Convolutions and fluctuations: free, finite, quantized.

Transcription

1 Convolutions and fluctuations: free, finite, quantized. Vadim Gorin MIT (Cambridge) and IITP (Moscow) February 2018

2 Hermitian matrix operations a 1 0 b a b 2 0 A =. 0.. B = a N 0 b N U, V Haar random in Unitary(N; R / C / H) C = UAU + VBV uniformly random eigenvectors

3 Hermitian matrix operations a a 2 0 A = a N b b 2 0 B = b N U, V Haar random in Unitary(N; R / C / H) C = UAU + VBV uniformly random eigenvectors or C = (UAU ) (VBV ) or C = P k (UAU )P k Question: What can you say about eigenvalues of C?

4 Hermitian matrix operations a a 2 0 A = a N b b 2 0 B = b N U, V Haar random in Unitary(N; R / C / H) C = UAU + VBV uniformly random eigenvectors 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

5 As β(= 1, 2, 4) Theorem. (Gorin Marcus 17) Eigenvalues of C crystallize (=they become deterministic) as β : lim β C = UAU + VBV N (z c i ) = 1 N! lim C = β (UAU ) (VBV ) N (z c i ) = 1 N! lim C = P k(uau )P k β k (z c i ) N k σ S(N) σ S(N) z N k N (z a i b σ(i) ) N (z a i b σ(i) ) N (z a i ) Finite free convolutions and projection

6 As N Theorem. (Voiculescu, 80s) At β = 1, 2 empirical measure of eigenvalues of C becomes deterministic as N. 1 N 1 N µ A = lim δ ai µ B = lim δ bi N N N N µ(dx) G µ (z) = z x R µ (z) = ( ) 1 1 G µ (z) z, S µ(z) = z 1 + z µ C = lim N 1 N N ( ) 1 1 zg µ (z) lim N C = UAU + VBV R µc (z) = R µa (z) + R µb (z) lim N C = (UAU ) (VBV ) S µc (z) = S µa (z) S µb (z) lim k(uau )P k R µc (z) = N N k R µ A (z) Free convolutions and projection δ ci

7 As N Theorem. (Voiculescu, 80s) At β = 1, 2 empirical measure of eigenvalues of C becomes deterministic as N. 1 N 1 N µ A = lim δ ai µ B = lim δ bi N N N N µ(dx) G µ (z) = z x R µ (z) = ( ) 1 1 G µ (z) z, S µ(z) = z 1 + z µ C = lim N 1 N N ( ) 1 1 zg µ (z) lim N C = UAU + VBV R µc (z) = R µa (z) + R µb (z) lim N C = (UAU ) (VBV ) S µc (z) = S µa (z) S µb (z) lim k(uau )P k R µc (z) = N N k R µ A (z) Free convolutions and projection Conjecture. Same is true for any fixed β > 0. δ ci

8 Discretization T λ irreducible (linear) representations of U(N; C) λ 1 > λ 2 > > λ N, λ i Z. T λ T ν = κ c κ λ,ν T κ Littlewood Richardson coefficients cλ,ν κ intractable as N.

9 Discretization T λ irreducible (linear) representations of U(N; C) λ 1 > λ 2 > > λ N, λ i Z. T λ T ν = κ c κ λ,ν T κ Littlewood Richardson coefficients c κ λ,ν intractable as N. Random κ through P(κ) = dim(t κ)c κ λ,ν dim T λ dim T ν. Semi-classical limit degenerates representations of a Lie group into orbital measures on its Lie algebra T λ T ν UAU + VBV

10 Discretization Theorem. (Gorin Bufetov 13; following Biane in 90s) Empirical measure of ν becomes deterministic as N. T λ T ν = κ 1 N µ λ = lim δ N N G µ (z) = µ(dx) z x, c κ λ,ν T κ, P(κ) = dim(t κ)c κ λ,ν dim T λ dim T ν ( λi N ) 1, µ κ = lim N N N δ ( κi Scaling is important! ( ) 1 1 Rquant µ (z) = G µ (z) 1 exp( z) R quant µ κ (z) = R quant µ λ (z) + R quant µ ν (z) Quantized free convolution N )

