The Ginibre ensemble GinUE ( N ) is the space Mat N ( C ) circular - PowerPoint PPT Presentation

Resolvent Behaviour of R -diagonal operators Todd Kemp CLE Moore Instructor MIT Banf and only Banf January 17, 2008 1 / 11

The Ginibre Ensemble • GinUE ( N ) The Ginibre ensemble GinUE ( N ) is the space Mat N ( C ) • circular equipped with the probability measure • R -diagonal • Properties α N e − X ∗ X dX. • Haagerup Ineq. • ∗ -Pairings • Resolvent • Blow-Up Alternatively, it is the set of N × N random matrices whose entries • Moments are all i.i.d. complex normals (Re,Im i.i.d. N (0 , 1) ). • References 2 / 11

The Ginibre Ensemble • GinUE ( N ) The Ginibre ensemble GinUE ( N ) is the space Mat N ( C ) • circular equipped with the probability measure • R -diagonal • Properties α N e − X ∗ X dX. • Haagerup Ineq. • ∗ -Pairings • Resolvent • Blow-Up Alternatively, it is the set of N × N random matrices whose entries • Moments are all i.i.d. complex normals (Re,Im i.i.d. N (0 , 1) ). Renormalize it • References √ by 1 / 2 N , and . . . Matlab code : X=randn(4000); Y=randn(4000) ; C=(X+iY)/sqrt(8000) ; E=eig(C) ; plot(E,’b.’); 2 / 11

The Ginibre Ensemble • GinUE ( N ) The Ginibre ensemble GinUE ( N ) is the space Mat N ( C ) • circular equipped with the probability measure • R -diagonal • Properties α N e − X ∗ X dX. • Haagerup Ineq. • ∗ -Pairings • Resolvent • Blow-Up Alternatively, it is the set of N × N random matrices whose entries • Moments are all i.i.d. complex normals (Re,Im i.i.d. N (0 , 1) ). Renormalize it • References √ by 1 / 2 N , and . . . Matlab code : 1 X=randn(4000); 0.5 Y=randn(4000) ; C=(X+iY)/sqrt(8000) ; 0 E=eig(C) ; −0.5 plot(E,’b.’); −1 −1 −0.5 0 0.5 1 2 / 11

Voiculescu’s Circular Operator The circular operator c is the limit (in the sense of free probability) of • GinUE ( N ) • circular the renormalized GinUE ( N ) as N → ∞ . • R -diagonal • Properties 1 It can also be realized as c = 2 ( s 1 + is 2 ) , where s 1 , s 2 are free √ • Haagerup Ineq. • ∗ -Pairings semicircular operators. • Resolvent • Blow-Up • Moments • References 3 / 11

Voiculescu’s Circular Operator The circular operator c is the limit (in the sense of free probability) of • GinUE ( N ) • circular the renormalized GinUE ( N ) as N → ∞ . • R -diagonal • Properties 1 It can also be realized as c = 2 ( s 1 + is 2 ) , where s 1 , s 2 are free √ • Haagerup Ineq. • ∗ -Pairings semicircular operators. • Resolvent • Blow-Up It is my favourite operator. • Moments • References 3 / 11

Voiculescu’s Circular Operator The circular operator c is the limit (in the sense of free probability) of • GinUE ( N ) • circular the renormalized GinUE ( N ) as N → ∞ . • R -diagonal • Properties 1 It can also be realized as c = 2 ( s 1 + is 2 ) , where s 1 , s 2 are free √ • Haagerup Ineq. • ∗ -Pairings semicircular operators. • Resolvent • Blow-Up It is my favourite operator. • Moments • References From a combinatorial standpoint, it is characterized by its extremely simple free cumulants: among all cumulants in c, c ∗ , the only non-zero ones are κ 2 [ c, c ∗ ] = κ 2 [ c ∗ , c ] = 1 . 3 / 11

Voiculescu’s Circular Operator The circular operator c is the limit (in the sense of free probability) of • GinUE ( N ) • circular the renormalized GinUE ( N ) as N → ∞ . • R -diagonal • Properties 1 It can also be realized as c = 2 ( s 1 + is 2 ) , where s 1 , s 2 are free √ • Haagerup Ineq. • ∗ -Pairings semicircular operators. • Resolvent • Blow-Up It is my favourite operator. • Moments • References From a combinatorial standpoint, it is characterized by its extremely simple free cumulants: among all cumulants in c, c ∗ , the only non-zero ones are κ 2 [ c, c ∗ ] = κ 2 [ c ∗ , c ] = 1 . Quick advertisement : pick up Lectures on the Combinatorics of Free Probability by A. Nica and R. Speicher for everything you need to know about free cumulants. 3 / 11

R -Diagonal Operators Let u be a Haar unitary operator (renormalized limit of Haar unitary • GinUE ( N ) • circular ensemble). As Jamie Mingo showed us on Monday, the only • R -diagonal non-zero free cumulants of u, u ∗ are of the form • Properties • Haagerup Ineq. • ∗ -Pairings κ 2 n [ u, u ∗ , . . . , u, u ∗ ] = κ 2 n [ u ∗ , u, . . . , u ∗ , u ] = ( − 1) n C n − 1 . • Resolvent • Blow-Up • Moments • References 4 / 11

