etc.) Applied to p t for which f ( p t ) = ( t � p 2 t ) over the widow t 2 [ � a , 0]: ⇣ ⌘ P ( TW � < � a ) � c � P (1 , � a ) p t 2 [0 , 2] for all t 2 [ � a , 0] R 0 R 0 h β � a ( t � b 2 t ) db t � β � a ( t � b t ) 2 dt i = c � E (1 , � a ) 1 A e 4 8 with A = { b t 2 [0 , 2] , t 2 [ � a , 0] } . Brian Rider (Temple University) Operator limits of random matrices 20 / 20
Left-tail lower bound con’t Cameron-Martin-Girsanov: Let P denote the measure induced on continuous paths by the solution of x t = p σ b t + R t · f ( x s ) ds . Over finite time windows this will be absolutely continuous to Brownian motion measure with dP R T R T � 1 S f ( b t ) db t � 1 S f 2 ( b t ) dt F [ S , T ] = e � σ 2 σ dBM � (assuming nice enough f , both processes started from the same place, etc.) Applied to p t for which f ( p t ) = ( t � p 2 t ) over the widow t 2 [ � a , 0]: ⇣ ⌘ P ( TW � < � a ) � c � P (1 , � a ) p t 2 [0 , 2] for all t 2 [ � a , 0] R 0 R 0 h β � a ( t � b 2 t ) db t � β � a ( t � b t ) 2 dt i = c � E (1 , � a ) 1 A e 4 8 with A = { b t 2 [0 , 2] , t 2 [ � a , 0] } . R t R t 0 f 0 ( b t ) db t + 1 0 f 00 ( b t ) dt finish the job. Exercise: Granted Itˆ o’s rule f ( b t ) � f ( b 0 ) = 2 Brian Rider (Temple University) Operator limits of random matrices 20 / 20
Operator limits of random matrices II. Stochastic Airy: proofs and extensions Brian Rider Temple University Brian Rider (Temple University) Operator limits of random matrices 1 / 16
Task for the hour (1) Show that Stochastic Airy H � = � d 2 2 p β b 0 ( x ) dx 2 + x + (on R + with Dirichlet boundaries) can be made sensible. Brian Rider (Temple University) Operator limits of random matrices 2 / 16
Task for the hour (1) Show that Stochastic Airy H � = � d 2 2 p β b 0 ( x ) dx 2 + x + (on R + with Dirichlet boundaries) can be made sensible. (2) Show the β -Hermite matrix H n , � , with g 1 g 2 p n β , p n β , . . . on diagonal and χ � ( n � 1) p n β , χ � ( n � 2) p n β , . . . on the o ff -diagonals satisfies n 2 / 3 (2 I � H n , � ) ! H � in some operator sense . (3) Payo ff s for other beta ensembles. Brian Rider (Temple University) Operator limits of random matrices 2 / 16
Return to the quadratic form Advertised that � TW � can be defined as the infimum of Z 1 Z 1 2 [( f 0 ) 2 ( x ) + xf 2 ( x )] dx + f 2 ( x ) db x h f , H � f i = p β 0 0 R 1 R 1 f 2 ( x ) = 1, 0 [( f 0 ) 2 ( x ) + xf 2 ( x )] dx < 1 (i.e., over f satisfying f (0) = 0, 0 f 2 L ). Brian Rider (Temple University) Operator limits of random matrices 3 / 16
Return to the quadratic form Advertised that � TW � can be defined as the infimum of Z 1 Z 1 2 [( f 0 ) 2 ( x ) + xf 2 ( x )] dx + f 2 ( x ) db x h f , H � f i = p β 0 0 R 1 R 1 f 2 ( x ) = 1, 0 [( f 0 ) 2 ( x ) + xf 2 ( x )] dx < 1 (i.e., over f satisfying f (0) = 0, 0 f 2 L ). To start need a lower bound. Rough idea is that it would be nice to replace b 0 x with “( ∆ b ) x ”, and you almost can. Brian Rider (Temple University) Operator limits of random matrices 3 / 16
Return to the quadratic form Advertised that � TW � can be defined as the infimum of Z 1 Z 1 2 [( f 0 ) 2 ( x ) + xf 2 ( x )] dx + f 2 ( x ) db x h f , H � f i = p β 0 0 R 1 R 1 f 2 ( x ) = 1, 0 [( f 0 ) 2 ( x ) + xf 2 ( x )] dx < 1 (i.e., over f satisfying f (0) = 0, 0 f 2 L ). To start need a lower bound. Rough idea is that it would be nice to replace b 0 x with “( ∆ b ) x ”, and you almost can. Decompose Z x +1 b x = ¯ b x + ( b x � ¯ ¯ b x ) , b x = b y dy x and then Z 1 Z 1 f 2 ( x )¯ f 0 ( x ) f ( x )(¯ h f , b 0 f i = b 0 x dx + 2 b x � b x ) dx . 0 0 and least for smooth compactly supported f . Brian Rider (Temple University) Operator limits of random matrices 3 / 16