11 Discretization Theorem. (Gorin Bufetov 13; following Biane in 90s) Empirical measure of ν becomes deterministic as N. T λ T ν = κ 1 N µ λ = lim δ N N G µ (z) = µ(dx) z x, c κ λ,ν T κ, P(κ) = dim(t κ)c κ λ,ν dim T λ dim T ν ( λi N ) 1, µ κ = lim N N N δ ( κi Scaling is important! ( ) 1 1 Rquant µ (z) = G µ (z) 1 exp( z) R quant µ κ (z) = R quant µ λ (z) + R quant µ ν (z) Quantized free convolution Restrictions to smaller subgroups lead to projection. N )

12 Zoo of operations T λ T ν = c κ λ,ν Tκ semi-classical T λ U(k) = c κ λ Tκ A, B 1 semi-classical C = (UAU ) (V BV ) C = UAU + V BV C = P k(uau )P k B = Pk β β (z ci) = 1 N! (z ai b σ(i) ) A, B 1 β (z ci) = 1 N! (z aib σ(i) ) B = Pk R quant µκ (z) = R quant µλ (z) + R quant µν (z) semi-classical N R µc (z) = R µa (z) + R µb (z) N (z ci) N k z N k (z ai) R quant µκ (z) = N k Rquant µλ (z) S µc (z) = S µa (z) S µb (z) A, B 1 semi-classical N R µc (z) = N k RµA(z) 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.

13 Zoo of operations T λ T ν = c κ λ,ν Tκ semi-classical T λ U(k) = c κ λ Tκ A, B 1 semi-classical C = (UAU ) (V BV ) C = UAU + V BV C = P k(uau )P k B = Pk β β (z ci) = 1 N! (z ai b σ(i) ) A, B 1 β (z ci) = 1 N! (z aib σ(i) ) B = Pk R quant µκ (z) = R quant µλ (z) + R quant µν (z) semi-classical N R µc (z) = R µa (z) + R µb (z) N (z ci) N k z N k (z ai) R quant µκ (z) = N k Rquant µλ (z) S µc (z) = S µa (z) S µb (z) A, B 1 semi-classical N R µc (z) = N k RµA(z) 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.

14 N N matrix UAU M 11 M 12 M 13 M 14 M 21 M 22 M 23 M 24 M 31 M 32 M 33 M 34 M 41 M 42 M 43 M 44 Matrix corners The most explicit case. a 1 a 2 a 3 a 4 x 3 1 x 3 2 x 3 3 x 2 1 x 2 2 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=1 1 i<j k (x k i x k j ) 2 β k k+1 a=1 b=1 x 1 1 xa k x k+1 b β/2 1 A basis of extension from β = 1, 2, 4 to general β > 0. Consistent with (Hermite/Laguerre/Jacobi) β log gases

15 N N matrix UAU M 11 M 12 M 13 M 14 M 21 M 22 M 23 M 24 M 31 M 32 M 33 M 34 M 41 M 42 M 43 M 44 N 1 k=1 1 i<j k Matrix corners (x k i x k j ) 2 β k a=1 b=1 a 1 a 2 a 3 a 4 k+1 x 3 1 x 3 2 x 3 3 x 2 1 x 2 2 Multivariate Bessel Function N k B a1,...,a N (z 1,..., z N ) = E exp z k k=1 x 1 1 xa k x k+1 b β/2 1 x k i k 1 j=1 x k 1 j

16 N N matrix UAU M 11 M 12 M 13 M 14 M 21 M 22 M 23 M 24 M 31 M 32 M 33 M 34 M 41 M 42 M 43 M 44 N 1 k=1 1 i<j k Matrix corners (x k i x k j ) 2 β k a=1 b=1 a 1 a 2 a 3 a 4 k+1 x 3 1 x 3 2 x 3 3 x 2 1 x 2 2 Multivariate Bessel Function N k B a1,...,a N (z 1,..., z N ) = E exp z k k=1 x 1 1 xa k x k+1 b β/2 1 x k i k 1 j=1 x k 1 j Diagonal matrix elements at β = 1, 2, 4 Proposition. B a1,...,a N (z 1,..., z N ) is symmetric in z 1,..., z N.

