Convolutions and fluctuations: free, finite, quantized. Vadim Gorin MIT (Cambridge) and IITP (Moscow) February 2018
Hermitian matrix operations 0 0 a 1 b 1 0 a 2 0 0 b 2 0 A = B = ... ... 0 0 0 0 0 0 a N b N U , V − Haar–random in Unitary ( N ; R / C / H ) C = UAU ∗ + VBV ∗ տ ր uniformly random eigenvectors
Hermitian matrix operations 0 0 a 1 b 1 0 a 2 0 0 b 2 0 A = B = ... ... 0 0 0 0 0 0 a N b N U , V − Haar–random in Unitary ( N ; R / C / H ) C = UAU ∗ + VBV ∗ տ ր uniformly random eigenvectors ւ ց ց or C = ( UAU ∗ ) · ( VBV ∗ ) or C = P k ( UAU ∗ ) P k Question: What can you say about eigenvalues of C ?
Hermitian matrix operations 0 0 a 1 b 1 0 a 2 0 0 b 2 0 A = B = ... ... 0 0 0 0 0 0 a N b N U , V − Haar–random in Unitary ( N ; R / C / H ) C = UAU ∗ + VBV ∗ տ ր uniformly random eigenvectors ւ ց ց or C = ( UAU ∗ ) · ( VBV ∗ ) or C = P k ( UAU ∗ ) P k Question: What can you say about eigenvalues of C ? I) As β (= 1 , 2 , 4) → ∞ II) As N → ∞ III) In discretization
As β (= 1 , 2 , 4) → ∞ Theorem. (Gorin–Marcus–17) Eigenvalues of C crystallize (=they become deterministic) as β → ∞ : β →∞ C = UAU ∗ + VBV ∗ lim N N ( z − c i ) = 1 � � � ( z − a i − b σ ( i ) ) N ! i =1 i =1 σ ∈ S ( N ) β →∞ C = ( UAU ∗ ) · ( VBV ∗ ) lim N N ( z − c i ) = 1 � � � ( z − a i b σ ( i ) ) N ! i =1 i =1 σ ∈ S ( N ) β →∞ C = P k ( UAU ∗ ) P k lim k N ( z − c i ) ∼ ∂ N − k � � ( z − a i ) ∂ z N − k i =1 i =1 Finite free convolutions and projection
As N → ∞ Theorem. (Voiculescu, 80s) At β = 1 , 2 empirical measure of eigenvalues of C becomes deterministic as N → ∞ . N N N 1 1 1 � � � µ A = lim δ a i µ B = lim δ b i µ C = lim δ c i N N N N →∞ N →∞ N →∞ i =1 i =1 i =1 � µ ( dx ) G µ ( z ) = z − x � − 1 − 1 z � − 1 � � R µ ( z ) = G µ ( z ) z , S µ ( z ) = 1 − zG µ ( z ) 1 + z N →∞ C = UAU ∗ + VBV ∗ lim R µ C ( z ) = R µ A ( z ) + R µ B ( z ) N →∞ C = ( UAU ∗ ) · ( VBV ∗ ) lim S µ C ( z ) = S µ A ( z ) · S µ B ( z ) N →∞ C = P k ( UAU ∗ ) P k R µ C ( z ) = N lim k R µ A ( z ) Free convolutions and projection
As N → ∞ Theorem. (Voiculescu, 80s) At β = 1 , 2 empirical measure of eigenvalues of C becomes deterministic as N → ∞ . N N N 1 1 1 � � � µ A = lim δ a i µ B = lim δ b i µ C = lim δ c i N N N N →∞ N →∞ N →∞ i =1 i =1 i =1 � µ ( dx ) G µ ( z ) = z − x � − 1 − 1 z � − 1 � � R µ ( z ) = G µ ( z ) z , S µ ( z ) = 1 − zG µ ( z ) 1 + z N →∞ C = UAU ∗ + VBV ∗ lim R µ C ( z ) = R µ A ( z ) + R µ B ( z ) N →∞ C = ( UAU ∗ ) · ( VBV ∗ ) lim S µ C ( z ) = S µ A ( z ) · S µ B ( z ) N →∞ C = P k ( UAU ∗ ) P k R µ C ( z ) = N lim k R µ A ( z ) Free convolutions and projection Conjecture. Same is true for any fixed β > 0 .
