Properties of Brown Measure • Dedication The Brown measure has some nice properties analogous to the • Citations spectral measure, but not all: Brown Measure � � • Brown Measure z k µ a ( dzd ¯ z k µ a ( dzd ¯ τ ( a k ) = and τ ( a ∗ k ) = z ) ¯ z ) • • Circular • Properties C C but you cannot max and match . • Convergence • Regularize � • Spectrum τ (log | a − λ | ) = L ( a − λ ) = log | z − λ | µ a ( dzd ¯ z ) for • • L p Inverse • L p Spectrum C large λ , and this characterizes µ a . In particular, the ∗ -distribution • Support of a determines µ a – but with a log discontinuity. Brownian Motion Segal–Bargmann supp µ a ⊆ Spec( a ) • (can be a strict subset). Brown Measure Support Let A N be a sequence of matrices with a as limit in ∗ -distribution. Since the Brown measure µ A N is the empirical spectral distribution of A N , it is natural to expect that ESD( A N ) → µ a . 7 / 35
Properties of Brown Measure • Dedication The Brown measure has some nice properties analogous to the • Citations spectral measure, but not all: Brown Measure � � • Brown Measure z k µ a ( dzd ¯ z k µ a ( dzd ¯ τ ( a k ) = and τ ( a ∗ k ) = z ) ¯ z ) • • Circular • Properties C C but you cannot max and match . • Convergence • Regularize � • Spectrum τ (log | a − λ | ) = L ( a − λ ) = log | z − λ | µ a ( dzd ¯ z ) for • • L p Inverse • L p Spectrum C large λ , and this characterizes µ a . In particular, the ∗ -distribution • Support of a determines µ a – but with a log discontinuity. Brownian Motion Segal–Bargmann supp µ a ⊆ Spec( a ) • (can be a strict subset). Brown Measure Support Let A N be a sequence of matrices with a as limit in ∗ -distribution. Since the Brown measure µ A N is the empirical spectral distribution of A N , it is natural to expect that ESD( A N ) → µ a . The log discontinuity often makes this exceedingly difficult to prove. 7 / 35
Convergence of the Brown Measure • Dedication Let { a, a n } n ∈ N be a uniformly bounded set of operators in some • Citations W ∗ -probability spaces, with a n → a in ∗ -distribution. We would Brown Measure hope that µ a n → µ a . Without some very fine information about the • Brown Measure • Circular spectral measure of | a n − λ | near the edge of Spec( a n ) , the best • Properties • Convergence that can be said in general is the following. • Regularize • Spectrum • L p Inverse • L p Spectrum • Support Brownian Motion Segal–Bargmann Brown Measure Support 8 / 35
Convergence of the Brown Measure • Dedication Let { a, a n } n ∈ N be a uniformly bounded set of operators in some • Citations W ∗ -probability spaces, with a n → a in ∗ -distribution. We would Brown Measure hope that µ a n → µ a . Without some very fine information about the • Brown Measure • Circular spectral measure of | a n − λ | near the edge of Spec( a n ) , the best • Properties • Convergence that can be said in general is the following. • Regularize • Spectrum • L p Inverse Proposition. Suppose that µ a n → µ weakly for some probability • L p Spectrum measure µ on C . Then • Support Brownian Motion � � log | z − λ | µ ( dzd ¯ z ) ≤ log | z − λ | µ a ( dzd ¯ z ) Segal–Bargmann C C Brown Measure Support for all λ ∈ C ; and equality holds for sufficiently large λ . 8 / 35
Convergence of the Brown Measure • Dedication Let { a, a n } n ∈ N be a uniformly bounded set of operators in some • Citations W ∗ -probability spaces, with a n → a in ∗ -distribution. We would Brown Measure hope that µ a n → µ a . Without some very fine information about the • Brown Measure • Circular spectral measure of | a n − λ | near the edge of Spec( a n ) , the best • Properties • Convergence that can be said in general is the following. • Regularize • Spectrum • L p Inverse Proposition. Suppose that µ a n → µ weakly for some probability • L p Spectrum measure µ on C . Then • Support Brownian Motion � � log | z − λ | µ ( dzd ¯ z ) ≤ log | z − λ | µ a ( dzd ¯ z ) Segal–Bargmann C C Brown Measure Support for all λ ∈ C ; and equality holds for sufficiently large λ . Corollary. Let V a be the unbounded connected component of C \ supp µ a . Then supp µ ⊆ C \ V a . (In particular, if supp µ a is simply-connected, then supp µ ⊆ supp µ a .) 8 / 35
Brown Measure via Regularization • Dedication � The function L ( a − λ ) = R log t µ | a | ( dt ) is essentially impossible • Citations to compute with. But we can use regularity properties of the spectral Brown Measure resolution to approach it in a different way. Define • Brown Measure • Circular • Properties L ǫ ( a ) = 1 2 τ (log( a ∗ a + ǫ )) , • Convergence ǫ > 0 . • Regularize • Spectrum • L p Inverse • L p Spectrum • Support Brownian Motion Segal–Bargmann Brown Measure Support 9 / 35
Brown Measure via Regularization • Dedication � The function L ( a − λ ) = R log t µ | a | ( dt ) is essentially impossible • Citations to compute with. But we can use regularity properties of the spectral Brown Measure resolution to approach it in a different way. Define • Brown Measure • Circular • Properties L ǫ ( a ) = 1 2 τ (log( a ∗ a + ǫ )) , • Convergence ǫ > 0 . • Regularize • Spectrum • L p Inverse The function λ �→ L ǫ ( a − λ ) is C ∞ ( C ) , and is subharmonic. • L p Spectrum • Support Define a ( λ ) = 1 Brownian Motion h ǫ 2 π ∇ 2 λ L ǫ ( a − λ ) . Segal–Bargmann Then h ǫ Brown Measure Support a is a smooth probability density on C , and ǫ ↓ 0 h ǫ µ a ( dλ ) = lim a ( λ ) dλ. 9 / 35
Brown Measure via Regularization • Dedication � The function L ( a − λ ) = R log t µ | a | ( dt ) is essentially impossible • Citations to compute with. But we can use regularity properties of the spectral Brown Measure resolution to approach it in a different way. Define • Brown Measure • Circular • Properties L ǫ ( a ) = 1 2 τ (log( a ∗ a + ǫ )) , • Convergence ǫ > 0 . • Regularize • Spectrum • L p Inverse The function λ �→ L ǫ ( a − λ ) is C ∞ ( C ) , and is subharmonic. • L p Spectrum • Support Define a ( λ ) = 1 Brownian Motion h ǫ 2 π ∇ 2 λ L ǫ ( a − λ ) . Segal–Bargmann Then h ǫ Brown Measure Support a is a smooth probability density on C , and ǫ ↓ 0 h ǫ µ a ( dλ ) = lim a ( λ ) dλ. It is not difficult to explicitly calculate the density h ǫ a for fixed ǫ > 0 . 9 / 35