17 N N matrix UAU M 11 M 12 M 13 M 14 M 21 M 22 M 23 M 24 M 31 M 32 M 33 M 34 M 41 M 42 M 43 M 44 N 1 k=1 1 i<j k Matrix corners (x k i x k j ) 2 β k a=1 b=1 a 1 a 2 a 3 a 4 k+1 x 3 1 x 3 2 x 3 3 x 2 1 x 2 2 Multivariate Bessel Function N k B a1,...,a N (z 1,..., z N ) = E exp z k k=1 x 1 1 xa k x k+1 b β/2 1 x k i k 1 j=1 Laplace transform for full matrix at β = 1, 2, 4. B a1,...,a N (z 1,..., z N ) = E exp [Trace(UAU Z)], x k 1 j Z {z i } eigenvalues

18 N N matrix UAU M 11 M 12 M 13 M 14 M 21 M 22 M 23 M 24 M 31 M 32 M 33 M 34 M 41 M 42 M 43 M 44 Matrix corners Multivariate Bessel Function N k B a1,...,a N (z 1,..., z N ) = E exp z k k=1 (Harish Chandra; Itzykson Zuber) At β = 2: a 1 a 2 a 3 a 4 x 3 1 x 3 2 x 3 3 x 2 1 x 2 2 x k i x 1 1 k 1 j=1 B a1,...,a N (z 1,..., z N ) det[ exp(a i z j ) ] N i,j=1 (a i a j )(z i z j ) i<j x k 1 j

19 [ N N matrix UAU M 11 M 12 M 13 M 14 M 21 M 22 M 23 M 24 M 31 M 32 M 33 M 34 M 41 M 42 M 43 M 44 Matrix corners Multivariate Bessel Function N k B a1,...,a N (z 1,..., z N ) = E exp z k k=1 a 1 a 2 a 3 a 4 x 3 1 x 3 2 x 3 3 x 2 1 x 2 2 x k i x 1 1 k 1 j=1 x k 1 j B a1,...,a N (z 1,..., z N ) ] (z i z j ) β/2 is an eigenfunction of z 2 i i<j rational Calogero Sutherland Hamiltonian for any β > 0 N 2 β(2 β) (z i z j ) 2 i<j

20 N N matrix UAU M 11 M 12 M 13 M 14 M 21 M 22 M 23 M 24 M 31 M 32 M 33 M 34 M 41 M 42 M 43 M 44 Matrix corners k=1 a 1 a 2 a 3 a 4 x 3 1 x 3 2 x 3 3 x 2 1 x 2 2 N k B a1,...,a N (z 1,..., z N ) = E exp z k x 1 1 x k i k 1 x k 1 j j=1 The probability law of interest is readily reconstructed: B a1,...,a N (z 1,..., z k, 0,..., 0) = B x k 1,x2 k (z,...,xk 1,..., z k )P(dx1 k, dx2 k,..., dxk k ) k

21 N N matrix UAU M 11 M 12 M 13 M 14 M 21 M 22 M 23 M 24 M 31 M 32 M 33 M 34 M 41 M 42 M 43 M 44 Matrix corners k=1 a 1 a 2 a 3 a 4 x 3 1 x 3 2 x 3 3 x 2 1 x 2 2 N k B a1,...,a N (z 1,..., z N ) = E exp z k x 1 1 x k i k 1 x k 1 j j=1 The probability law of interest is readily reconstructed: B a1,...,a N (z 1,..., z k, 0,..., 0) = B x k 1,x2 k (z,...,xk 1,..., z k )P(dx1 k, dx2 k,..., dxk k ) k Equivalently, we have a family of observables EB x k 1,x2 k (z,...,xk 1,..., z k ) = B a1,...,a k N (z 1,..., z k, 0,..., 0)