Discretization T λ irreducible (linear) representations of U ( N ; C ) λ 1 > λ 2 > · · · > λ N , λ i ∈ Z . � c κ T λ ⊗ T ν = λ,ν T κ κ Littlewood–Richardson coefficients c κ λ,ν intractable as N → ∞ .
Discretization T λ irreducible (linear) representations of U ( N ; C ) λ 1 > λ 2 > · · · > λ N , λ i ∈ Z . � c κ T λ ⊗ T ν = λ,ν T κ κ Littlewood–Richardson coefficients c κ λ,ν intractable as N → ∞ . dim( T κ ) c κ λ,ν Random κ through P ( κ ) = . dim T λ · dim T ν Semi-classical limit degenerates representations of a Lie group into orbital measures on its Lie algebra → UAU ∗ + VBV ∗ T λ ⊗ T ν −
Discretization Theorem. (Gorin–Bufetov–13; following Biane in 90s) Empirical measure of ν becomes deterministic as N → ∞ . dim( T κ ) c κ λ,ν � c κ T λ ⊗ T ν = λ,ν T κ , P ( κ ) = dim T λ · dim T ν κ N N 1 � λ i � 1 � κ i � � � µ λ = lim δ , µ κ = lim δ N N N N N →∞ N →∞ i =1 i =1 տ ր Scaling is important! � µ ( dx ) � − 1 1 � G µ ( z ) = z − x , R quant ( z ) = G µ ( z ) − µ 1 − exp( − z ) R quant ( z ) = R quant ( z ) + R quant ( z ) µ κ µ λ µ ν Quantized free convolution
Discretization Theorem. (Gorin–Bufetov–13; following Biane in 90s) Empirical measure of ν becomes deterministic as N → ∞ . dim( T κ ) c κ λ,ν � c κ T λ ⊗ T ν = λ,ν T κ , P ( κ ) = dim T λ · dim T ν κ N N 1 � λ i � 1 � κ i � � � µ λ = lim δ , µ κ = lim δ N N N N N →∞ N →∞ i =1 i =1 տ ր Scaling is important! � µ ( dx ) � − 1 1 � G µ ( z ) = z − x , R quant ( z ) = G µ ( z ) − µ 1 − exp( − z ) R quant ( z ) = R quant ( z ) + R quant ( z ) µ κ µ λ µ ν Quantized free convolution Restrictions to smaller subgroups lead to projection .
Zoo of operations T λ ⊗ T ν = � c κ λ,ν T κ C = ( UAU ∗ ) · ( V BV ∗ ) U ( k ) = � c κ A, B ≈ 1 � T λ λ T κ � semi-classical semi-classical B = P k C = UAU ∗ + V BV ∗ β → ∞ C = P k ( UAU ∗ ) P k β → ∞ β → ∞ � ( z − c i ) = 1 � � ( z − a i b σ ( i ) ) N ! � ( z − c i ) = 1 � � ( z − a i − b σ ( i ) ) N ! A, B ≈ 1 B = P k ∂ N − k � ( z − c i ) ∼ � ( z − a i ) R quant ( z ) = R quant ( z ) + R quant ∂z N − k ( z ) µ κ µ λ µ ν semi-classical S µ C ( z ) = S µ A ( z ) · S µ B ( z ) N → ∞ N → ∞ N → ∞ R quant ( z ) = N k R quant ( z ) µ κ µ λ R µ C ( z ) = R µ A ( z ) + R µ B ( z ) R µ C ( z ) = N k R µ A ( z ) semi-classical A, B ≈ 1 Operations on matrices and representations lead to a variety of Laws of Large Numbers , resulting in convolutions . They are all cross-related by limit transitions.