Key inequality For any c > 0 there is an almost surely finite C ( c , b ) with Z 1 Z 1 Z 1 � � � f 2 ( x ) db x � [( f 0 ) 2 ( x ) + xf 2 ( x )] dx + C ( c , b ) f 2 ( x ) dx . � c � � � 0 0 0 Brian Rider (Temple University) Operator limits of random matrices 4 / 16
Key inequality For any c > 0 there is an almost surely finite C ( c , b ) with Z 1 Z 1 Z 1 � � � f 2 ( x ) db x � [( f 0 ) 2 ( x ) + xf 2 ( x )] dx + C ( c , b ) f 2 ( x ) dx . � c � � � 0 0 0 Recall from above, first for “nice” test functions, Z 1 Z 1 Z 1 f 2 ( x )¯ f 0 ( x ) f ( x )(¯ f 2 ( x ) db x = b 0 x dx + 2 b x � b x ) dx , 0 0 0 then note the relative slow growth of the running Brownian increment: Brian Rider (Temple University) Operator limits of random matrices 4 / 16
Key inequality For any c > 0 there is an almost surely finite C ( c , b ) with Z 1 Z 1 Z 1 � � � f 2 ( x ) db x � [( f 0 ) 2 ( x ) + xf 2 ( x )] dx + C ( c , b ) f 2 ( x ) dx . � c � � � 0 0 0 Recall from above, first for “nice” test functions, Z 1 Z 1 Z 1 f 2 ( x )¯ f 0 ( x ) f ( x )(¯ f 2 ( x ) db x = b 0 x dx + 2 b x � b x ) dx , 0 0 0 then note the relative slow growth of the running Brownian increment: Exercise : There is an C ( b ) < ∞ (almost surely) so that | b x + y − b x | sup x > 0 sup ≤ C ( b ) . p log(1 + x ) 0 < y 1 It follows that | ¯ x | and | ¯ b 0 b x − b x | are similarly bounded. (Just uses that b x has independent homogeneous increments, and a bound on P 0 (sup x < 1 | b x | > c )). Brian Rider (Temple University) Operator limits of random matrices 4 / 16
Existence of the groundstate Let’s introduce the natural norm on L : Z 1 k f k 2 [( f 0 ) 2 ( x ) + (1 + x ) f 2 ( x )] dx . ⇤ = 0 Then what we have shown can be summarized as: there are constants c (deterministc) and C , C 0 (random) such that for all f 2 L c k f k 2 ⇤ � C k f k 2 2 h f , H � f i C 0 k f k 2 ⇤ . Brian Rider (Temple University) Operator limits of random matrices 5 / 16
Existence of the groundstate Let’s introduce the natural norm on L : Z 1 k f k 2 [( f 0 ) 2 ( x ) + (1 + x ) f 2 ( x )] dx . ⇤ = 0 Then what we have shown can be summarized as: there are constants c (deterministc) and C , C 0 (random) such that for all f 2 L c k f k 2 ⇤ � C k f k 2 2 h f , H � f i C 0 k f k 2 ⇤ . Now argue the existence of an eigenvalue/eigenvector pair: Brian Rider (Temple University) Operator limits of random matrices 5 / 16
Existence of the groundstate Let’s introduce the natural norm on L : Z 1 k f k 2 [( f 0 ) 2 ( x ) + (1 + x ) f 2 ( x )] dx . ⇤ = 0 Then what we have shown can be summarized as: there are constants c (deterministc) and C , C 0 (random) such that for all f 2 L c k f k 2 ⇤ � C k f k 2 2 h f , H � f i C 0 k f k 2 ⇤ . Now argue the existence of an eigenvalue/eigenvector pair: Let f n 2 L be a minimizing sequence, h f n , H � f n i ! ˜ • Λ 0 Brian Rider (Temple University) Operator limits of random matrices 5 / 16