The Density h ǫ a and the Spectrum of a Lemma. Let λ ∈ C , and denote a λ = a − λ . Then • Dedication • Citations a ( λ ) = 1 Brown Measure h ǫ ( a ∗ λ a λ + ǫ ) − 1 ( a λ a ∗ λ + ǫ ) − 1 � � πǫτ . • Brown Measure • Circular • Properties • Convergence • Regularize • Spectrum • L p Inverse • L p Spectrum • Support Brownian Motion Segal–Bargmann Brown Measure Support 10 / 35
The Density h ǫ a and the Spectrum of a Lemma. Let λ ∈ C , and denote a λ = a − λ . Then • Dedication • Citations a ( λ ) = 1 Brown Measure h ǫ ( a ∗ λ a λ + ǫ ) − 1 ( a λ a ∗ λ + ǫ ) − 1 � � πǫτ . • Brown Measure • Circular • Properties • Convergence From here it is easy to see why supp µ a ⊆ Spec( a ) . If λ ∈ Res( a ) • Regularize so that a − 1 ∈ A , we quickly estimate • Spectrum λ • L p Inverse • L p Spectrum ( a ∗ λ a λ + ǫ ) − 1 ( a λ a ∗ λ + ǫ ) − 1 �� � � � τ • Support � � ( a ∗ λ a λ + ǫ ) − 1 ( a λ a ∗ λ + ǫ ) − 1 � � Brownian Motion ≤ � Segal–Bargmann Brown Measure Support 10 / 35
The Density h ǫ a and the Spectrum of a Lemma. Let λ ∈ C , and denote a λ = a − λ . Then • Dedication • Citations a ( λ ) = 1 Brown Measure h ǫ ( a ∗ λ a λ + ǫ ) − 1 ( a λ a ∗ λ + ǫ ) − 1 � � πǫτ . • Brown Measure • Circular • Properties • Convergence From here it is easy to see why supp µ a ⊆ Spec( a ) . If λ ∈ Res( a ) • Regularize so that a − 1 ∈ A , we quickly estimate • Spectrum λ • L p Inverse • L p Spectrum ( a ∗ λ a λ + ǫ ) − 1 ( a λ a ∗ λ + ǫ ) − 1 �� � � � τ • Support � � ( a ∗ λ a λ + ǫ ) − 1 ( a λ a ∗ λ + ǫ ) − 1 � � Brownian Motion ≤ � Segal–Bargmann � ( a ∗ λ a λ + ǫ ) − 1 � � ( a λ a ∗ λ + ǫ ) − 1 � � � � ≤ � Brown Measure Support 10 / 35
The Density h ǫ a and the Spectrum of a Lemma. Let λ ∈ C , and denote a λ = a − λ . Then • Dedication • Citations a ( λ ) = 1 Brown Measure h ǫ ( a ∗ λ a λ + ǫ ) − 1 ( a λ a ∗ λ + ǫ ) − 1 � � πǫτ . • Brown Measure • Circular • Properties • Convergence From here it is easy to see why supp µ a ⊆ Spec( a ) . If λ ∈ Res( a ) • Regularize so that a − 1 ∈ A , we quickly estimate • Spectrum λ • L p Inverse • L p Spectrum ( a ∗ λ a λ + ǫ ) − 1 ( a λ a ∗ λ + ǫ ) − 1 �� � � � τ • Support � � ( a ∗ λ a λ + ǫ ) − 1 ( a λ a ∗ λ + ǫ ) − 1 � � Brownian Motion ≤ � Segal–Bargmann � ( a ∗ λ a λ + ǫ ) − 1 � � ( a λ a ∗ λ + ǫ ) − 1 � � � � ≤ � Brown Measure Support � ( a ∗ λ a λ ) − 1 � � ( a λ a ∗ λ ) − 1 � � � � ≤ � 10 / 35
The Density h ǫ a and the Spectrum of a Lemma. Let λ ∈ C , and denote a λ = a − λ . Then • Dedication • Citations a ( λ ) = 1 Brown Measure h ǫ ( a ∗ λ a λ + ǫ ) − 1 ( a λ a ∗ λ + ǫ ) − 1 � � πǫτ . • Brown Measure • Circular • Properties • Convergence From here it is easy to see why supp µ a ⊆ Spec( a ) . If λ ∈ Res( a ) • Regularize so that a − 1 ∈ A , we quickly estimate • Spectrum λ • L p Inverse • L p Spectrum ( a ∗ λ a λ + ǫ ) − 1 ( a λ a ∗ λ + ǫ ) − 1 �� � � � τ • Support � � ( a ∗ λ a λ + ǫ ) − 1 ( a λ a ∗ λ + ǫ ) − 1 � � Brownian Motion ≤ � Segal–Bargmann � ( a ∗ λ a λ + ǫ ) − 1 � � ( a λ a ∗ λ + ǫ ) − 1 � � � � ≤ � Brown Measure Support � ( a ∗ λ a λ ) − 1 � � ( a λ a ∗ λ ) − 1 � � � � ≤ � ≤� ( a − λ ) − 1 � 4 . 10 / 35
The Density h ǫ a and the Spectrum of a Lemma. Let λ ∈ C , and denote a λ = a − λ . Then • Dedication • Citations a ( λ ) = 1 Brown Measure h ǫ ( a ∗ λ a λ + ǫ ) − 1 ( a λ a ∗ λ + ǫ ) − 1 � � πǫτ . • Brown Measure • Circular • Properties • Convergence From here it is easy to see why supp µ a ⊆ Spec( a ) . If λ ∈ Res( a ) • Regularize so that a − 1 ∈ A , we quickly estimate • Spectrum λ • L p Inverse • L p Spectrum ( a ∗ λ a λ + ǫ ) − 1 ( a λ a ∗ λ + ǫ ) − 1 �� � � � τ • Support � � ( a ∗ λ a λ + ǫ ) − 1 ( a λ a ∗ λ + ǫ ) − 1 � � Brownian Motion ≤ � Segal–Bargmann � ( a ∗ λ a λ + ǫ ) − 1 � � ( a λ a ∗ λ + ǫ ) − 1 � � � � ≤ � Brown Measure Support � ( a ∗ λ a λ ) − 1 � � ( a λ a ∗ λ ) − 1 � � � � ≤ � ≤� ( a − λ ) − 1 � 4 . This is locally uniformly bounded in λ ; so taking ǫ ↓ 0 , the factor of ǫ in h ǫ a ( λ ) kills the term; we find µ a = 0 in a neighborhood of λ . 10 / 35
Invertibility in L p ( A , τ ) Recall that L p ( A , τ ) is the closure of A in the norm • Dedication • Citations Brown Measure � ( a ∗ a ) p/ 2 � � a � p p = τ ( | a | p ) = τ . • Brown Measure • Circular • Properties (It can be realized as a set of densely-defined unbounded operators, • Convergence • Regularize acting on the same Hilbert space as A ). The non-commutative • Spectrum • L p Inverse L p -norms satisfy the same H¨ older inequality as the classical ones. • L p Spectrum • Support Brownian Motion Segal–Bargmann Brown Measure Support 11 / 35
Invertibility in L p ( A , τ ) Recall that L p ( A , τ ) is the closure of A in the norm • Dedication • Citations Brown Measure � ( a ∗ a ) p/ 2 � � a � p p = τ ( | a | p ) = τ . • Brown Measure • Circular • Properties (It can be realized as a set of densely-defined unbounded operators, • Convergence • Regularize acting on the same Hilbert space as A ). The non-commutative • Spectrum • L p Inverse L p -norms satisfy the same H¨ older inequality as the classical ones. • L p Spectrum It is perfectly possible for a ∈ A to be invertible in L p ( A , τ ) without • Support Brownian Motion having a bounded inverse. That is: there can exist b ∈ L p ( A , τ ) \ A Segal–Bargmann with ab = ba = 1 (viewed as an equation in L p ( A , τ ) ). Brown Measure Support 11 / 35
Invertibility in L p ( A , τ ) Recall that L p ( A , τ ) is the closure of A in the norm • Dedication • Citations Brown Measure � ( a ∗ a ) p/ 2 � � a � p p = τ ( | a | p ) = τ . • Brown Measure • Circular • Properties (It can be realized as a set of densely-defined unbounded operators, • Convergence • Regularize acting on the same Hilbert space as A ). The non-commutative • Spectrum • L p Inverse L p -norms satisfy the same H¨ older inequality as the classical ones. • L p Spectrum It is perfectly possible for a ∈ A to be invertible in L p ( A , τ ) without • Support Brownian Motion having a bounded inverse. That is: there can exist b ∈ L p ( A , τ ) \ A Segal–Bargmann with ab = ba = 1 (viewed as an equation in L p ( A , τ ) ). Brown Measure Support The preceding proof (with very little change) shows that h ǫ a ( λ ) → 0 at any point λ where a − λ is invertible in L 4 ( A , τ ) . 11 / 35
