Multiplication law and S-transform for non-hermitian random matrices

Transcription

1 and S-transform for non-hermitian random matrices (in collaboration with Z. Burda and R.A. Janik, math-ph/ ) Mark Kac Complex Systems Research Center, Marian Smoluchowski Institute of Physics, Jagiellonian University, Kraków, Poland ESI workshop on Bialgebras in Free Probability April 5th, 20, Wien

2 Outline Classical probability calculus Addition law 2 Random matrices with one dimensional spectra Addition law (Hermitian ensembles) (Unitary ensembles) 3 Non-hermitian random matrices (two-dimensional spectra) 4 Summary Addition law

3 Probability: Addition law Knowing independent p a (x a ) and p b (x b ), we want to infer p(s), where s = x a + x b p(s) = dx a dx b p a (x a )p b (x b )δ(s (x a + x b )) = dxpa (x)p b (s x) Fourier transform f (k) = dke ikx p(x) = (ik) n n=0 n! p(x)x n dk = (ik) n n=0 n! m n unravels the convolution, f (k) = f a (k) f b (k) r(k) ln f (k) = (ik) n n= n! c n generates cumulants Addition law r(k) = r a (k) + r b (k), i.e. c n = c (a) n + c (b) n Ex.: Gaussian pdf p(x) = 2π e x2 2 N(0, ) ( only c 2 0) N a (0, ) N b (0, ) = N(0, 2)

4 Probability: Knowing independent p a (x a ) and p b (x b ), we want to infer p(r), where r = x a x b p(r) = dx a dx b p a (x a )p b (x b )δ(r x a x b ) = dx x p a( r x )p b(x) Mellin transform m(t) = dx 0 x x t p(x) factorizes above integral m(t) = m a (t) m b (t) Technical modification for negative random variables m(t) = m ++ (t) + m + (t) + m + (t) + m (t) [Epstein;948] Inverse Mellin transform p(r) = Γ m(t)t r dt yields the result Ex.: N a (0, ) N b (0, ) = π K 0(x) (MacDonald function), lim x 0 K 0 (x) ln x, lim x K 0 (x) π 2 x e x

5 RMT: Matrix-valued probability calculus (real eigenvalues) dµ(x ) P(X )dx = e NtrV (X ) dx Key question: statistics of the spectra of X, e.g. ρ(λ) = N trδ(λ X ) dµ({λ i }) = C β N e N j V (λ j ) i<j (λ i λ j ) β j dλ j, eigenvalues interact with each other (β = (2) for real(complex) matrices). Large N simplifications: theoretical (planar graphs dominate in the guise of t Hooft expansion), practical (8 ) Complex Green s function generates spectral moments M k : G(z) = N tr z N X = M k k=0 where z k+ M k = N trx k = ρ(λ)λ k dλ Analytic properties are crucial: ρ(λ) = π lim ɛ 0 IG(z) z=λ+iɛ

6 RMT: Matrix-valued probability calculus (real eigenvalues) What about spectral cumulants C k? R-transform [Voiculescu;986] R(z) = k= C kz k G(R(z) + z ) = z and R(G) + G = z Physicist proof : RMT is a QFT in dimensions, hence Z = dxe NtrV (X ) t Hooft double line notation for Wick expansion: Propagator < XX > /N, each vertex brings N, each loop brings N 2 Only planar Feynman graphs survive N limit 3 We define PI self energy Σ(z) as G(z) = z Σ(z) 4 Self-energy gets contributions from renormalized propagators /z G(z) and renormalized vertices C k, so Σ(z) = k= C kg(z) k = R(G(z)) (Schwinger-Dyson equation)

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8 R-transform: Addition law Knowing ρ A (λ) and ρ B (λ) for independent (free) RM ensembles, we want to infer ρ A+B (λ), i.e. calculate G A+B (z) = N dxa dx B P A (X A )P B (X B )tr z N (X A +X B ) Non-commutative convolution, since [X A, X B ] 0 Addition law: First, from the definition G(R(z) + z ) = z we read the corresponding transforms R A (z) and R B (z). Second we apply addition law R A+B (z) = R A (z) + R B (z). Third, we invert functionally R A+B (z) to get the desired result. For matrix analogue of the Gaussian distribution only C 2 0, i.e. R W (z) = C 2 z. For C 2 = /4 Green s function reads G W (z) = 2(z z 2 ), so ρ W (λ) = 2 π λ 2 W (0, ). Wigner semicircle Gaussian. Ex.: W A (0, ) W B (0, ) = W (0, 2), in analogy to the classical case.