Existence of the groundstate Let’s introduce the natural norm on L : Z 1 k f k 2 [( f 0 ) 2 ( x ) + (1 + x ) f 2 ( x )] dx . ⇤ = 0 Then what we have shown can be summarized as: there are constants c (deterministc) and C , C 0 (random) such that for all f 2 L c k f k 2 ⇤ � C k f k 2 2 h f , H � f i C 0 k f k 2 ⇤ . Now argue the existence of an eigenvalue/eigenvector pair: Let f n 2 L be a minimizing sequence, h f n , H � f n i ! ˜ • Λ 0 • The (a.s.) uniform bound on k f n k ⇤ produces a subsequence f n 0 ! f 0 occuring: weakly in H 1 , uniformly on compacts, and in L 2 . Brian Rider (Temple University) Operator limits of random matrices 5 / 16
Existence of the groundstate Let’s introduce the natural norm on L : Z 1 k f k 2 [( f 0 ) 2 ( x ) + (1 + x ) f 2 ( x )] dx . ⇤ = 0 Then what we have shown can be summarized as: there are constants c (deterministc) and C , C 0 (random) such that for all f 2 L c k f k 2 ⇤ � C k f k 2 2 h f , H � f i C 0 k f k 2 ⇤ . Now argue the existence of an eigenvalue/eigenvector pair: Let f n 2 L be a minimizing sequence, h f n , H � f n i ! ˜ • Λ 0 • The (a.s.) uniform bound on k f n k ⇤ produces a subsequence f n 0 ! f 0 occuring: weakly in H 1 , uniformly on compacts, and in L 2 . From here can conclude h f 0 , H � f 0 i = ˜ Λ 0 . (And ˜ • Λ 0 = Λ 0 = � TW � .) Brian Rider (Temple University) Operator limits of random matrices 5 / 16
Higher eigenvalues and more We can now define Λ 1 < Λ 2 < · · · by Rayleigh-Ritz, for example ˜ Λ 1 := f 2 L , f ? f 0 h f , H � f i . inf The same type of argument will show a pair (˜ Λ 1 , f 1 ) exists. Then can check it is an eigenvalue/eigenvector (and announce the former = Λ 1 ). Brian Rider (Temple University) Operator limits of random matrices 6 / 16
Higher eigenvalues and more We can now define Λ 1 < Λ 2 < · · · by Rayleigh-Ritz, for example ˜ Λ 1 := f 2 L , f ? f 0 h f , H � f i . inf The same type of argument will show a pair (˜ Λ 1 , f 1 ) exists. Then can check it is an eigenvalue/eigenvector (and announce the former = Λ 1 ). A couple cute points. With A = � d 2 dx 2 + x the usual Airy operator what we have can yield.. Brian Rider (Temple University) Operator limits of random matrices 6 / 16
Higher eigenvalues and more We can now define Λ 1 < Λ 2 < · · · by Rayleigh-Ritz, for example ˜ Λ 1 := f 2 L , f ? f 0 h f , H � f i . inf The same type of argument will show a pair (˜ Λ 1 , f 1 ) exists. Then can check it is an eigenvalue/eigenvector (and announce the former = Λ 1 ). A couple cute points. With A = � d 2 dx 2 + x the usual Airy operator what we have can yield.. Exercise : For any ✏ > 0 there is a random C so that − CI + (1 − ✏ ) A ≤ H β ≤ (1 + ✏ ) A + CI in the sense of operators (quadratic forms). Brian Rider (Temple University) Operator limits of random matrices 6 / 16
Higher eigenvalues and more We can now define Λ 1 < Λ 2 < · · · by Rayleigh-Ritz, for example ˜ Λ 1 := f 2 L , f ? f 0 h f , H � f i . inf The same type of argument will show a pair (˜ Λ 1 , f 1 ) exists. Then can check it is an eigenvalue/eigenvector (and announce the former = Λ 1 ). A couple cute points. With A = � d 2 dx 2 + x the usual Airy operator what we have can yield.. Exercise : For any ✏ > 0 there is a random C so that − CI + (1 − ✏ ) A ≤ H β ≤ (1 + ✏ ) A + CI in the sense of operators (quadratic forms). 2 ⇡ k ) 2 / 3 + o (1), show that Exercise : Granted the classical asymptotics � k ( A ) = ( 3 k � 2 / 3 Λ k → (3 2 ⇡ ) 2 / 3 with probability one. Brian Rider (Temple University) Operator limits of random matrices 6 / 16