22 N N matrix UAU M 11 M 12 M 13 M 14 M 21 M 22 M 23 M 24 M 31 M 32 M 33 M 34 M 41 M 42 M 43 M 44 Matrix addition + N k B a1,...,a N (z 1,..., z N ) = E exp z k k=1 k=1 N N matrix VBV M 11 M 12 M 13 M 14 M 21 M 22 M 23 M 24 M 31 M 32 M 33 M 34 M 41 M 42 M 43 M 44 N k B b1,...,b N (z 1,..., z N ) = E exp z k C = UAU + VBV Question: EB c1,...,c N (z 1,..., z N ) =? x k i y k i k 1 x k 1 j j=1 k 1 y k 1 j j=1

23 N N matrix UAU M 11 M 12 M 13 M 14 M 21 M 22 M 23 M 24 M 31 M 32 M 33 M 34 M 41 M 42 M 43 M 44 Matrix addition + N k B a1,...,a N (z 1,..., z N ) = E exp z k k=1 k=1 N N matrix VBV M 11 M 12 M 13 M 14 M 21 M 22 M 23 M 24 M 31 M 32 M 33 M 34 M 41 M 42 M 43 M 44 N k B b1,...,b N (z 1,..., z N ) = E exp z k C = UAU + VBV x k i y k i k 1 x k 1 j j=1 k 1 y k 1 j j=1 EB c1,...,c N (z 1,..., z N ) = B a1,...,a N (z 1,..., z N ) B b1,...,b N (z 1,..., z N )

24 N N matrix UAU M 11 M 12 M 13 M 14 M 21 M 22 M 23 M 24 M 31 M 32 M 33 M 34 M 41 M 42 M 43 M 44 Matrix addition + N k B a1,...,a N (z 1,..., z N ) = E exp z k k=1 k=1 N N matrix VBV M 11 M 12 M 13 M 14 M 21 M 22 M 23 M 24 M 31 M 32 M 33 M 34 M 41 M 42 M 43 M 44 N k B b1,...,b N (z 1,..., z N ) = E exp z k C = UAU + VBV x k i y k i k 1 x k 1 j j=1 k 1 y k 1 j j=1 EB c1,...,c N (z 1,..., z N ) = B a1,...,a N (z 1,..., z N ) B b1,...,b N (z 1,..., z N ) Any β > 0! Caveat: positivity in E is open outside β = 1, 2, 4.

25 Operations through symmetric functions qc = UAU + VBV C = P k (UAU )P k C = (UAU ) (VBV ) Multivariate Bessel EB C ( z) = B A ( z) B B ( z) Multivariate Bessel EB C ( z) = B A ( z, 0 N k ) Heckman Opdam hypergeometric EHO C ( z) = HO A ( z) HO B ( z) T λ T ν = κ T λ U(k) = κ c κ λ,ν T κ c κ λ T κ Schur polynomials ] E [ sκ( z) s κ(1 N ) = s λ( z) s λ (1 N ) Schur polynomials ] E [ sκ( z) s κ(1 k ) = s λ( z,1 N k ) s λ (1 N ) s ν( z) s ν(1 N )

26 Operations through symmetric functions qc = UAU + VBV C = P k (UAU )P k C = (UAU ) (VBV ) Multivariate Bessel EB C ( z) = B A ( z) B B ( z) Multivariate Bessel EB C ( z) = B A ( z, 0 N k ) Heckman Opdam hypergeometric EHO C ( z) = HO A ( z) HO B ( z) T λ T ν = κ c κ λ,ν T κ Schur polynomials ] E [ sκ( z) s κ(1 N ) = s λ( z) s λ (1 N ) s ν( z) s ν(1 N ) T λ U(k) = κ c κ λ T κ Schur polynomials ] E [ sκ( z) s κ(1 k ) = s λ( z,1 N k ) s λ (1 N ) These all are degenerations of Macdonald polynomials.