Invertibility in L p ( A , τ ) Recall that L p ( A , τ ) is the closure of A in the norm • Dedication • Citations Brown Measure � ( a ∗ a ) p/ 2 � � a � p p = τ ( | a | p ) = τ . • Brown Measure • Circular • Properties (It can be realized as a set of densely-defined unbounded operators, • Convergence • Regularize acting on the same Hilbert space as A ). The non-commutative • Spectrum • L p Inverse L p -norms satisfy the same H¨ older inequality as the classical ones. • L p Spectrum It is perfectly possible for a ∈ A to be invertible in L p ( A , τ ) without • Support Brownian Motion having a bounded inverse. That is: there can exist b ∈ L p ( A , τ ) \ A Segal–Bargmann with ab = ba = 1 (viewed as an equation in L p ( A , τ ) ). Brown Measure Support The preceding proof (with very little change) shows that h ǫ a ( λ ) → 0 at any point λ where a − λ is invertible in L 4 ( A , τ ) . Definition. The L p ( A , τ ) resolvent Res p,τ ( a ) is the interior of the set of λ ∈ C for which a − λ has an inverse in L p ( A , τ ) . The L p ( A , τ ) spectrum Spec p,τ ( a ) is C \ Res p,τ ( a ) . 11 / 35
The L p ( A , τ ) Spectrum From H¨ older’s inequality, we have the inclusions • Dedication • Citations Spec p,τ ( a ) ⊆ Spec q,τ ( a ) ⊆ Spec( a ) Brown Measure • Brown Measure • Circular for 1 ≤ p ≤ q < ∞ . Without including the closure in the definition, • Properties • Convergence these inclusions can be strict; with the closure, my (wild) conjecture • Regularize is that Spec 1 ,τ ( a ) = Spec( a ) for all a . • Spectrum • L p Inverse • L p Spectrum • Support Brownian Motion Segal–Bargmann Brown Measure Support 12 / 35
The L p ( A , τ ) Spectrum From H¨ older’s inequality, we have the inclusions • Dedication • Citations Spec p,τ ( a ) ⊆ Spec q,τ ( a ) ⊆ Spec( a ) Brown Measure • Brown Measure • Circular for 1 ≤ p ≤ q < ∞ . Without including the closure in the definition, • Properties • Convergence these inclusions can be strict; with the closure, my (wild) conjecture • Regularize is that Spec 1 ,τ ( a ) = Spec( a ) for all a . • Spectrum • L p Inverse • L p Spectrum As noted, supp µ a ⊆ Spec 4 ,τ ( a ) . • Support Brownian Motion Segal–Bargmann Brown Measure Support 12 / 35
The L p ( A , τ ) Spectrum From H¨ older’s inequality, we have the inclusions • Dedication • Citations Spec p,τ ( a ) ⊆ Spec q,τ ( a ) ⊆ Spec( a ) Brown Measure • Brown Measure • Circular for 1 ≤ p ≤ q < ∞ . Without including the closure in the definition, • Properties • Convergence these inclusions can be strict; with the closure, my (wild) conjecture • Regularize is that Spec 1 ,τ ( a ) = Spec( a ) for all a . • Spectrum • L p Inverse • L p Spectrum As noted, supp µ a ⊆ Spec 4 ,τ ( a ) . But we can do better. Recall that • Support π Brownian Motion ǫ h ǫ ( a ∗ λ a λ + ǫ ) − 1 ( a λ a ∗ λ + ǫ ) − 1 � � a ( λ ) = τ . Segal–Bargmann Brown Measure Support ıvely set ǫ = 0 on the right-hand-side, we get (heuristically) If we na¨ ( a ∗ λ a λ ) − 1 ( a λ a ∗ λ ) − 1 ) ( a ∗ λ ) − 1 ( a λ ) − 2 ( a ∗ λ ) − 1 � � � � τ = τ 12 / 35
The L p ( A , τ ) Spectrum From H¨ older’s inequality, we have the inclusions • Dedication • Citations Spec p,τ ( a ) ⊆ Spec q,τ ( a ) ⊆ Spec( a ) Brown Measure • Brown Measure • Circular for 1 ≤ p ≤ q < ∞ . Without including the closure in the definition, • Properties • Convergence these inclusions can be strict; with the closure, my (wild) conjecture • Regularize is that Spec 1 ,τ ( a ) = Spec( a ) for all a . • Spectrum • L p Inverse • L p Spectrum As noted, supp µ a ⊆ Spec 4 ,τ ( a ) . But we can do better. Recall that • Support π Brownian Motion ǫ h ǫ ( a ∗ λ a λ + ǫ ) − 1 ( a λ a ∗ λ + ǫ ) − 1 � � a ( λ ) = τ . Segal–Bargmann Brown Measure Support ıvely set ǫ = 0 on the right-hand-side, we get (heuristically) If we na¨ ( a ∗ λ a λ ) − 1 ( a λ a ∗ λ ) − 1 ) ( a ∗ λ ) − 1 ( a λ ) − 2 ( a ∗ λ ) − 1 � � � � τ = τ ( a − 2 λ ) ∗ a − 2 = � a − 2 λ � 2 � � = τ 2 . λ 12 / 35
The L p ( A , τ ) Spectrum From H¨ older’s inequality, we have the inclusions • Dedication • Citations Spec p,τ ( a ) ⊆ Spec q,τ ( a ) ⊆ Spec( a ) Brown Measure • Brown Measure • Circular for 1 ≤ p ≤ q < ∞ . Without including the closure in the definition, • Properties • Convergence these inclusions can be strict; with the closure, my (wild) conjecture • Regularize is that Spec 1 ,τ ( a ) = Spec( a ) for all a . • Spectrum • L p Inverse • L p Spectrum As noted, supp µ a ⊆ Spec 4 ,τ ( a ) . But we can do better. Recall that • Support π Brownian Motion ǫ h ǫ ( a ∗ λ a λ + ǫ ) − 1 ( a λ a ∗ λ + ǫ ) − 1 � � a ( λ ) = τ . Segal–Bargmann Brown Measure Support ıvely set ǫ = 0 on the right-hand-side, we get (heuristically) If we na¨ ( a ∗ λ a λ ) − 1 ( a λ a ∗ λ ) − 1 ) ( a ∗ λ ) − 1 ( a λ ) − 2 ( a ∗ λ ) − 1 � � � � τ = τ ( a − 2 λ ) ∗ a − 2 = � a − 2 λ � 2 � � = τ 2 . λ Note, this is not equal to � a − 1 λ � 4 4 when a λ is not normal. 12 / 35
The L 2 2 ,τ Spectrum Proposition. Let a ∈ A , and suppose a 2 is invertible in L 2 ( A , τ ) . • Dedication • Citations Then for all ǫ > 0 , Brown Measure • Brown Measure ( a ∗ a + ǫ ) − 1 ( aa ∗ + ǫ ) − 1 � ≤ � a − 2 � 2 • Circular � τ 2 . • Properties • Convergence • Regularize (The proof is trickier than you might think.) • Spectrum • L p Inverse • L p Spectrum • Support Brownian Motion Segal–Bargmann Brown Measure Support 13 / 35
The L 2 2 ,τ Spectrum Proposition. Let a ∈ A , and suppose a 2 is invertible in L 2 ( A , τ ) . • Dedication • Citations Then for all ǫ > 0 , Brown Measure • Brown Measure ( a ∗ a + ǫ ) − 1 ( aa ∗ + ǫ ) − 1 � ≤ � a − 2 � 2 • Circular � τ 2 . • Properties • Convergence • Regularize (The proof is trickier than you might think.) • Spectrum • L p Inverse 2 ,τ resolvent of a , Res 2 Definition. The L 2 2 ,τ ( a ) , is the interior of the • L p Spectrum set of λ ∈ C for which ( a − λ ) 2 is invertible in L 2 ( A , τ ) . The L 2 • Support 2 ,τ Brownian Motion spectrum of a is Spec 2 2 ,τ ( a ) = C \ Res 2 2 ,τ ( a ) . Segal–Bargmann Brown Measure Support 13 / 35
The L 2 2 ,τ Spectrum Proposition. Let a ∈ A , and suppose a 2 is invertible in L 2 ( A , τ ) . • Dedication • Citations Then for all ǫ > 0 , Brown Measure • Brown Measure ( a ∗ a + ǫ ) − 1 ( aa ∗ + ǫ ) − 1 � ≤ � a − 2 � 2 • Circular � τ 2 . • Properties • Convergence • Regularize (The proof is trickier than you might think.) • Spectrum • L p Inverse 2 ,τ resolvent of a , Res 2 Definition. The L 2 2 ,τ ( a ) , is the interior of the • L p Spectrum set of λ ∈ C for which ( a − λ ) 2 is invertible in L 2 ( A , τ ) . The L 2 • Support 2 ,τ Brownian Motion spectrum of a is Spec 2 2 ,τ ( a ) = C \ Res 2 2 ,τ ( a ) . Segal–Bargmann Theorem. supp µ a ⊆ Spec 2 2 ,τ ( a ) . Brown Measure Support 13 / 35