Convergence proof: setup 1 1 Bring back the matrix model H n , � with p � n g k and p � n χ � ( n � k ) on the diagonals/o ff diagonals. No controversy to declare: k v k =1 h v , ˆ H n , � = n 2 / 3 (2 I � H n , � ) . ˆ TW � ( n ) := min H n , � v i , Brian Rider (Temple University) Operator limits of random matrices 7 / 16
Convergence proof: setup 1 1 Bring back the matrix model H n , � with p � n g k and p � n χ � ( n � k ) on the diagonals/o ff diagonals. No controversy to declare: k v k =1 h v , ˆ H n , � = n 2 / 3 (2 I � H n , � ) . ˆ TW � ( n ) := min H n , � v i , Now write: n n ( v k +1 � v k ) 2 + h v , ˆ H n , � v i = n 2 / 3 X X η n , k v k v k +1 k =0 k =0 n X y (1) k + y (2) n , k v 2 2 + n , k v k v k +1 p β k =0 in which v 0 = v n +1 = 0 and n , k = � 1 y (1) p β n 1 / 6 ( p 2 n 1 / 6 g k 2 η n , k = β n � E χ � ( n � k ) ) , and y (2) n , k a centered/scaled χ � ( n � k ) . Brian Rider (Temple University) Operator limits of random matrices 7 / 16
Convergence proof: improved heuristics Want to show k v k =1 h v , ˆ min H n , � v i ! inf f 2 L h f , H � f i . Brian Rider (Temple University) Operator limits of random matrices 8 / 16
Convergence proof: improved heuristics Want to show k v k =1 h v , ˆ min H n , � v i ! inf f 2 L h f , H � f i . Embed the discrete minimization problem in L 2 : any v 2 R n is identified with a piecewise constant f v ( x ) = v ( d n 1 / 3 x e ) for x 2 [0 , d n 2 / 3 e ], f v = 0 otherwise. Brian Rider (Temple University) Operator limits of random matrices 8 / 16
Convergence proof: improved heuristics Want to show k v k =1 h v , ˆ min H n , � v i ! inf f 2 L h f , H � f i . Embed the discrete minimization problem in L 2 : any v 2 R n is identified with a piecewise constant f v ( x ) = v ( d n 1 / 3 x e ) for x 2 [0 , d n 2 / 3 e ], f v = 0 otherwise. With this point of view better to consider n n ( v k +1 � v k ) 2 + n � 1 / 3 h v , ˆ H n , � v i = n 1 / 3 X X η n , k v k v k +1 k =0 k =0 n X y (1) k + y (2) p β n � 1 / 3 n , k v 2 2 + n , k v k v k +1 k =0 A calculation shows: d n 1 / 3 x e d n 1 / 3 x e η n , k ! x 2 ( y (1) n , k + y (2) X X n � 1 / 3 p β n � 1 / 3 2 n , k ) ) 2 p β b x . 2 , k =1 k =1 Brian Rider (Temple University) Operator limits of random matrices 8 / 16
Convergence proof: An actual estimate Need to show the discrete quadratic form is bounded below, as n ! 1 . Brian Rider (Temple University) Operator limits of random matrices 9 / 16
Convergence proof: An actual estimate Need to show the discrete quadratic form is bounded below, as n ! 1 . Very much as in the proof that Stochastic Airy is well defined: show the noise part of the form can be controlled by deterministic part: e.g., for any c > 0, � � n n � � X y (1) X � n � 1 / 3 n , k v 2 v 2 k n � 1 / 3 � c k v k n , ⇤ + C n � � k � � k =0 k =1 where C n = C n ( y (1) , c ) is a tight random sequence and n n n 1 / 3 ( v k +1 � v k ) 2 + k v k 2 X X kn � 2 / 3 v 2 n , ⇤ = k k =0 k =0 is the analog of our k · k 2 ⇤ norm from before. And similarly for the y (2) noise term. Brian Rider (Temple University) Operator limits of random matrices 9 / 16
Convergence proof: Now what? The bound just described gives (also similar to the continuum): ` 2 h v , ˆ c k v k 2 n , ⇤ � C n k v k 2 H n , � v i C 0 n k v k 2 n , ⇤ for tight C n and C 0 n . Brian Rider (Temple University) Operator limits of random matrices 10 / 16
