Free probabilities and the large N limit, IV. Loop equations and all-order asymptotic expansions... Gaëtan Borot

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1 Free probabilities and the large N limit, IV March 27th 2014 Loop equations and all-order asymptotic expansions... Gaëtan Borot MPIM Bonn & MIT based on joint works with Alice Guionnet, MIT Karol Kozlowski, Dijon

2 Loop equations and all-order asymptotic expansions Model and results 2. Schwinger-Dyson equations 3. Sketch of proof of the main result 4. Conclusion

3 } The β ensembles Probability measure on dµ A N = 1 Z A N exp N N A N R N T (λ i ) 1 i<j N λ i λ j β N 1 A (λ i )dλ i β>0 It is the measure induced on eigenvalues of a random matrix M N Tr T (M) dm e Wigner, Dyson, Mehta (50s-60s) β =1 β =2 β =4 real symmetric matrices hermitian matrices quaternionic self-dual matrices M = triagonal all β>0, T polynomial of even degree Dumitriu, Edelman 02 Krishnapur, Rider, Virág 13

4 Mean-field models Probability measure on dµ N = 1 Z N exp N 2 T 0 (L (λ) N ) A N R N 1 i<j N λ i λ j β N 1 A (λ i )dλ i β>0 where L (λ) N = 1 N N δ λ i is the (random) empirical measure Exemples in Chern-Simons theory O(n) model on random lattices T (µ) = 0 T (µ) = n 0 2 dµ(x)dµ(y) m β m ln dµ(x)dµ(y) ln x + y sinh[α m(x y)] α m (x y) Here, we take T (µ) = 0 T r T (x 1,...,x r ) real-analytic on A r dµ(x i )

5 We would like to study when N... the (random) empirical measure L (λ) N = 1 N N δ λ i N f(λ i)=n f(ξ)dl (λ) N (ξ) what kind of random variable is? the partition function Z N = A N exp N 2 T 0 (L (λ) N ) 1 i<j N λ i λ j β N dλ i the k-point correlators N dl (λ) W k (x 1,...,x k ) = Cumulant N (ξ 1) x 1 ξ 1,..., (λ) N dl N (ξ k) x k ξ k

6 The leading order... is given by a continuous approximation Define the energy functional on a proba. measure T (µ) = k r dµ(x i ) T (x 1,...,x k )+ β r 2 µ dµ(x 1 )dµ(x 2 )ln x 1 x 2 Assumption 1 : uniqueness of maximizer µ eq Characterization : exists a constant C such that for x A µ eq -everywhere T (µ eq )[δ x ] C Assumption 2 : local strict concavity at µ eq for any ν = finite signed measure of mass 0 T (µ eq )[ν, ν] =D 2 [ν] [0, + ] and = 0 iff ν =0

7 The leading order... is given by a continuous approximation Define the energy functional on a proba measure T (µ) = k r dµ(x i ) T (x 1,...,x k )+ β r 2 Assumption 1 : uniqueness of maximizer Assumption 2 : local strict concavity at µ eq µ eq µ dµ(x 1 )dµ(x 2 )ln x 1 x 2 Lemma L (λ) N µ eq almost surely and in expectation Z N =exp N 2 T (µ eq )+o(1)

8 Large deviations for a single particle A particle at position x feels the effective potential J(x) =T (µ eq )[δ x ] sup T (µ eq )[δ ξ ] ξ A Lemma For any closed F A P i, λ i F exp N sup x F J(x)+o(1) J(x) 0 One can restrict to a compact neighborhood of J(x) =0 B A Z B N = Z A N (1 + o(e cn )) B x A

9 Large deviations of empirical measure Natural distance T (µ eq )[ν, ν] =D 2 [ν] [0, + ] but D[L (λ) N µ eq] =+ because of atoms and log singularity Let us pick a nice regularization L (λ) N = 1 N δ λi L N (λ) N idea from Maïda, Maurel-Segala Lemma If T P N D[ L (λ) N is smooth, we have for N large enough µ eq] >t exp N ln N N 2 t 2 /2

10 The equilibrium measure T real-analytic = µ eq } S = is supported on a finite number of segments g [a h,b h ] h=0 α S is a hard edge if dµ eq (x) = 1 S(x)dx 2π M(x) α A α soft, is a soft edge otherwise x α 1/2 α hard x α 1/2 µeq/dx 1-cut regime x transition through critical points µeq/dx (g + 1)-cuts regime x We say that µ eq is off-critical when M(x) > 0 on A