The L 2 2 ,τ Spectrum Proposition. Let a ∈ A , and suppose a 2 is invertible in L 2 ( A , τ ) . • Dedication • Citations Then for all ǫ > 0 , Brown Measure • Brown Measure ( a ∗ a + ǫ ) − 1 ( aa ∗ + ǫ ) − 1 � ≤ � a − 2 � 2 • Circular � τ 2 . • Properties • Convergence • Regularize (The proof is trickier than you might think.) • Spectrum • L p Inverse 2 ,τ resolvent of a , Res 2 Definition. The L 2 2 ,τ ( a ) , is the interior of the • L p Spectrum set of λ ∈ C for which ( a − λ ) 2 is invertible in L 2 ( A , τ ) . The L 2 • Support 2 ,τ Brownian Motion spectrum of a is Spec 2 2 ,τ ( a ) = C \ Res 2 2 ,τ ( a ) . Segal–Bargmann Theorem. supp µ a ⊆ Spec 2 2 ,τ ( a ) . Brown Measure Support Another wild conjecture: this is actually equality. (That depends on showing that, if a 2 is not invertible in L 2 ( A , τ ) , the above quantity blows up at rate Ω(1 /ǫ ) . This appears to be what happens in the case that a is normal, which would imply Spec 2 2 ,τ ( a ) = Spec 4 ,τ ( a ) = Spec( a ) in that case.) 13 / 35
• Dedication • Citations Brown Measure Brownian Motion • BM on Lie Groups • U & GL • Free + BM • Free × BM Brownian Motion on U( N ) , • Free Unitary BM • Transforms GL( N, C ) , and the Large- N Limit • Free Mult. BM • GL Spectrum Segal–Bargmann Brown Measure Support 14 / 35
Brownian Motion on Lie Groups On any Riemannian manifold M , there’s a Laplace operator ∆ M . • Dedication And where there’s a Laplacian, there’s a Brownian motion: the • Citations Markov process ( B x t ) t ≥ 0 on M with generator 1 2 ∆ M , started at Brown Measure B x 0 = x ∈ M . Brownian Motion • BM on Lie Groups • U & GL • Free + BM • Free × BM • Free Unitary BM • Transforms • Free Mult. BM • GL Spectrum Segal–Bargmann Brown Measure Support 15 / 35
Brownian Motion on Lie Groups On any Riemannian manifold M , there’s a Laplace operator ∆ M . • Dedication And where there’s a Laplacian, there’s a Brownian motion: the • Citations Markov process ( B x t ) t ≥ 0 on M with generator 1 2 ∆ M , started at Brown Measure B x 0 = x ∈ M . Brownian Motion • BM on Lie Groups • U & GL Let Γ be a (matrix) Lie group. Any inner product on Lie(Γ) = T I Γ • Free + BM gives rise to a unique left-invariant Riemannian metric, and • Free × BM • Free Unitary BM corresponding Laplacian ∆ Γ . On Γ we canonically start the • Transforms Brownian motion ( B t ) t ≥ 0 at I ∈ Γ . • Free Mult. BM • GL Spectrum Segal–Bargmann Brown Measure Support 15 / 35
Brownian Motion on Lie Groups On any Riemannian manifold M , there’s a Laplace operator ∆ M . • Dedication And where there’s a Laplacian, there’s a Brownian motion: the • Citations Markov process ( B x t ) t ≥ 0 on M with generator 1 2 ∆ M , started at Brown Measure B x 0 = x ∈ M . Brownian Motion • BM on Lie Groups • U & GL Let Γ be a (matrix) Lie group. Any inner product on Lie(Γ) = T I Γ • Free + BM gives rise to a unique left-invariant Riemannian metric, and • Free × BM • Free Unitary BM corresponding Laplacian ∆ Γ . On Γ we canonically start the • Transforms Brownian motion ( B t ) t ≥ 0 at I ∈ Γ . • Free Mult. BM • GL Spectrum There is a beautiful relationship between the Brownian motion W t on Segal–Bargmann the Lie algebra Lie(Γ) and the Brownian motion B t : the rolling map Brown Measure Support � t dB t = B t ◦ dW t , B t = I + B t ◦ dW t . i.e. 0 Here ◦ denotes the Stratonovich stochastic integral. This can always be converted into an Itˆ o integral; but the answer depends on the structure of the group Γ (and the chosen inner product). 15 / 35
Brownian Motion on U( N ) and GL( N, C ) Fix the reverse normalized Hilbert–Schmidt inner product on • Dedication M N ( C ) for all matrix Lie algebras: • Citations Brown Measure � A, B � = N Tr( B ∗ A ) . Brownian Motion • BM on Lie Groups • U & GL Let X t = X N and Y t = Y N be independent Hermitian Brownian • Free + BM t t • Free × BM motions of variance t/N . • Free Unitary BM • Transforms • Free Mult. BM • GL Spectrum Segal–Bargmann Brown Measure Support 16 / 35
Brownian Motion on U( N ) and GL( N, C ) Fix the reverse normalized Hilbert–Schmidt inner product on • Dedication M N ( C ) for all matrix Lie algebras: • Citations Brown Measure � A, B � = N Tr( B ∗ A ) . Brownian Motion • BM on Lie Groups • U & GL Let X t = X N and Y t = Y N be independent Hermitian Brownian • Free + BM t t • Free × BM motions of variance t/N . • Free Unitary BM • Transforms The Brownian motion on Lie(U( N )) is iX t ; the Brownian motion • Free Mult. BM U t on U( N ) satisfies • GL Spectrum Segal–Bargmann dU t = iU t dX t − 1 Brown Measure Support 2 U t dt. 16 / 35
Brownian Motion on U( N ) and GL( N, C ) Fix the reverse normalized Hilbert–Schmidt inner product on • Dedication M N ( C ) for all matrix Lie algebras: • Citations Brown Measure � A, B � = N Tr( B ∗ A ) . Brownian Motion • BM on Lie Groups • U & GL Let X t = X N and Y t = Y N be independent Hermitian Brownian • Free + BM t t • Free × BM motions of variance t/N . • Free Unitary BM • Transforms The Brownian motion on Lie(U( N )) is iX t ; the Brownian motion • Free Mult. BM U t on U( N ) satisfies • GL Spectrum Segal–Bargmann dU t = iU t dX t − 1 Brown Measure Support 2 U t dt. The Brownian motion on Lie(GL( N, C )) = M N ( C ) is Z t = 2 − 1 / 2 i ( X t + iY t ) ; the Brownian motion G t on GL( N, C ) satisfies dG t = G t dZ t . 16 / 35
1 1 Free Additive Brownian Motion If X t = X N • Dedication is a Hermitian Brownian motion process, then at each t • Citations time t > 0 it is a GUE N with entries of variance t/N . Wigner’s law Brown Measure then shows that the empirical spectral distribution of X N converges Brownian Motion t 1 � (4 t − x 2 ) + dx . • BM on Lie Groups to the semicircle law ς t = 2 πt • U & GL • Free + BM • Free × BM • Free Unitary BM • Transforms • Free Mult. BM • GL Spectrum Segal–Bargmann Brown Measure Support 17 / 35