27 Macdonald polynomials P λ (x 1,..., x N ; q, t): homogenous, symmetric, leading term N [T i;q f ](x 1,..., x N ) = f (x 1,..., x i 1, qx i, x i+1,..., x N ) x λ i i D = N j i tx i x j x i x j T i;q [ N ] DP λ (x 1,..., x N ; q, t) = q λ i t N i P λ (x 1,..., x N ; q, t)

28 Macdonald polynomials P λ (x 1,..., x N ; q, t): homogenous, symmetric, leading term N [T i;q f ](x 1,..., x N ) = f (x 1,..., x i 1, qx i, x i+1,..., x N ) x λ i i D = N j i tx i x j x i x j T i;q [ N ] DP λ (x 1,..., x N ; q, t) = q λ i t N i P λ (x 1,..., x N ; q, t) P λ (x 1,..., x N ) P ν (x 1,..., x N ) P λ (1, t,..., t N 1 ) P ν (1, t,..., t N 1 ) = κ c κ λ,ν P κ (x 1,..., x N ) P κ (1, t,..., t N 1 ) Conjecture. cλ,ν κ 0 whenever 0 < q, t < 1. Known cases: q = t; q = 0; t = 0; q 1, t = q β/2, β = 1, 2, 4.

29 Macdonald polynomials P λ (x 1,..., x N ) P ν (x 1,..., x N ) P λ (1, t,..., t N 1 ) P ν (1, t,..., t N 1 ) = κ c κ λ,ν P κ (x 1,..., x N ) P κ (1, t,..., t N 1 ) Tλ Tν = c κ λ,ν Tκ semi-classical Tλ U(k) = c κ λ Tκ A, B 1 semi-classical C = (UAU ) (V BV ) C = UAU + V BV C = Pk(UAU )Pk B = Pk β β (z ci) = 1 N! (z ai b σ(i)) A, B 1 β (z ci) = 1 N! (z aib σ(i)) B = Pk R quant µκ (z) = R quant µλ (z) + R quant µν (z) (z ci) N k z N k (z ai) semi-classical N (z) = RµA(z) + RµB(z) RµC N R quant µκ (z) = N k Rquant µλ (z) (z) = SµA(z) SµB(z) SµC A, B 1 semi-classical N RµC (z) = N k RµA(z) κ cκ λ,ν = 1 and they define a distribution on κ s. Everything we saw so far is a limit of this distribution.

30 Macdonald polynomials P λ (x 1,..., x N ) P ν (x 1,..., x N ) P λ (1, t,..., t N 1 ) P ν (1, t,..., t N 1 ) = κ c κ λ,ν P κ (x 1,..., x N ) P κ (1, t,..., t N 1 ) Our analysis of cλ,ν κ random κ: 1. Direct combinatorial limits of Macdonald polynomials. 2. Applying difference/differential operators in x 1,..., x N to the defining identity. 3. Contour integral expressions for P λ(x 1,1,t,...,t N 2 ) P λ (1,t,...,t N 1 ). Then N analysis of the most general Macdonald case is open.

31 Beyond the Law of Large Numbers P λ (x 1,..., x N ) P ν (x 1,..., x N ) P λ (1, t,..., t N 1 ) P ν (1, t,..., t N 1 ) = κ c κ λ,ν P κ (x 1,..., x N ) P κ (1, t,..., t N 1 ) Information is lost by considering deterministic limits of random κ. What can be recovered?

32 Matrix corners again Back to the most explicit case. N N matrix UAU M 11 M 12 M 13 M 14 M 21 M 22 M 23 M 24 M 31 M 32 M 33 M 34 M 41 M 42 M 43 M 44 a 1 a 2 a 3 a 4 x 3 1 x 3 2 x 3 3 x 2 1 x 2 2 With (x N 1,..., x N N ) = (a 1,..., a N ), the joint law of particles is N 1 k=1 1 i<j k (x k i x k j ) 2 β k k+1 a=1 b=1 What is happening as β? x 1 1 xa k x k+1 b β/2 1

33 N N matrix UAU M 11 M 12 M 13 M 14 M 21 M 22 M 23 M 24 M 31 M 32 M 33 M 34 M 41 M 42 M 43 M 44 Matrix corners again a 1 a 2 a 3 a 4 x 3 1 x 3 2 x 3 3 x 2 1 x 2 2 With (x N 1,..., x N N ) = (a 1,..., a N ), the joint law of particles is N 1 k=1 1 i<j k N 1 k=1 1 i<j k (x k i x k j ) 2 β k k+1 a=1 b=1 As β, particles maximize (x k i x k j ) 2 k k+1 a=1 b=1 x 1 1 xa k x k+1 b β/2 1 xa k x k+1 b max