Zoo of operations T λ ⊗ T ν = � c κ λ,ν T κ C = ( UAU ∗ ) · ( V BV ∗ ) U ( k ) = � c κ A, B ≈ 1 � T λ λ T κ � semi-classical semi-classical B = P k C = UAU ∗ + V BV ∗ β → ∞ C = P k ( UAU ∗ ) P k β → ∞ β → ∞ � ( z − c i ) = 1 � � ( z − a i b σ ( i ) ) N ! � ( z − c i ) = 1 � � ( z − a i − b σ ( i ) ) N ! A, B ≈ 1 B = P k ∂ N − k � ( z − c i ) ∼ � ( z − a i ) R quant ( z ) = R quant ( z ) + R quant ∂z N − k ( z ) µ κ µ λ µ ν semi-classical S µ C ( z ) = S µ A ( z ) · S µ B ( z ) N → ∞ N → ∞ N → ∞ R quant ( z ) = N k R quant ( z ) µ κ µ λ R µ C ( z ) = R µ A ( z ) + R µ B ( z ) R µ C ( z ) = N k R µ A ( z ) semi-classical A, B ≈ 1 Operations on matrices and representations lead to a variety of Laws of Large Numbers , resulting in convolutions . They are all cross-related by limit transitions. We present a unifying framework for the operations.
Matrix corners The most explicit case. N × N matrix UAU ∗ a 1 a 2 a 3 a 4 M 11 M 12 M 13 M 14 x 3 x 3 x 3 1 2 3 M 21 M 22 M 23 M 24 M 31 M 32 M 33 M 34 x 2 x 2 1 2 M 41 M 42 M 43 M 44 x 1 1 Theorem. (Gelfand–Naimark–50s; Baryshnikov, Neretin – 00s) With ( x N 1 , . . . , x N N ) = ( a 1 , . . . , a N ) , the joint law of particles is N − 1 k k +1 � � � � ( x k i − x k j ) 2 − β | x k a − x k +1 | β/ 2 − 1 b a =1 k =1 1 ≤ i < j ≤ k b =1 • A basis of extension from β = 1 , 2 , 4 to general β > 0 . • Consistent with (Hermite/Laguerre/Jacobi) β log–gases
Matrix corners N × N matrix UAU ∗ a 1 a 2 a 3 a 4 M 11 M 12 M 13 M 14 x 3 x 3 x 3 M 21 M 22 M 23 M 24 1 2 3 x 2 x 2 M 31 M 32 M 33 M 34 1 2 M 41 M 42 M 43 M 44 x 1 1 N − 1 k k +1 � � ( x k i − x k j ) 2 − β � � | x k a − x k +1 | β/ 2 − 1 b k =1 1 ≤ i < j ≤ k a =1 b =1 Multivariate Bessel Function N k k − 1 � � � x k − 1 x k B a 1 ,..., a N ( z 1 , . . . , z N ) = E exp z k · i − j k =1 i =1 j =1
Matrix corners N × N matrix UAU ∗ a 1 a 2 a 3 a 4 M 11 M 12 M 13 M 14 x 3 x 3 x 3 M 21 M 22 M 23 M 24 1 2 3 x 2 x 2 M 31 M 32 M 33 M 34 1 2 M 41 M 42 M 43 M 44 x 1 1 N − 1 k k +1 � � ( x k i − x k j ) 2 − β � � | x k a − x k +1 | β/ 2 − 1 b k =1 1 ≤ i < j ≤ k a =1 b =1 Multivariate Bessel Function N k k − 1 � � � x k − 1 x k B a 1 ,..., a N ( z 1 , . . . , z N ) = E exp z k · i − j k =1 i =1 j =1 ր Diagonal matrix elements at β = 1 , 2 , 4 Proposition. B a 1 ,..., a N ( z 1 , . . . , z N ) is symmetric in z 1 , . . . , z N .
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