9 S-transform: Knowing ρ A (λ) and ρ B (λ) for independent (free) RM ensembles, we want to infer ρ A B (λ), i.e. calculate G A B (z) = N dxa dx B P A (X A )P B (X B )tr z N (X A X B ) In general, for hermitian matrices, (H H 2 ) H H 2, so spectra are complex. For unitary random matrices, (U U 2 ) U U 2 =, spectra on the unit circle λ = e iθ, analytic methods applicable. S-transform [Voiculescu;987] S(z)G( +z z S(z)) = z : S A B (z) = S A (z) S B (z). S-transforms and R-transforms are related, alike Fourier and Mellin transforms are related. Alternative version for multiplication law [Janik;997] R A B (g) = R A (g B ) R B (g A ) where g A = gr A (g B ) and g B = R B (g A )g

10 S-transform - preliminary: relation to R-transform S(z) = +z z χ(z), where χ(zg(z) ) = z. If z yg(y), then S(yG(y) ) =. Since y G(y) G(y) = z Σ(y), we get S(G(y)Σ(y)) = Σ(y). Since Σ(y) = R(G(y)), we arrive (after taking reciprocals of both sides) at S(G(y)R(G(y)) = R(G(y)). Finally, changing variables again z = G(y) we arrive at R(z) = S(zR(z)). Note that S transform can be defined only if R(0) 0, (non-vanishing first moment) Last equation can be inverted: Let us define y = zr(z). Then S(y) = R( y R(z) ) = y R( y ), where z is recursively eliminated ) R( R(... ) ad infinitum S(y) = R(yS(y)). Mutually inverse maps z = ys(y) and y = zr(z)

11 S-transform - diagrammatics We consider ( 2N ) 2N block matrices ( ) 0 A (AB) H = H B 0 2k 2k 0 = 0 (BA) 2k ( ) G (w) G We define G(w) = 2 (w) = G 2 (w) G 22 (w) N tr b2 w H ( ) ( ) A B tr A tr B Block-trace definition tr b2 = C D tr C tr D Note that G AB (z = w 2 ) = N tr = G (w) z AB w ( ) Σ (w) Σ Similarly, we define Σ(w) = 2 (w), where Σ 2 (w) Σ 22 (w) G(w) = (w 2 Σ(w))

12 S-transform - alternative formulation From the ( flow of indices (H 2 A, H 2 ) B) we get 0 R Σ(w) = A (G 2 (w)) where R B (G 22 (w)) 0 G(w) = (w 2 Σ(w)) ( ) G (w) G 2 (w) = G(w) = (w G 2 (w) G 22 (w) 2 Σ(w)) = ( ) w R A (G 2 (w)) R B (G 22 (w)) w Inverting matrix we get (w 2 = z) G AB (z) = z Σ AB (z) = G (w) w = z R A (G 2 (w))r B (G 2 (w)) G 2 = G AB R A (G 2 ), G 2 = G AB R B (G 2 ) Using G AB (z) = z R AB (G AB ) we get the multiplication law.

13 Relation to canonical form of S G AB g, G 2 g A, G 2 g B Algorithm: Three equations with three complex variables: R A B (g) = R A (g B ) R B (g A ) where g A = gr A (g B ) and g B = R B (g A )g To unravel equations, we define y = gr AB (g). Then g B = gr B (g A ) = g R AB(g) R A (g B ) = y R A (g B ) = y y R A ( R A (...) ( ) ( ) ( )) y R AB ) y = R A ) y R B ( y R AB R AB (...) ( R y A R A (...) But one has to assume R i (0) 0 for i = A, B, AB ( ) R y B R B (...) Taking reciprocal of the above equation we arrive at S AB (y) = S A (y) S B (y) )

14 Non-hermitian case - electrostatic analogy (Dyson gas) Analytic methods break down, since spectra are complex ρ(z, z) = N i δ(2) (z λ i ). Potential Φ(z, z) = lim ɛ 0 lim N N tr ln[(z N X )( z N X ) + ɛ 2 N ] Poisson law 2 Φ z z = πρ(z, z), since δ(2) (z) = lim ɛ 0 π Electric field G(z, z) = Φ z Gauss law π zg(z, z) = ρ(z, z) [Brown;986],[Sommers,Crisanti,Sompolinsky,Stein;988] Bad news: G(z, z) = lim ɛ 0 lim N N tr z X (z N X )( z N X )+ɛ 2 No similarity to the hermitian case G(z) = N tr z N X ɛ 2 ( z 2 +ɛ 2 ) 2 Important in applications (dissipation, directed percolation, lagged correlations..), interesting in mathematics [Biane,Lehner;999]