Convergence proof: Now what? The bound just described gives (also similar to the continuum): ` 2 h v , ˆ c k v k 2 n , ⇤ � C n k v k 2 H n , � v i C 0 n k v k 2 n , ⇤ for tight C n and C 0 n . • Can select a subsequence of eigenvalue and (normalized) eigenvectors ( λ 0 ( n 0 ) , v n 0 ) such that you have the convergence v n 0 ! f ⇤ 2 L 2 \ H 1 . λ 0 ( n 0 ) ! λ ⇤ , Brian Rider (Temple University) Operator limits of random matrices 10 / 16
Convergence proof: Now what? The bound just described gives (also similar to the continuum): ` 2 h v , ˆ c k v k 2 n , ⇤ � C n k v k 2 H n , � v i C 0 n k v k 2 n , ⇤ for tight C n and C 0 n . • Can select a subsequence of eigenvalue and (normalized) eigenvectors ( λ 0 ( n 0 ) , v n 0 ) such that you have the convergence v n 0 ! f ⇤ 2 L 2 \ H 1 . λ 0 ( n 0 ) ! λ ⇤ , • In fact will have λ 0 ( n 0 ) = h v n 0 , ˆ H n 0 , � v n 0 i ! h f ⇤ , H � f ⇤ i Brian Rider (Temple University) Operator limits of random matrices 10 / 16
Convergence proof: Now what? The bound just described gives (also similar to the continuum): ` 2 h v , ˆ c k v k 2 n , ⇤ � C n k v k 2 H n , � v i C 0 n k v k 2 n , ⇤ for tight C n and C 0 n . • Can select a subsequence of eigenvalue and (normalized) eigenvectors ( λ 0 ( n 0 ) , v n 0 ) such that you have the convergence v n 0 ! f ⇤ 2 L 2 \ H 1 . λ 0 ( n 0 ) ! λ ⇤ , • In fact will have λ 0 ( n 0 ) = h v n 0 , ˆ H n 0 , � v n 0 i ! h f ⇤ , H � f ⇤ i • Gives at least λ ⇤ = h f ⇤ , H � f ⇤ i � Λ 0 = � TW � for any such limit point... (and that limit point is an eigenvalue of H � ...) Brian Rider (Temple University) Operator limits of random matrices 10 / 16
Other ensembles: Wishart matrices These are the random matrices of form MM † for M = n ⇥ m with iid entries. Brian Rider (Temple University) Operator limits of random matrices 11 / 16
Other ensembles: Wishart matrices These are the random matrices of form MM † for M = n ⇥ m with iid entries. In the real, complex, quaternion Gaussian cases the eigenvalue laws are again determinantal or Pfa ffi an processes. Well known see Tracy-Widom fluctuations for the largest eigenvalues (work of Johnstone, Johansson...) Brian Rider (Temple University) Operator limits of random matrices 11 / 16
Other ensembles: Wishart matrices These are the random matrices of form MM † for M = n ⇥ m with iid entries. In the real, complex, quaternion Gaussian cases the eigenvalue laws are again determinantal or Pfa ffi an processes. Well known see Tracy-Widom fluctuations for the largest eigenvalues (work of Johnstone, Johansson...) The appropriate general beta version is to take the density on n positive points with joint density: for β > 0 and κ > n � 1 n β | λ j � λ k | � ⇥ 2 ( � n +1) � 1 e � β Y Y P � 2 n � k . n , ( λ 1 , . . . , λ n ) / λ k j < k k =1 (When β = 1 , 2 , 4 and κ = m 2 Z this realizes the MM † real, complex, or quaternion Gaussian Wishart ensemble.) Brian Rider (Temple University) Operator limits of random matrices 11 / 16
β -Laguerre There is again a tridiagonal matrix model, due to Dumitriu-Edelman. Let B = B n , � , be the random upper bidiagonal 2 3 χ � χ � ( n � 1) χ � ( � 1) χ � ( n � 2) 6 7 1 6 7 ... B = p β n 6 7 , 6 7 6 7 χ � ( � n +2) χ � 4 5 χ � ( � n +1) with all variables independent. Then the eigenvalues of W = BB † have joint density P � n , . Brian Rider (Temple University) Operator limits of random matrices 12 / 16