34 N N matrix UAU M 11 M 12 M 13 M 14 M 21 M 22 M 23 M 24 M 31 M 32 M 33 M 34 M 41 M 42 M 43 M 44 N 1 k=1 1 i<j k Matrix corners again As β, particles maximize (x k i x k j ) 2 k k+1 a=1 b=1 a 1 a 2 a 3 a 4 x 3 1 x 3 2 x 3 3 x 2 1 x 2 2 x 1 1 xa k x k+1 b max Proposition. The optimal configuration is given by k (z x i k ) N k z N k N (z a j ), k = 1, 2,..., N. j=1

35 Matrix corners again With (x N 1,..., x N N ) = (a 1,..., a N ), the joint law of particles is N 1 k=1 1 i<j k k (z x i k ) N k (x k i x k j ) 2 β k z N k j=1 k+1 a=1 b=1 xa k x k+1 b β/2 1 N (z a j ), k = 1, 2,..., N. Proposition. ξj i := lim β(x i j x j i ) has Gaussian density β N 1 exp k=1 1 i<j k (ξi k ξj k)2 k 2(xi k xj k )2 k+1 a=1 b=1 (ξ k a ξ k+1 b ) 2 4(x k a x k+1 b ) 2 A version of discrete Gaussian Free Field..

36 Central Limit Theorems for fluctuations 1. lim [C = P k(uau )P k ]. (Gorin Marcus 17): discrete GFF β 2. lim [C = β (UAU )(+ or )(VBV )]. Open problem 3. lim N [ T λ U(k) = κ c κ λ T κ ]. (Petrov 12, Bufetov Gorin 15): Fluctuations are given by Gaussian Free Field (= covariance given by the Green function of the Laplace operator in suitable complex structure); bijection with random lozenge tilings. 4. lim [C = P k(uau )P k ]. Same methods and results apply. N 5. lim [C = N UAU + VBV ]. (Collins Mingo Sniady Speicher 04) Second order freeness. Covariance similar to GFF (??). 6. lim [C = N (UAU ) (VBV )]. (Vasilchuk 16) Similar (??). 7. Conjecture: [ The last 3 answers extend to general β > lim T λ T ν = ] cλ,ν κ T κ. (Bufetov Gorin 15) CLT with N κ covariance similar to GFF. Conceptual explanation?

37 N N matrix UAU M 11 M 12 M 13 M 14 M 21 M 22 M 23 M 24 M 31 M 32 M 33 M 34 M 41 M 42 M 43 M 44 The simplest open problem a 1 a 2 a 3 a 4 x 3 1 x 3 2 x 3 3 x 2 1 x 2 2 With (x N 1,..., x N N ) = (a 1,..., a N ), the joint law of particles is N 1 k=1 1 i<j k (x k i x k j ) 2 β k k+1 a=1 b=1 Fix arbitrary β > 0 and send N x 1 1 xa k x k+1 b β/2 1 Conjecture. The Law of Large Numbers is β independent. Conjecture. The Central Limit Theorem as N is β independent after rescaling by β. Difficulty: Negative powers in the interaction.

38 Summary 1. Operations on matrices and representations possess Laws of Large Numbers as N or β leading to various convolutions. 2. Fluctuations are described by Gaussian fields. Whenever formula less identification is made, it is GFF. 3. Macdonald polynomials multiplication is the mother of all. 4. Many open questions remain! Tλ Tν = c κ λ,ν Tκ semi-classical Tλ U(k) = c κ λ Tκ A, B 1 semi-classical C = (UAU ) (V BV ) C = UAU + V BV C = Pk(UAU )Pk B = Pk β β (z ci) = 1 N! (z ai b σ(i)) A, B 1 β (z ci) = 1 N! (z aib σ(i)) B = Pk R quant µκ (z) = R quant µλ (z) + R quant µν (z) (z ci) N k z N k (z ai) semi-classical N (z) = RµA(z) + RµB(z) RµC N R quant µκ (z) = N k Rquant µλ (z) (z) = SµA(z) SµB(z) SµC A, B 1 semi-classical N RµC (z) = N k RµA(z)