15 Non-hermitian case - Remedy: tr ln A = ln det A ( tr ln[(z N X )( z N X ) + ɛ 2 zn X iɛ N ] = ln det N iɛ N z N X ) Duplication trick [Janik,MAN,Papp,Zahed;996],[Feinberg,Zee;997] ( ) ( ) A B tr A tr B tr b = C D tr C tr D ( ) ( z iɛ X 0 Z N = iɛ z N Z N X = 0 X 2 by 2 objects G(Z) = N tr b Z N X ( G G = G G ) )

16 Benefits of the duplication trick Self-energy Σ is a 2 by 2 matrix G(Z) = Z Σ(Z) Analogue of R-transform exists R(G) + G = Z Addition law holds R A+B (Z) = R A (Z) + R B (Z) Upper-left corner of G, i.e. G = G(z, z), so π zg = ρ(λ) Condition G G = 0 provides the equation for the boundary of eigenvalues Ex.: W a (0, ) i W b (0, ) = Ginibre-Girko ensemble (uniform distribution bounded by the circle z = 2)

17 Hidden algebraic structure unveiled Each generic 2 by 2( matrix Q ) which has appeared before has z i w the structure Q = iw z Q is a quaternion ( ) q0 + iq Q = q iσ i q i = 3 i(q iq 2 ) i(q + iq 2 ) q 0 iq 3 One can exploit the whole space of Q, instead of staying infinitesimaly close (ɛ) in transverse directions (, 2) Algorithm how to embed Greens functions and R-transforms in quaternion space for any hermitian H, any H + ih 2 [Jarosz,MAN;2004], any unitary U [Jarosz,Goerlich;2005], and several other cases. ( ) ( ) a i b 0 i b Examples: R GUE (Q) = Q, R G G = ib ā ib 0

18 Correspondence of the addition laws Hermitian case real spectra Green s function G(z) is complex G(R(z) + z ) = z Addition law R A+B (z) = R A (z) + R B (z) Analytic functions (cuts and poles as singularities) Non-hermitian case complex spectra Green s function G(Q) is a quaternion G(R(Q) + Q ) = Q Addition law R A+B (Q) = R A (Q) + R B (Q) Matrix-valued non-analytic functions

19 Does nonhermitian S transform exist? Despite doubts if such construction is possible at all, several recent results on products of random matrices were suggesting the possibility that such law may exist, e.g.: Nonhermitian diffusion [Janik,Jurkiewicz,Gudowska- Nowak,MAN;2003],[Lohmayer,Neuberger,Wettig;2008], [Warcho l;200] Products of centered complex matrices [Girko,Vladimirova;2009], [Burda, Janik, Wac law;200],[burda,jarosz,livan,man,świȩch;200] Time-lagged correlations [Biely,Thurner;2007],[Jarosz;200] Additive laws for unitary ensembles [Goerlich,Jarosz;2004] Multiplication for vanishing mean ensembles [Rao,Speicher;2007]

20 for non-hermitian random matrices reads [Burda,Janik,MAN;20a] : R A B (G A B ) = R L A (G B) R R B (G A) where G A = [ G A B R L A (G B) ] L and G B = [ R R B (G ] R A)G A B where for generic Q we have Q L = UQU (Q R = U QU) and U = e i φ 2 σ 3 2 with φ = Arg z. Note that order matters, since matrices (quaternions) do not commute. Three (matrix-valued) equations for three quaternion variables. In the case when [G, R] = 0, addition law gets reduced to S-transform by Voiculescu, i.e to the mutiplication law for complex functions R A B (G A B ) = R A (G B ) R B (G A ) where G A = G A B R A (G B ) and G B = R B (G A )G A B

21 Elements of the construction (I) We have product of random matrices and spectrum is complex, so we combine both duplication tricks: 0 A 0 0 H = B B 0 0 A 0 G(W) 4N 4N G G 2 G G 2 G 2 G 22 G 2 G 2 2 G G 2 G G 2 G 2 G 22 G 2 G 2 2 where W = diag(w, w, w, w) 4 4 = N tr b4 W H Similarly Σ(W) = R(G(W)) where G(W) = (W Σ(W)) are all 4 by 4 matrices

22 Elements of the construction (II) Flow of indices yields Σ = 0 Σ AA Σ A Ā 0 Σ BB 0 0 Σ B B Σ BB Σ ĀA Σ ĀĀ Σ B B 0 Above ( eight elements ) ( can be grouped ΣAA Σ AĀ RAA (G = B ) R (G ) AĀ B) Σ ĀA Σ ĀĀ R (G ĀA B) R (G = R A (G B ) ( ) ( ĀĀ B) ΣBB Σ B B RBB (G = A ) R B B(G ) A ) R BB (G A) R B B (G = R B (G A ) A) Σ BB Σ B B Matrices ( G A and G B ) are unknown ( (alike ) g A, g B ) G2 G G A = G2 G G B = 2 2 G 22 G 2 G G 2