β -Laguerre There is again a tridiagonal matrix model, due to Dumitriu-Edelman. Let B = B n , � , be the random upper bidiagonal 2 3 χ � χ � ( n � 1) χ � ( � 1) χ � ( n � 2) 6 7 1 6 7 ... B = p β n 6 7 , 6 7 6 7 χ � ( � n +2) χ � 4 5 χ � ( � n +1) with all variables independent. Then the eigenvalues of W = BB † have joint density P � n , . Exercise: For M an m × n matrix of independent real/complex Gaussians show there are U and V unitary with UMV = the advertised B . Brian Rider (Temple University) Operator limits of random matrices 12 / 16
Tracy-Widom( β ) for β -Laguerre Previous procedure gives: Theorem (Ram´ ırez, R, Vir´ ag) Let λ 1 � λ 2 . . . denote the ordered β -Laguerre eigenvalues and set ( p n κ ) 1 / 3 µ n , = ( p n + p κ ) 2 , and σ n , = ( p n + p κ ) 4 / 3 . Then for any k , as n ! 1 with arbitrary κ = κ n > n � 1 we have ⇣ ⌘ ⇣ ⌘ σ n , ( µ n , � λ ` ) ` =1 ,..., k ) Λ 0 , Λ 1 , . . . , Λ k � 1 , the ordered eigenvalues for Stochastic Airy. Brian Rider (Temple University) Operator limits of random matrices 13 / 16
An application: Spikes Johnstone raised the question: What happens to Tracy-Widom for non-null Wishart ensembles? Or, what is λ max for M Σ M † for “general” Σ 6 = I ? Even in the “spiked” case: Σ = Σ r � I n � r , for Σ r = diag( c 1 , . . . , c r ). In 2005 Baik, Ben Arous, and P´ ech´ e, found a phase transition (in the complex case), here for r = 1: Brian Rider (Temple University) Operator limits of random matrices 14 / 16
An application: Spikes Johnstone raised the question: What happens to Tracy-Widom for non-null Wishart ensembles? Or, what is λ max for M Σ M † for “general” Σ 6 = I ? Even in the “spiked” case: Σ = Σ r � I n � r , for Σ r = diag( c 1 , . . . , c r ). In 2005 Baik, Ben Arous, and P´ ech´ e, found a phase transition (in the complex case), here for r = 1: ⇣ ⌘ If c < c : P σ n ( λ max � µ n ) t ! F 2 ( t ) . Brian Rider (Temple University) Operator limits of random matrices 14 / 16
An application: Spikes Johnstone raised the question: What happens to Tracy-Widom for non-null Wishart ensembles? Or, what is λ max for M Σ M † for “general” Σ 6 = I ? Even in the “spiked” case: Σ = Σ r � I n � r , for Σ r = diag( c 1 , . . . , c r ). In 2005 Baik, Ben Arous, and P´ ech´ e, found a phase transition (in the complex case), here for r = 1: ⇣ ⌘ If c < c : P σ n ( λ max � µ n ) t ! F 2 ( t ) . R t ⇣ ⌘ �1 e � x 2 / 2 dx σ 0 n ( λ max � µ 0 If c > c : P n ) t ! p 2 ⇡ . Brian Rider (Temple University) Operator limits of random matrices 14 / 16
An application: Spikes Johnstone raised the question: What happens to Tracy-Widom for non-null Wishart ensembles? Or, what is λ max for M Σ M † for “general” Σ 6 = I ? Even in the “spiked” case: Σ = Σ r � I n � r , for Σ r = diag( c 1 , . . . , c r ). In 2005 Baik, Ben Arous, and P´ ech´ e, found a phase transition (in the complex case), here for r = 1: ⇣ ⌘ If c < c : P σ n ( λ max � µ n ) t ! F 2 ( t ) . R t ⇣ ⌘ �1 e � x 2 / 2 dx σ 0 n ( λ max � µ 0 If c > c : P n ) t ! p 2 ⇡ . ⇣ ⌘ If c = c � wn � 1 / 3 : P σ n ( λ max � µ n ) t ! F ( t , w ) = F 2 ( t ) f ( t , w ) where f can again be described in terms of Painlev´ e II. Brian Rider (Temple University) Operator limits of random matrices 14 / 16