1 1 Free Additive Brownian Motion If X t = X N • Dedication is a Hermitian Brownian motion process, then at each t • Citations time t > 0 it is a GUE N with entries of variance t/N . Wigner’s law Brown Measure then shows that the empirical spectral distribution of X N converges Brownian Motion t 1 � (4 t − x 2 ) + dx . In fact, it converges • BM on Lie Groups to the semicircle law ς t = 2 πt • U & GL as a process . • Free + BM • Free × BM • Free Unitary BM • Transforms • Free Mult. BM • GL Spectrum Segal–Bargmann Brown Measure Support 17 / 35
1 1 Free Additive Brownian Motion If X t = X N • Dedication is a Hermitian Brownian motion process, then at each t • Citations time t > 0 it is a GUE N with entries of variance t/N . Wigner’s law Brown Measure then shows that the empirical spectral distribution of X N converges Brownian Motion t 1 � (4 t − x 2 ) + dx . In fact, it converges • BM on Lie Groups to the semicircle law ς t = 2 πt • U & GL as a process . • Free + BM • Free × BM A process ( x t ) t ≥ 0 (in a W ∗ -probability space with trace τ ) is a free • Free Unitary BM • Transforms additive Brownian motion if its increments are freely independent • Free Mult. BM • GL Spectrum — x t − x s is free from { x r : r ≤ s } — and x t − x s has the Segal–Bargmann semicircular distribution ς t − s , for all t > s . Brown Measure Support 17 / 35
Free Additive Brownian Motion If X t = X N • Dedication is a Hermitian Brownian motion process, then at each t • Citations time t > 0 it is a GUE N with entries of variance t/N . Wigner’s law Brown Measure then shows that the empirical spectral distribution of X N converges Brownian Motion t 1 � (4 t − x 2 ) + dx . In fact, it converges • BM on Lie Groups to the semicircle law ς t = 2 πt • U & GL as a process . • Free + BM • Free × BM A process ( x t ) t ≥ 0 (in a W ∗ -probability space with trace τ ) is a free • Free Unitary BM • Transforms additive Brownian motion if its increments are freely independent • Free Mult. BM • GL Spectrum — x t − x s is free from { x r : r ≤ s } — and x t − x s has the Segal–Bargmann semicircular distribution ς t − s , for all t > s . It can be constructed on Brown Measure Support the free Fock space over L 2 ( R + ) : x t = l ( 1 [0 ,t ] ) + l ∗ ( 1 [0 ,t ] ) . 17 / 35
Free Additive Brownian Motion If X t = X N • Dedication is a Hermitian Brownian motion process, then at each t • Citations time t > 0 it is a GUE N with entries of variance t/N . Wigner’s law Brown Measure then shows that the empirical spectral distribution of X N converges Brownian Motion t 1 � (4 t − x 2 ) + dx . In fact, it converges • BM on Lie Groups to the semicircle law ς t = 2 πt • U & GL as a process . • Free + BM • Free × BM A process ( x t ) t ≥ 0 (in a W ∗ -probability space with trace τ ) is a free • Free Unitary BM • Transforms additive Brownian motion if its increments are freely independent • Free Mult. BM • GL Spectrum — x t − x s is free from { x r : r ≤ s } — and x t − x s has the Segal–Bargmann semicircular distribution ς t − s , for all t > s . It can be constructed on Brown Measure Support the free Fock space over L 2 ( R + ) : x t = l ( 1 [0 ,t ] ) + l ∗ ( 1 [0 ,t ] ) . In 1991, Voiculescu showed that the Hermitian Brownian motion ( X N t ) t ≥ 0 converges to ( x t ) t ≥ 0 in finite-dimensional non-commutative distributions: 1 N Tr( P ( X t 1 , . . . , X t n )) → τ ( P ( x t 1 , . . . , x t n )) ∀ P. 17 / 35
Free Unitary and Free Multiplicative Brownian Motion • Dedication There is now a well-developed theory of free stochastic differential • Citations equations. Initially constructed in the free Fock space setting (by Brown Measure K¨ ummerer and Speicher in the early 1990s), it was used by Biane in Brownian Motion 1997 to define “free versions” of U t and G t . • BM on Lie Groups • U & GL • Free + BM • Free × BM • Free Unitary BM • Transforms • Free Mult. BM • GL Spectrum Segal–Bargmann Brown Measure Support 18 / 35
Free Unitary and Free Multiplicative Brownian Motion • Dedication There is now a well-developed theory of free stochastic differential • Citations equations. Initially constructed in the free Fock space setting (by Brown Measure K¨ ummerer and Speicher in the early 1990s), it was used by Biane in Brownian Motion 1997 to define “free versions” of U t and G t . • BM on Lie Groups • U & GL • Free + BM Let x t , y t be freely independent free additive Brownian motions, and • Free × BM z t = 2 − 1 / 2 i ( x t + iy t ) . The free unitary Brownian motion is the • Free Unitary BM • Transforms process started at u 0 = 1 defined by • Free Mult. BM • GL Spectrum du t = iu t dx t − 1 2 u t dt. Segal–Bargmann Brown Measure Support The free multiplicative Brownian motion is the process started at g 0 = 1 defined by dg t = g t dz t . 18 / 35
Free Unitary and Free Multiplicative Brownian Motion • Dedication There is now a well-developed theory of free stochastic differential • Citations equations. Initially constructed in the free Fock space setting (by Brown Measure K¨ ummerer and Speicher in the early 1990s), it was used by Biane in Brownian Motion 1997 to define “free versions” of U t and G t . • BM on Lie Groups • U & GL • Free + BM Let x t , y t be freely independent free additive Brownian motions, and • Free × BM z t = 2 − 1 / 2 i ( x t + iy t ) . The free unitary Brownian motion is the • Free Unitary BM • Transforms process started at u 0 = 1 defined by • Free Mult. BM • GL Spectrum du t = iu t dx t − 1 2 u t dt. Segal–Bargmann Brown Measure Support The free multiplicative Brownian motion is the process started at g 0 = 1 defined by dg t = g t dz t . It is natural to expect that these processes should be the large- N limits of the U( N ) and GL( N, C ) Brownian motions. 18 / 35
Free Unitary Brownian Motion • Dedication Theorem. [Biane, 1997] For all non-commutative (Laurent) • Citations polynomials P in n variables and times t 1 , . . . , t n ≥ 0 , Brown Measure Brownian Motion 1 N Tr( P ( U N t 1 , . . . , U N t n )) → τ ( P ( u t 1 , . . . , u t n )) a.s. • BM on Lie Groups • U & GL • Free + BM • Free × BM • Free Unitary BM • Transforms • Free Mult. BM • GL Spectrum Segal–Bargmann Brown Measure Support 19 / 35