11 } Finite size corrections : we assume... Uniqueness of maximizer Local strict concavity at µ eq µ eq V = V 0 +(1/N )V 1 + V 0 V 1 real analytic on A complex analytic on A Control of large deviations J(x) < 0 for x A \ S µ eq is off-critical f = test function, analytic on A

12 Result in the 1-cut regime 1/N asymptotic expansion Z N = N γn+γ exp m 2 N m F [m] + O(N ) µeq/dx x γ,γ depend only on β and the nature of the edges Central limit theorem N f(λ i ) N A f(ξ)dµ eq (ξ) (non-centered) gaussian

13 Result in the (g + 1)-cuts regime Oscillatory asymptotic expansion µeq/dx 1 eq 1 eq 2 eq eq 1 2 x Z N = N γn+γ (D N Θ N eq)(f [ 1] F [ 2] )exp where D N = p 0 1 p! 1,..., p 1 m 1,...,m p 2 i (m i+ i )>0 N i (m i+ i ) m 2 p N m F [m] + O(N ) F [m i],( i ) eq i! i w acts as a differential operator on the Siegel theta function Θ µ (w Q) = m Z g e w (m+µ)+ 1 2 (m+µ) Q (m+µ) (Pseudo)-periodicity come from µ = N eq mod Z g

14 Result in the (g + 1)-cuts regime No central limit theorem in general... E e isx N [f] N E e is(k[f]g 1+m[f]) E e isu[f] G 2 µeq/dx 1 eq 1 eq 2 eq eq 1 2 G 2 + N eq Z g x gaussian (non-centered) + discrete Gaussian, centered at µ = N eq mod Z g Corollary N f(λ i ) N A f(ξ)dµ eq (ξ) converges in law along subsequences

15 History of β ensembles : 1-cut regime β =2 If 1/N expansion exists, then and F {m} can be computed by the moment method Ambjørn, Chekhov, Kristjansen, Makeenko, 90s Z N = N γn+γ exp N 2m F {m} m 1 Rewriting of F {m} in terms of a universal topological recursion Eynard, 04 Existence of 1/N expansion by - analysis of SD equations - RH techniques - analysis of int. system Albeverio, Pastur, Shcherbina 01 Ercolani, McLaughlin 02 Bleher, Its, 05

16 History of β ensembles : 1-cut regime β>0 if 1/N expansion exists, then and F [m] computed by a β-topological recursion Chekhov, Eynard 06 Z N = N γn+γ exp N m F [m] m 2 Central limit theorem Johansson 98 Existence of 1/N expansion (analysis of SD eqn) Borot, Guionnet 11

17 History of β ensembles : multi-cut regime β =2 numerous observations of oscillatory behavior physicists, 90s Riemann-Hilbert techniques up to o(1) Deift, Kriecherbauer, McLaughlin, Venakides, Zhou,... heuristic derivation up to o(1) Bonnet, David, Eynard 00 generalization to all orders Eynard 07 observation of no CLT Pastur 06 β>0 Proof of no CLT and asymptotics of Z A N up to o(1) Shcherbina 12 General proof Borot, Guionnet 13

18 History of mean-field models dµ N = 1 Z N exp N 2 T 0 (L (λ) N ) with r-body interaction 0 1 i<j N T (µ) = λ i λ j β N T (x 1,...,x r ) r 1 A (λ i )dλ i dµ(x i ) same results for mean field models Borot, Guionnet, Kozlowski 13 computation of expansion by a topological recursion Borot, 13

19 Large-N asymptotic expansions in 1-d repulsive particle systems 1. Model and results 2. Schwinger-Dyson equations 3. Sketch of proof of the main result 4. Conclusion

20 What are Schwinger-Dyson equations? = relations between expectation values from integration by parts In the model dµ N = 1 Z N exp we find for any smooth test function and smooth functional O N 2 T 0 (L (λ) N ) h 1 i<j N λ i λ j β N 1 A (λ i )dλ i E i Nh(λ i ) T0 (L (λ) N )[δ λ i ]+β i<j h(λ i ) h(λ j ) λ i λ j + i h (λ i ) O(L (λ) N ) + i N 1 h(λ i ) O (L (λ) N )[δ λ i ] + boundary = 0

21 What are Schwinger-Dyson equations? Remind the k-points correlators N dl (λ) W k (x 1,...,x k ) = Cumulant N (ξ 1) x 1 ξ 1,..., (λ) N dl N (ξ k) x k ξ k Choose h z (x) = 1 z x for z,z i C \ A and O z2,...,z k (L (λ) N )= k i=2 dl (λ) N (ξ i) z i ξ i family of functional relations between W 1,...,W r+k 1 indexed by k 1