An application: Spikes Johnstone raised the question: What happens to Tracy-Widom for non-null Wishart ensembles? Or, what is λ max for M Σ M † for “general” Σ 6 = I ? Even in the “spiked” case: Σ = Σ r � I n � r , for Σ r = diag( c 1 , . . . , c r ). In 2005 Baik, Ben Arous, and P´ ech´ e, found a phase transition (in the complex case), here for r = 1: ⇣ ⌘ If c < c : P σ n ( λ max � µ n ) t ! F 2 ( t ) . R t ⇣ ⌘ �1 e � x 2 / 2 dx σ 0 n ( λ max � µ 0 If c > c : P n ) t ! p 2 ⇡ . ⇣ ⌘ If c = c � wn � 1 / 3 : P σ n ( λ max � µ n ) t ! F ( t , w ) = F 2 ( t ) f ( t , w ) where f can again be described in terms of Painlev´ e II. That β = 2 is absolutely critical to the analysis. Brian Rider (Temple University) Operator limits of random matrices 14 / 16
Spiked beta ensemble Can still tri-diagonalize. Get the same product of random bidiagonal B matrices, but with a multiplicative shift by p c in the (1 , 1) entry. ( Exercise ?) Brian Rider (Temple University) Operator limits of random matrices 15 / 16
Spiked beta ensemble Can still tri-diagonalize. Get the same product of random bidiagonal B matrices, but with a multiplicative shift by p c in the (1 , 1) entry. ( Exercise ?) Theorem (Bloemendal-Vir´ ag) At criticality, the appropriately scaled B c B † c with c = c � wn � 1 / 3 , converges in the now familiar operator sense to H � = � d 2 2 p β b 0 ( x ) , dx 2 + x + but subject now to f 0 (0) = wf (0) at the origin. Brian Rider (Temple University) Operator limits of random matrices 15 / 16
Spiked beta ensemble Can still tri-diagonalize. Get the same product of random bidiagonal B matrices, but with a multiplicative shift by p c in the (1 , 1) entry. ( Exercise ?) Theorem (Bloemendal-Vir´ ag) At criticality, the appropriately scaled B c B † c with c = c � wn � 1 / 3 , converges in the now familiar operator sense to H � = � d 2 2 p β b 0 ( x ) , dx 2 + x + but subject now to f 0 (0) = wf (0) at the origin. So have a “general beta spiked” Tracy-Widom law TW � , w , with TW � = TW � , 1 Brian Rider (Temple University) Operator limits of random matrices 15 / 16
PDE for TW β , w distributions Can again use the Riccati trick. The Robin boundary condition means that any x 7! ψ ( x , λ ) satisfying H � ψ = λψ is subject to ( ψ (0 , λ ) , ψ 0 (0 , λ )) = (1 , w ), or p (0 , λ ) = 0 (0 , � ) (0 , � ) = w . Brian Rider (Temple University) Operator limits of random matrices 16 / 16
PDE for TW β , w distributions Can again use the Riccati trick. The Robin boundary condition means that any x 7! ψ ( x , λ ) satisfying H � ψ = λψ is subject to ( ψ (0 , λ ) , ψ 0 (0 , λ )) = (1 , w ), or p (0 , λ ) = 0 (0 , � ) (0 , � ) = w . The upshot is: P ( TW � , w λ ) = P � , w ( p never explodes) , p β db t + ( t � p 2 2 with p our friend from before: dp t = t ) dt , now begun at place w at time λ . Brian Rider (Temple University) Operator limits of random matrices 16 / 16
PDE for TW β , w distributions Can again use the Riccati trick. The Robin boundary condition means that any x 7! ψ ( x , λ ) satisfying H � ψ = λψ is subject to ( ψ (0 , λ ) , ψ 0 (0 , λ )) = (1 , w ), or p (0 , λ ) = 0 (0 , � ) (0 , � ) = w . The upshot is: P ( TW � , w λ ) = P � , w ( p never explodes) , p β db t + ( t � p 2 2 with p our friend from before: dp t = t ) dt , now begun at place w at time λ . Now view F ( λ , w ) = F � ( λ , w ) = P ( TW � , w λ ) as a hitting distribution for the “space-time” Markov process ( p t , t ). By general theory any such function is killed by the generator : ∂ 2 F ∂ F ∂λ + 2 ∂ 2 w + ( λ � w 2 ) ∂ F ∂ w = 0 . β This PDE has been used by Rumanov to find the first Painlev´ e formulas for TW � outside of β = 1 , 2 , 4 - for β = 6! Brian Rider (Temple University) Operator limits of random matrices 16 / 16