Free Unitary Brownian Motion • Dedication Theorem. [Biane, 1997] For all non-commutative (Laurent) • Citations polynomials P in n variables and times t 1 , . . . , t n ≥ 0 , Brown Measure Brownian Motion 1 N Tr( P ( U N t 1 , . . . , U N t n )) → τ ( P ( u t 1 , . . . , u t n )) a.s. • BM on Lie Groups • U & GL • Free + BM Biane also computed the moments of u t , and its spectral measure • Free × BM • Free Unitary BM ν t : it has a density (smooth on the interior of its support), supported • Transforms • Free Mult. BM on a compact arc for t < 4 , and fully supported on U for t ≥ 4 . • GL Spectrum Segal–Bargmann Brown Measure Support 19 / 35
Free Unitary Brownian Motion • Dedication Theorem. [Biane, 1997] For all non-commutative (Laurent) • Citations polynomials P in n variables and times t 1 , . . . , t n ≥ 0 , Brown Measure Brownian Motion 1 N Tr( P ( U N t 1 , . . . , U N t n )) → τ ( P ( u t 1 , . . . , u t n )) a.s. • BM on Lie Groups • U & GL • Free + BM Biane also computed the moments of u t , and its spectral measure • Free × BM • Free Unitary BM ν t : it has a density (smooth on the interior of its support), supported • Transforms • Free Mult. BM on a compact arc for t < 4 , and fully supported on U for t ≥ 4 . • GL Spectrum Segal–Bargmann 600 Brown Measure Support 500 400 300 200 100 0 -3 -2 -1 0 1 2 3 19 / 35
Analytic Transforms Related to u t Biane’s approach to understanding the measure ν t was through its • Dedication moment-generating function • Citations Brown Measure uz � � m n ( ν t ) z n Brownian Motion ψ t ( z ) = 1 − uz ν t ( du ) = • BM on Lie Groups U n ≥ 1 • U & GL • Free + BM • Free × BM (the second = holds for | z | < 1 ; the integral converges for • Free Unitary BM 1 /z / ∈ supp ν t ). • Transforms • Free Mult. BM • GL Spectrum Segal–Bargmann Brown Measure Support 20 / 35
Analytic Transforms Related to u t Biane’s approach to understanding the measure ν t was through its • Dedication moment-generating function • Citations Brown Measure uz � � m n ( ν t ) z n Brownian Motion ψ t ( z ) = 1 − uz ν t ( du ) = • BM on Lie Groups U n ≥ 1 • U & GL • Free + BM • Free × BM (the second = holds for | z | < 1 ; the integral converges for • Free Unitary BM 1 /z / ∈ supp ν t ). Then define • Transforms • Free Mult. BM ψ t ( z ) • GL Spectrum χ t ( z ) = 1 + ψ t ( z ) . Segal–Bargmann Brown Measure Support The function χ t is injective on D , and has a one-sided inverse f t : f t ( χ t ( z )) = z for z ∈ D (but χ t ◦ f t is only the identity on a certain region in C ; more on this later). 20 / 35
Analytic Transforms Related to u t Biane’s approach to understanding the measure ν t was through its • Dedication moment-generating function • Citations Brown Measure uz � � m n ( ν t ) z n Brownian Motion ψ t ( z ) = 1 − uz ν t ( du ) = • BM on Lie Groups U n ≥ 1 • U & GL • Free + BM • Free × BM (the second = holds for | z | < 1 ; the integral converges for • Free Unitary BM 1 /z / ∈ supp ν t ). Then define • Transforms • Free Mult. BM ψ t ( z ) • GL Spectrum χ t ( z ) = 1 + ψ t ( z ) . Segal–Bargmann Brown Measure Support The function χ t is injective on D , and has a one-sided inverse f t : f t ( χ t ( z )) = z for z ∈ D (but χ t ◦ f t is only the identity on a certain region in C ; more on this later). Using the SDE for u t and some clever complex analysis, Biane showed that 1+ z t 1 − z . f t ( z ) = ze 2 20 / 35
The Large- N Limit of G N t In 1997 Biane conjectured a similar large- N limit should hold for the Brownian motion on GL( N, C ) , but the ideas of his U N proof (spectral theorem, t representation theory of U( N ) ) did not translate well to the a.s. non-normal process G N t . 21 / 35
The Large- N Limit of G N t In 1997 Biane conjectured a similar large- N limit should hold for the Brownian motion on GL( N, C ) , but the ideas of his U N proof (spectral theorem, t representation theory of U( N ) ) did not translate well to the a.s. non-normal process G N t . Theorem. [K, 2014 (2016)] For all non-commutative Laurent polynomials P in 2 n variables, and times t 1 , . . . , t n ≥ 0 , 1 P ( G N t 1 , ( G N t 1 ) ∗ , . . . , G N t n , ( G N t n ) ∗ ) P ( g t 1 , g ∗ t 1 , . . . , g t n , g ∗ � � � � N Tr → τ t n ) a.s. 21 / 35
The Large- N Limit of G N t In 1997 Biane conjectured a similar large- N limit should hold for the Brownian motion on GL( N, C ) , but the ideas of his U N proof (spectral theorem, t representation theory of U( N ) ) did not translate well to the a.s. non-normal process G N t . Theorem. [K, 2014 (2016)] For all non-commutative Laurent polynomials P in 2 n variables, and times t 1 , . . . , t n ≥ 0 , 1 P ( G N t 1 , ( G N t 1 ) ∗ , . . . , G N t n , ( G N t n ) ∗ ) P ( g t 1 , g ∗ t 1 , . . . , g t n , g ∗ � � � � N Tr → τ t n ) a.s. The proof required several new ingredients: a detailed understanding of the Laplacian on GL( N, C ) , and concentration of measure for trace polynomials. Putting these together with an iteration scheme from the SDE, together with requisite covariance estimates, yielded the proof. 21 / 35
The Large- N Limit of G N t In 1997 Biane conjectured a similar large- N limit should hold for the Brownian motion on GL( N, C ) , but the ideas of his U N proof (spectral theorem, t representation theory of U( N ) ) did not translate well to the a.s. non-normal process G N t . Theorem. [K, 2014 (2016)] For all non-commutative Laurent polynomials P in 2 n variables, and times t 1 , . . . , t n ≥ 0 , 1 P ( G N t 1 , ( G N t 1 ) ∗ , . . . , G N t n , ( G N t n ) ∗ ) P ( g t 1 , g ∗ t 1 , . . . , g t n , g ∗ � � � � N Tr → τ t n ) a.s. The proof required several new ingredients: a detailed understanding of the Laplacian on GL( N, C ) , and concentration of measure for trace polynomials. Putting these together with an iteration scheme from the SDE, together with requisite covariance estimates, yielded the proof. This is convergence of the (multi-time) ∗ -distribution, of a non-normal matrix process. What about the eigenvalues? 21 / 35
The Eigenvalues of Brownian Motion GL( N, C ) Because U N and u t are normal, their ∗ -distributions encode their ESDs, so t the bulk eigenvalue behavior is fully understood. 22 / 35
The Eigenvalues of Brownian Motion GL( N, C ) Because U N and u t are normal, their ∗ -distributions encode their ESDs, so t the bulk eigenvalue behavior is fully understood. The GL( N, C ) Brownian motion G N eigenvalues are much more challenging. t 22 / 35