22 The master operator Decompose with W eq (z) = W 1 (z) =N W eq (z)+δ 1 W 1 (z) dµeq (ξ) z ξ Schwinger-Dyson equations can be recast (K + δk)[δ 1 W 1 ](z) =A 1 + boundary (K + δk)[w n (,z 2,...,z n )](z) =A n + boundary with : K[f](z) =2W eq (z)f(z)+ 2 f(λ)dλ β T 0 (µ eq ) z λ δk[f](z) =2δ 1 W 1 (z)f(z)+n 1 (1 2/β) z f(z)+

23 Asymptotic analysis Introduce norms f Γ = sup f(z) z Ext(Γ) Large deviations of empirical measure Nδ 1 W 1 Γ1 C 1 (N ln N) 1/2 W k Γk C k (N ln N) k/2 Γ 1 Γ 2 Γ 3 A Large deviation of single eigenvalue : boundary effects o(e cn ) Rigidity of SD equations : if Nδ 1 W 1 Γi1 c 1 ηn κ N +1 W k Γik c k η k N κ N + N 2 k K invertible and K 1 [f] Γi+1 c f Γi = Nδ 1 W 1 Γi1+2 c1 ηn (η N /N )κ N +1 W k Γik c +2 k η k N (η N /N )κ N + N 2 k

24 Asymptotic analysis Large deviations of empirical measure + Rigidity of SD equations Corollary If K invertible and K 1 [f] Γi+1 c f Γi we have, for any W k = M 1 m=k 2 M 0 N m W [m] k + O(N M ;Γ M,k ) an asymptotic expansion Remark : (g + 1) cuts c = nb. critical conditions dim Ker K = g + c

25 Large-N asymptotic expansions in 1-d repulsive particle systems 1. Model and results 2. Schwinger-Dyson equations 3. Sketch of proof of the main result 4. Conclusion

26 Scheme of the proof Models with fixed filling fractions Initial model (multi-cut regime) 1. Eq. measure and regularity (potential theory) 2. Invertibility of K (functional + cx analysis) same large deviations estimates same Schwinger-Dyson equations 3. Expansion of correlators interpolation series analysis 4. Expansion of partition fn. 5. Expansion of partition fn.

27 Conditioning on the filling fractions From large deviations on single eigenvalue : up to o(e cn ), we can choose A 0 A 1 A 2 x A = g h=0 A h We will study µ (A 0,...,A g ) (N 0,...,N g ) = µa N conditioned to have } N 0 first λ s in A 0 N 1 next λ s in A 1 etc. The partition function decomposes Z A N = N 0 + N g =N N! g h=0 N h! Z(A 0,...,A g ) (N 0,...,N g ) h = N h /N are the filling fractions

28 Equilibrium measures... Assumption 1 : uniqueness of maximizer (= T (µ) = r dµ(x i ) T (x 1,...,x r )+ β 2 among all proba. measures µ eq ) of dµ(x 1 )dµ(x 2 )ln x 1 x 2 Let eq,h = µ eq [A h ] be the equilibrium filling fraction Assumption 2 : local strict concavity at µ eq Lemma 1 For close enough to eq T has a unique maximizer (= µ eq, ) over proba. measure with µ[a h ]= h

29 Equilibrium measures... Assumption 1 : uniqueness of maximizer (= T (µ) = k dµ(x i ) T (x 1,...,x k )+ β 2 among all proba. measures µ eq ) of dµ(x 1 )dµ(x 2 )ln x 1 x 2 Let eq,h = µ eq [A h ] be the equilibrium filling fraction Assumption 2 : local strict concavity at Assumption 3 : Assumption 4 : Lemma 2 T µ eq µ eq is analytic has (g + 1) cuts and is off-critical For close enough to eq µ eq; has (g + 1) cuts and is off-critical The edges depend smoothly on The density of µ eq; depends smoothly on away from edges

30 Equilibrium measures... Assumption 1 : uniqueness of maximizer (= T (µ) = k dµ(x i ) T (x 1,...,x k )+ β 2 among all proba. measures µ eq ) of dµ(x 1 )dµ(x 2 )ln x 1 x 2 Let eq,h = µ eq [A h ] be the equilibrium filling fraction Assumption 2 : local strict concavity at Assumption 3 : Assumption 4 : Lemma 3 For T µ eq close enough to is analytic has (g + 1) cuts and is off-critical eq µ eq the large deviation estimates also holds uniformly in the conditioned model with filling fractions

31 The return of the master operator The correlators W k W k; in the initial model in the conditioned model satisfy the same Schwinger-Dyson equations We have = W k; (z 1,...,z k ) A h1 A hk k dz i 2iπ = δ k,1 N h1 we need the restriction K 0; of to the codim. = g subspace f, h, f(z)dz =0 Ah K Lemma 4 For close enough to eq K 0; is continuously invertible, and K 1 0; depends smoothly on