Operator limits of random matrices III. Hard edge Brian Rider Temple University Brian Rider (Temple University) Operator limits of random matrices 1 / 17
Back to complex Wishart Have n ⇥ m matrices M of independent complex Gaussians and form the appropriately scaled 1 n MM † . Brian Rider (Temple University) Operator limits of random matrices 2 / 17
Back to complex Wishart Have n ⇥ m matrices M of independent complex Gaussians and form the appropriately scaled 1 n MM † . The Marchenko-Pastur law replaces the semi-circle: if say m n ! � � 1, n 1 ( � � ` )( r � � ) d � X p � λ k ( � ) ! 2 ⇡� n k =1 where ` = (1 � p � ) 2 and r = (1 + p � ) 2 Brian Rider (Temple University) Operator limits of random matrices 2 / 17
Back to complex Wishart Have n ⇥ m matrices M of independent complex Gaussians and form the appropriately scaled 1 n MM † . The Marchenko-Pastur law replaces the semi-circle: if say m n ! � � 1, n 1 ( � � ` )( r � � ) d � X p � λ k ( � ) ! 2 ⇡� n k =1 where ` = (1 � p � ) 2 and r = (1 + p � ) 2 When � > 1 both edges are “soft”, and see Tracy-Widom fluctuations. Brian Rider (Temple University) Operator limits of random matrices 2 / 17
Back to complex Wishart Have n ⇥ m matrices M of independent complex Gaussians and form the appropriately scaled 1 n MM † . The Marchenko-Pastur law replaces the semi-circle: if say m n ! � � 1, n 1 ( � � ` )( r � � ) d � X p � λ k ( � ) ! 2 ⇡� n k =1 where ` = (1 � p � ) 2 and r = (1 + p � ) 2 When � > 1 both edges are “soft”, and see Tracy-Widom fluctuations. When � = 1, then ` = 0 and eigenvalues feel the “hard edge” of the origin. Brian Rider (Temple University) Operator limits of random matrices 2 / 17
Back to complex Wishart Have n ⇥ m matrices M of independent complex Gaussians and form the appropriately scaled 1 n MM † . The Marchenko-Pastur law replaces the semi-circle: if say m n ! � � 1, n 1 ( � � ` )( r � � ) d � X p � λ k ( � ) ! 2 ⇡� n k =1 where ` = (1 � p � ) 2 and r = (1 + p � ) 2 When � > 1 both edges are “soft”, and see Tracy-Widom fluctuations. When � = 1, then ` = 0 and eigenvalues feel the “hard edge” of the origin. In fact, if m = n + a as n " 1 there is a one-parameter family of limit laws for � min (also due Tracy-Widom). Brian Rider (Temple University) Operator limits of random matrices 2 / 17
Hard edge kernel/process Using the determinantal structure: with m � n ⌘ a as n ! 1 it holds, ⇣ ⌘ n 2 � min � t P ! det L 2 [0 , t ) ( I � K Bessel ) where K Bessel ( x , y ) = J a ( p x ) p yJ 0 a ( p y ) � J a ( p y ) p xJ 0 a ( p x ) , x � y and J a is the Bessel function of first kind. (The Fredholm determinant itself can be expressed in terms of Painlev´ e V). Brian Rider (Temple University) Operator limits of random matrices 3 / 17
Hard edge kernel/process Using the determinantal structure: with m � n ⌘ a as n ! 1 it holds, ⇣ ⌘ n 2 � min � t P ! det L 2 [0 , t ) ( I � K Bessel ) where K Bessel ( x , y ) = J a ( p x ) p yJ 0 a ( p y ) � J a ( p y ) p xJ 0 a ( p x ) , x � y and J a is the Bessel function of first kind. (The Fredholm determinant itself can be expressed in terms of Painlev´ e V). Defines the “hard-edge” process for each a . Brian Rider (Temple University) Operator limits of random matrices 3 / 17
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