The Eigenvalues of Brownian Motion GL( N, C ) Because U N and u t are normal, their ∗ -distributions encode their ESDs, so t the bulk eigenvalue behavior is fully understood. The GL( N, C ) Brownian motion G N eigenvalues are much more challenging. t t = 1 22 / 35
The Eigenvalues of Brownian Motion GL( N, C ) Because U N and u t are normal, their ∗ -distributions encode their ESDs, so t the bulk eigenvalue behavior is fully understood. The GL( N, C ) Brownian motion G N eigenvalues are much more challenging. t t = 2 22 / 35
The Eigenvalues of Brownian Motion GL( N, C ) Because U N and u t are normal, their ∗ -distributions encode their ESDs, so t the bulk eigenvalue behavior is fully understood. The GL( N, C ) Brownian motion G N eigenvalues are much more challenging. t t = 4 22 / 35
• Dedication • Citations Brown Measure Brownian Motion Segal–Bargmann • SBT • Free SBT • Σ t The Segal–Bargmann Transform Brown Measure Support 23 / 35
The Unitary Segal–Bargmann Transform • Dedication The Segal–Bargmann (Hall) Transform is a map from functions on • Citations U( N ) to holomorphic functions on GL( N, C ) . It is defined by the Brown Measure analytic continuation of the action of the heat operator: Brownian Motion Segal–Bargmann � � t B N 2 ∆ U( N ) f t f = e C . • SBT • Free SBT • Σ t Brown Measure Support 24 / 35
The Unitary Segal–Bargmann Transform • Dedication The Segal–Bargmann (Hall) Transform is a map from functions on • Citations U( N ) to holomorphic functions on GL( N, C ) . It is defined by the Brown Measure analytic continuation of the action of the heat operator: Brownian Motion Segal–Bargmann � � t B N 2 ∆ U( N ) f t f = e C . • SBT • Free SBT • Σ t Writing out what this integral formula means in probabilistic terms, Brown Measure Support here is a nice way to express it: let F already be a holomorphic function on GL( N ) , C ) , and let f = F | U( N ) . Let U t and G t be independent Brownian motions on U( N ) and GL( N, C ) . Then ( B t f )( G t ) = E [ F ( G t U t ) | G t ] . 24 / 35
The Unitary Segal–Bargmann Transform • Dedication The Segal–Bargmann (Hall) Transform is a map from functions on • Citations U( N ) to holomorphic functions on GL( N, C ) . It is defined by the Brown Measure analytic continuation of the action of the heat operator: Brownian Motion Segal–Bargmann � � t B N 2 ∆ U( N ) f t f = e C . • SBT • Free SBT • Σ t Writing out what this integral formula means in probabilistic terms, Brown Measure Support here is a nice way to express it: let F already be a holomorphic function on GL( N ) , C ) , and let f = F | U( N ) . Let U t and G t be independent Brownian motions on U( N ) and GL( N, C ) . Then ( B t f )( G t ) = E [ F ( G t U t ) | G t ] . This extends beyond f that already possess an analytic continuation; it defines an isometric isomorphism B N t : L 2 (U( N ) , U t ) → H L 2 (GL( N, C ) , G t ) . 24 / 35
The Free Unitary Segal–Bargmann Transform In 1997, Biane introduced a free version of the Unitary SBT, which • Dedication can be described in similar terms: acting on, say, polynomials f in a • Citations single variable, G t f is defined by Brown Measure Brownian Motion ( G t f )( g t ) = τ [ f ( g t u t ) | g t ] . Segal–Bargmann • SBT • Free SBT He conjectured that G t is the large- N limit of B N in an appropriate • Σ t t sense; this was proven by Driver, Hall, and me in 2013. (It was for Brown Measure Support this work that we invented trace polynomial concentration.) 25 / 35
The Free Unitary Segal–Bargmann Transform In 1997, Biane introduced a free version of the Unitary SBT, which • Dedication can be described in similar terms: acting on, say, polynomials f in a • Citations single variable, G t f is defined by Brown Measure Brownian Motion ( G t f )( g t ) = τ [ f ( g t u t ) | g t ] . Segal–Bargmann • SBT • Free SBT He conjectured that G t is the large- N limit of B N in an appropriate • Σ t t sense; this was proven by Driver, Hall, and me in 2013. (It was for Brown Measure Support this work that we invented trace polynomial concentration.) Biane proved directly (and it follows from the large- N limit) that G t extends to an isometric isomorphism G t : L 2 ( U , ν t ) → A t where A t is a certain reproducing-kernel Hilbert space of holomorphic functions. The norm on A t is given by � F � 2 A t = τ ( | F ( g t ) | 2 ) = τ ( F ( g t ) ∗ F ( g t )) = � F ( g t ) � 2 2 . 25 / 35
The Range of the Free Segal–Bargmann Transform The functions F ∈ A t are not all entire functions. They are • Dedication • Citations holomorphic on a bounded region Σ t Brown Measure Brownian Motion Σ t = C \ χ t ( C \ supp ν t ) Segal–Bargmann • SBT 1+ z t • Free SBT 1 − z . where (recall) χ t is the (right-)inverse of f t ( z ) = ze 2 • Σ t Brown Measure Support 26 / 35
The Range of the Free Segal–Bargmann Transform • Dedication • Citations Brown Measure Brownian Motion 5 Segal–Bargmann • SBT • Free SBT • Σ t Brown Measure Support 0 - 5 0 5 10 t = 1 26 / 35
The Range of the Free Segal–Bargmann Transform • Dedication • Citations Brown Measure Brownian Motion 5 Segal–Bargmann • SBT • Free SBT • Σ t Brown Measure Support 0 - 5 0 5 10 t = 2 26 / 35
The Range of the Free Segal–Bargmann Transform • Dedication • Citations Brown Measure Brownian Motion 5 Segal–Bargmann • SBT • Free SBT • Σ t Brown Measure Support 0 - 5 0 5 10 t = 3 . 9 26 / 35
The Range of the Free Segal–Bargmann Transform • Dedication • Citations Brown Measure Brownian Motion 5 Segal–Bargmann • SBT • Free SBT • Σ t Brown Measure Support 0 - 5 0 5 10 t = 4 26 / 35
The Range of the Free Segal–Bargmann Transform • Dedication • Citations Brown Measure Brownian Motion Segal–Bargmann • SBT • Free SBT • Σ t Brown Measure Support t = 4 26 / 35
The Range of the Free Segal–Bargmann Transform • Dedication • Citations Brown Measure Brownian Motion 5 Segal–Bargmann • SBT • Free SBT • Σ t Brown Measure Support 0 - 5 - 5 0 5 10 t = 4 . 01 26 / 35
• Dedication • Citations Brown Measure Brownian Motion Segal–Bargmann Brown Measure Support • Main Theorem The Brown Measure of Free • Proof • Simulations • Simulations Multiplicative Brownian Motion • Implicit • The Brown Measure • Simulations • Questions 27 / 35