23 Elements of the construction (III) Note that G AB (z, z) = G MM (Z) = G (W) w, where M = AB. It is possible ( iff ) ΣMM Σ Σ M M M Σ MM Σ = ( M M ΣAA w w Σ ) ( ) AĀ w Σ w w Σ BB w Σ B B ĀA ΣĀĀ w w Σ Σ L A BB Σ ΣR B B B Recalling that Σ A = R A (G B ) and Σ A = R A (G B ) we have R M (G M ) = [R A (G B )] L [R B (G A )] R. Tedious calculations allow to close the construction G A = [G M [R A (G B )] L] L ] R G B = [[R B (G A )] R G M

24 Elements of the construction (IV) Formally, one can define now two matrix-valued S-transforms via equations S (LEFT ) (Y) = R L ( [S (LEFT ) (Y)Y] R) S (RIGHT ) (Y) = R R ( [YS (RIGHT ) (Y)] L) reads [ ([R(G)G] L)] R [ = S (LEFT ) M S (RIGHT ) S (RIGHT ) M B (GR M (G)) S (LEFT ) A (R M (G)G) ( [GR(G)] R)] L =

25 Algorithm at work Write ( down known ) R A (G B ) and ( R B (G A ), where a i b a i G A = and G ib ā B = b ) ib ā 2 Modify R A R L A, R B R R B 3 R M = R L A RR B 4 Write down consistency conditions (Z R M )G R A = [R A(G B )] L and G L B (Z R M) = [R B (G A )] R 5 Solve (4) for a, b, a, b and read out G M Note that if A and B are identical free ensembles, pair (3) reduces to one equation, since a = a, b = b.

26 Example : GUE times GUE We ask for the spectrum of H A H B, where both ensembles are free (independent) GUE. Then for both ensembles R A (G) = R B (G) = G, in analogy to R(z) = z. ( ) ( ) 2 R L A (G) = w a i b a i b w w w ib, R R w B w ā (G) = w ib w ā 3 R M = R L A RR B 4 Consistency conditions provide matrix equation for a, b. 5 Two solutions: a = b = 0 or a = 0, b 2 = w w = z z 6 Holomorphic solution G(z) = z, nonholomorphic G(z, z) = z z, they match on boundary z = Spectral density ρ(z, z) = π zg(z, z) = 2π x 2 +y 2 So W A (0, ) W B (0, ) is a Halloween hat law [Girko,Vladimirowa;2009], [Burda,Janik,Wac law;200] on unit disc.

27 Example : Visualization Note qualitative similarity to N a (0, ) N b (0, ) = π K 0(x).

28 Example 2: Shifted Ginibre-Girko Ensemble times shifted Ginibre-Girko Ensemble We ask for the spectrum of ( + X A )( + X B ), where both ensembles X A, X B are free (independent) GGE, i.e. spectrum of + X is uniform unit disc centered at x =, y = 0. ( ) ( ) a i b i b R +X = ib ā ib Algorithm yields the boundary and the spectral density Boundary belongs to the family of parametric curves appearing in non-hermitian diffusion, density involves the solution of biquadratic equation [Gudowska-Nowak,Janik,Jurkiewicz,MAN;2003]. Boundary reads r = + 2 cos φ, where z = re iφ. Curve known as Pascal limaçon (after Etienne Pascal (588-65), father of Blaise Pascal), but actually known already to Albrecht Dürer (Underweysung der Messung, 525)

29 Limaçon: visualization by Dürer

30 Limaçon: visualization five centuries later

31 Limac on in 3D ! 0

32 Limaçon: numerical crosscheck of spectral desnity at y =

33 Challenges Mathematical arguments why described construction is possible Geometric interpretation of the left and right rotation (connection to SU(2)), X L = UXU, X R = U XU, where U = e i σ 3 φ 2 2. Less academic examples [Burda,Janik,MAN;20b]

34 Summary type of randomness Addition law Random variables Random matrices (-d) Random matrices (2-d)

35 Quaternization [Jarosz,Nowak;2004] For any hermitian H, knowing G(z) and R(z), we write down qg(q) qg q G(Q) = q q 2 G(q) G( q q q Q R(Q) = qr(q) qr q q q 2 R(q) R( q q q Q where q, q are eigenvalues of the quaternion Q. Note that R(G(Q) + /Q) = Q In analogy to G th (z) = t G H( z t ) and R th = tr H (tz) for t real we have now for complex t G tx (Q) = G X (T Q)T and R tx (Q) = T R X (QT ), where T = diag(t, t) Similar formulae for unitary ensembles [Jarosz, Goerlich;2005], in particular for CUE we get ( ) a i b 0 i b 4 b 2 R CUE = 2 b 2 ib ā ib 4 b b 2