R -Diagonal Operators Let u be a Haar unitary operator (renormalized limit of Haar unitary • GinUE ( N ) • circular ensemble). As Jamie Mingo showed us on Monday, the only • R -diagonal non-zero free cumulants of u, u ∗ are of the form • Properties • Haagerup Ineq. • ∗ -Pairings κ 2 n [ u, u ∗ , . . . , u, u ∗ ] = κ 2 n [ u ∗ , u, . . . , u ∗ , u ] = ( − 1) n C n − 1 . • Resolvent • Blow-Up • Moments Definition. a is R -diagonal if its only non-zero free cumulants are • References of the forms κ 2 n [ a, a ∗ , . . . , a, a ∗ ] κ 2 n [ a ∗ , a, . . . , a ∗ , a ] . Alternate characterization. a is R -diagonal if, given u Haar unitary ∗ -free from a , ua ∼ a. 4 / 11

Properties of R -Diagonal Operators Matrix models: ensembles of the form α N e − V ( X ∗ X ) dX . • GinUE ( N ) • • circular • R -diagonal If a, b are R -diagonal and ∗ -free, then a + b and a n are • Properties • • Haagerup Ineq. R -diagonal; if x is anything ∗ -free from a , ax is R -diagonal. • ∗ -Pairings • Resolvent • Blow-Up • Never normal except scalar multiples of Haar unitaries. • Moments • References Brown measure of a can be computed explicitly from the • S -transform of a ∗ a ; rotationally-invariant, analytic density. • Have continuous families of invariant subspaces. Maximize free entropy ( χ and χ ∗ ) under distribution constraints. • • Satisfy a strong Haagerup inequality. 5 / 11

Strong Haagerup Inequality Theorem. Let a 1 , . . . , a d be ∗ -free R -diagonal operators. If T is • GinUE ( N ) • circular spanned by words of length n in a 1 , . . . , a d (and not a ∗ 1 , . . . , a ∗ d ), • R -diagonal • Properties then � T � ≤ α √ n � T � 2 , • Haagerup Ineq. • ∗ -Pairings • Resolvent where α is a constant depending on sup( � a j � / � a j � 2 ) . • Blow-Up • Moments • References 6 / 11

Strong Haagerup Inequality Theorem. Let a 1 , . . . , a d be ∗ -free R -diagonal operators. If T is • GinUE ( N ) • circular spanned by words of length n in a 1 , . . . , a d (and not a ∗ 1 , . . . , a ∗ d ), • R -diagonal • Properties then � T � ≤ α √ n � T � 2 , • Haagerup Ineq. • ∗ -Pairings • Resolvent where α is a constant depending on sup( � a j � / � a j � 2 ) . • Blow-Up • Moments • References E.g. for Haar unitaries u 1 , . . . , u d , this is a statement about the free group: if f : F d → C is supported on words of length n in the generators (and not their inverses), then √ � f � ∗ ≤ √ e n + 1 � f � 2 . (If the inverses are included, the constant is ( n + 1) ; this is the classical Haagerup inequality.) 6 / 11

Non-Crossing ∗ -Pairings E.g. with n = 3 , r = 4 : u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u 1 u 5 u 2 u 3 u 2 u 1 u 1 u 3 u 3 u 4 u 5 u 4 2 3 3 1 2 1 4 5 4 3 5 1 7 / 11

Non-Crossing ∗ -Pairings E.g. with n = 3 , r = 4 : u − 1 u − 1 3 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u 1 u 5 u 2 u 3 u 2 u 1 u 1 u 3 u 3 u 4 u 5 u 4 2 3 1 2 1 4 5 4 3 5 1 7 / 11

Non-Crossing ∗ -Pairings E.g. with n = 3 , r = 4 : u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u 1 u 5 u 2 u 3 u 2 u 1 1 u 1 u 3 u 3 u 4 u 5 u 4 2 3 3 1 2 4 5 4 3 5 1 7 / 11

Non-Crossing ∗ -Pairings E.g. with n = 3 , r = 4 : u − 1 u − 1 3 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u 1 u 5 u 2 u 3 u 2 u 1 1 u 1 u 3 u 3 u 4 u 5 u 4 2 3 1 2 4 5 4 3 5 1 7 / 11

Non-Crossing ∗ -Pairings E.g. with n = 3 , r = 4 : u − 1 u − 1 3 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u 1 u 5 u 2 u 3 u 2 u 1 1 u 1 u 3 u 3 u 4 u 5 u 4 2 3 1 2 4 5 4 3 5 1 7 / 11

Non-Crossing ∗ -Pairings E.g. with n = 3 , r = 4 : u − 1 u − 1 3 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u 1 u 5 u 2 u 3 u 2 u 1 1 u 1 u 3 u 3 u 4 u 5 u 4 2 3 1 2 4 5 4 3 5 1 7 / 11

Non-Crossing ∗ -Pairings E.g. with n = 3 , r = 4 : u − 1 u − 1 3 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u 1 u 5 u 2 u 3 u 2 u 1 1 u 1 u 3 u 3 u 4 u 5 u 4 2 3 1 2 4 5 4 3 5 1 7 / 11

Non-Crossing ∗ -Pairings E.g. with n = 3 , r = 4 : u − 1 u − 1 3 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u − 1 u 1 u 5 u 2 u 3 u 2 u 1 1 u 1 u 3 u 3 u 4 u 5 u 4 2 3 1 2 4 5 4 3 5 1 We get a non-crossing pairing π . 7 / 11