The Support of The Brown Measure of g t • Dedication Theorem. (Hall, K, 2018) • Citations Brown Measure supp µ g t ⊆ Σ t . Brownian Motion Segal–Bargmann Brown Measure Support • Main Theorem • Proof • Simulations • Simulations • Implicit • The Brown Measure • Simulations • Questions 28 / 35
The Support of The Brown Measure of g t • Dedication Theorem. (Hall, K, 2018) • Citations Brown Measure supp µ g t ⊆ Σ t . Brownian Motion Segal–Bargmann Proof. We show that Spec 2 2 ,τ ( g t ) = Σ t . Equivalently, from the Brown Measure Support definition of Σ t , we show that Res 2 2 ,τ ( g t ) = χ t ( C \ supp ν t ) . • Main Theorem • Proof • Simulations • Simulations • Implicit • The Brown Measure • Simulations • Questions 28 / 35
The Support of The Brown Measure of g t • Dedication Theorem. (Hall, K, 2018) • Citations Brown Measure supp µ g t ⊆ Σ t . Brownian Motion Segal–Bargmann Proof. We show that Spec 2 2 ,τ ( g t ) = Σ t . Equivalently, from the Brown Measure Support definition of Σ t , we show that Res 2 2 ,τ ( g t ) = χ t ( C \ supp ν t ) . • Main Theorem • Proof 2 ,τ ( g t ) iff ( g t − λ ) 2 is invertible in L 2 ( τ ) , i.e. Essentially, λ ∈ Res 2 • Simulations • Simulations • Implicit | ( g t − λ ) − 2 | 2 � � ∞ > τ • The Brown Measure • Simulations • Questions 28 / 35
The Support of The Brown Measure of g t • Dedication Theorem. (Hall, K, 2018) • Citations Brown Measure supp µ g t ⊆ Σ t . Brownian Motion Segal–Bargmann Proof. We show that Spec 2 2 ,τ ( g t ) = Σ t . Equivalently, from the Brown Measure Support definition of Σ t , we show that Res 2 2 ,τ ( g t ) = χ t ( C \ supp ν t ) . • Main Theorem • Proof 2 ,τ ( g t ) iff ( g t − λ ) 2 is invertible in L 2 ( τ ) , i.e. Essentially, λ ∈ Res 2 • Simulations • Simulations • Implicit | ( g t − λ ) − 2 | 2 � = � ( z − λ ) − 2 � 2 � ∞ > τ A t . • The Brown Measure • Simulations • Questions 28 / 35
The Support of The Brown Measure of g t • Dedication Theorem. (Hall, K, 2018) • Citations Brown Measure supp µ g t ⊆ Σ t . Brownian Motion Segal–Bargmann Proof. We show that Spec 2 2 ,τ ( g t ) = Σ t . Equivalently, from the Brown Measure Support definition of Σ t , we show that Res 2 2 ,τ ( g t ) = χ t ( C \ supp ν t ) . • Main Theorem • Proof 2 ,τ ( g t ) iff ( g t − λ ) 2 is invertible in L 2 ( τ ) , i.e. Essentially, λ ∈ Res 2 • Simulations • Simulations • Implicit | ( g t − λ ) − 2 | 2 � = � ( z − λ ) − 2 � 2 � ∞ > τ A t . • The Brown Measure • Simulations • Questions Recall that G t is an isometry from L 2 ( U , ν t ) onto A t . Can we find a function α λ t on U with G t ( α λ t )( z ) = ( z − λ ) − 2 ? 28 / 35
The Support of The Brown Measure of g t • Dedication Theorem. (Hall, K, 2018) • Citations Brown Measure supp µ g t ⊆ Σ t . Brownian Motion Segal–Bargmann Proof. We show that Spec 2 2 ,τ ( g t ) = Σ t . Equivalently, from the Brown Measure Support definition of Σ t , we show that Res 2 2 ,τ ( g t ) = χ t ( C \ supp ν t ) . • Main Theorem • Proof 2 ,τ ( g t ) iff ( g t − λ ) 2 is invertible in L 2 ( τ ) , i.e. Essentially, λ ∈ Res 2 • Simulations • Simulations • Implicit | ( g t − λ ) − 2 | 2 � = � ( z − λ ) − 2 � 2 � ∞ > τ A t . • The Brown Measure • Simulations • Questions Recall that G t is an isometry from L 2 ( U , ν t ) onto A t . Can we find a function α λ t on U with G t ( α λ t )( z ) = ( z − λ ) − 2 ? Using PDE techniques, we can compute that (( z − λ ) − 1 ) = 1 f t ( λ ) G − 1 f t ( λ ) − u. t λ 28 / 35
The Support of The Brown Measure of g t • Dedication G t : 1 f t ( λ ) 1 • Citations f t ( λ ) − u �→ z − λ. λ Brown Measure Brownian Motion 1 d 1 ( z − λ ) 2 = z − λ , using regularity properties of G t we have Since dλ Segal–Bargmann Brown Measure Support t ( u ) = d � 1 f t ( λ ) � • Main Theorem α λ . • Proof dλ λ f t ( λ ) − u • Simulations • Simulations • Implicit • The Brown Measure • Simulations • Questions 29 / 35
The Support of The Brown Measure of g t • Dedication G t : 1 f t ( λ ) 1 • Citations f t ( λ ) − u �→ z − λ. λ Brown Measure Brownian Motion 1 d 1 ( z − λ ) 2 = z − λ , using regularity properties of G t we have Since dλ Segal–Bargmann Brown Measure Support t ( u ) = d � 1 f t ( λ ) � • Main Theorem α λ . • Proof dλ λ f t ( λ ) − u • Simulations • Simulations The question is: for which λ is α λ t ∈ L 2 ( U , ν t ) ? I.e. • Implicit • The Brown Measure • Simulations � t ( u ) | 2 ν t ( du ) < ∞ . • Questions | α λ U 29 / 35
The Support of The Brown Measure of g t • Dedication G t : 1 f t ( λ ) 1 • Citations f t ( λ ) − u �→ z − λ. λ Brown Measure Brownian Motion 1 d 1 ( z − λ ) 2 = z − λ , using regularity properties of G t we have Since dλ Segal–Bargmann Brown Measure Support t ( u ) = d � 1 f t ( λ ) � • Main Theorem α λ . • Proof dλ λ f t ( λ ) − u • Simulations • Simulations The question is: for which λ is α λ t ∈ L 2 ( U , ν t ) ? I.e. • Implicit • The Brown Measure • Simulations � t ( u ) | 2 ν t ( du ) < ∞ . • Questions | α λ U The answer is: precisely when f t ( λ ) / ∈ supp ν t . I.e. Res 2 2 ,τ ( g t ) = f − 1 ( C \ supp ν t ) = χ t ( C \ supp ν t ) . t � 29 / 35
The Empirical Spectrum and Σ t Here is a simulation of eigenvalues of G ( N ) for N = 2000 , together t • Dedication with the boundary of Σ t , at t = 3 (produced in Mathematica). • Citations Brown Measure Brownian Motion 4 Segal–Bargmann Brown Measure Support • Main Theorem • Proof 2 • Simulations • Simulations • Implicit • The Brown Measure 0 • Simulations • Questions - 2 - 4 - 2 0 2 4 6 8 30 / 35
The Empirical Spectrum and Σ t N = 2000 , t = 2 , 3 . 9 , 4 , 4 . 1 . • Dedication • Citations Brown Measure Brownian Motion Segal–Bargmann Brown Measure Support • Main Theorem • Proof • Simulations • Simulations • Implicit • The Brown Measure • Simulations • Questions 31 / 35
Computing the Brown Measure Very recently, jointly with Driver and Hall, we have been able to push further and actually compute the Brown measure. 32 / 35
Computing the Brown Measure Very recently, jointly with Driver and Hall, we have been able to push further and actually compute the Brown measure. To describe it, we need an auxiliary implicit function ̺ = ̺ ( t, θ ) , determined by 2 − ̺ 2 + 2 � � � 1 − ̺ 2 1 − ̺ cos θ 1 − ̺ 2 log = t. ̺ 2 � This defines a real analytic function for | θ | < θ max ( t ) = cos − 1 (1 − t/ 2) ∧ π , which is precisely the argument range of Σ t : 32 / 35
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