32 Asymptotic expansion of correlators in the conditioned model Corollary For close enough to eq we have, for any W k; = M 1 m=k 2 M 0, an asymptotic expansion W [m] k; + O(N M ;Γ M,k ) depending smoothly on, with remainder uniform in

33 Partition function of the conditioned model Z (T 1) N; Z (T 0) N; =exp N 2 r t T t (x 1,...,x r ) r dl (λ),t t N (x i ) can be expressed in terms of W T t j; for the model with interaction T t If we can find a interpolating family (T t ) t [0,1] respecting uniformly our assumptions for which Z (T 0) is known N; we deduce an expansion Z (T 1) N; = Z(T 0) N; M 1 exp m= 2 N m F [m] + O(N M ) Idea : interpolate in the space of equilibrium measures (µ t eq;) t [0,1] (T t ) t [0,1]

34 An interpolation path... 0 t =1 1 2 convex linear combination with semi-circles a 0 b 0 a 1 b 1 a 2 b 2 t 0 semi-circles squeezing the supports t =1/2 0 semi-circles 1 2 a 0 b 0 a 1 b 1 a 2 b a 0 +b 0 2 Z A N [V t ] Z (T t) N; a 1 +b 1 2 a 2 +b 2 2 t 0 0 h<h g a h + b h a h b h 2 N 2 h h β g Selberg β Gaussian h=0 Selberg Gaussian integral integral over R N h

35 Sums and interferences - 1/3 We initially wanted to compute Z A N = Z N N 0 + N g =N N! g h=0 N h! Z(A 0,...,A g ) Z(N N;(N0 0,...,N /N,...,N g ) g /N ) From large deviations of empirical measures : A N! 1+O(e Z = g h=0 N h! ZA N (1 + O(e cn N Z cn N;N/N )) ) N N ln N For N N o(n), we just proved, with =(N h /N ) 1 h g N! g h=0 N Z h! ZA N = N γn+γ N; exp M 1 m= 2 k 2 N m k F [m] k ()+O(N K + M ) ) where F [m] depend smoothly on eq Extra lemma : ( F [ 2] ) eq =0 and ( F [ 2] ) eq < 0

36 Sums and interferences - 2/3 We plug the asymptotic formula and use a Taylor expansion at eq E.g. up to o(1) : Z N = N γn+γ e N 2 F [ 2] eq +NF [ 1] eq +F [0] eq N N eq ln N e 1 2 ( 2 F [ 2] ) eq (N N eq ) 2 +( F [ 1] ) eq (N N eq ) 1+O(e c (ln N) 3 /N ) It is the general term of a super-exponentially fast converging series : Z N = N γn+γ e N 2 F [ 2] eq +NF [ 1] eq +F [0] eq 2 ( 2 F [ 2] ) eq (N N eq ) 2 +( F [ 1] ) eq (N N eq ) 1+O(e c (ln N) 3 /N ) N Z g e 1 We recognize Θ Neq ( F [ 1] ) eq ( 2 F [ 2] ) eq

37 Sums and interferences - 3/3 Including higher orders yields terms of the form 1 p p! N Z g ( i F [m i] ) eq i! (N N eq ) ( i i) e 1 2 Q (N N eq) 2 +w (N N eq ) We recognize 1 p p! N Z g ( i F [m i] ) eq i! ( ( i i) w Θ Neq )(w Q) Here Q =( 2 F [ 2] ) eq and w =( F [ 1] ) eq We justified step by step the heuristics of Bonnet, David, Eynard 00, Eynard 07

38 Summary : the (g + 1)-cuts regime Oscillatory asymptotic expansion µeq/dx 1 eq 1 eq 2 eq eq 1 2 x Z N = N γn+γ (D N Θ Neq ) ( F [ 1] ) eq ( 2 F [ 2] ) eq exp where D N = p 0 1 p! 1,..., p 1 m 1,...,m p 2 i (m i+ i )>0 N i (m i+ i ) p m 2 N m F [m] + O(N ) ( i F [m i] ) eq i w i! acts as a differential operator on the Siegel theta function Θ µ (w Q) = m Z g e w (m+µ)+ 1 2 (m+µ) Q (m+µ) Moving characteristics Quadratic form µ = N eq mod Z g Q = Hessian =eq [T (µ eq; )]

39 All order asymptotics for β-ensembles in the multi-cut regime 1. Beta-ensembles and random matrices 2. Applications to orthogonal polynomials 3. Sketch of the proof of the main result 4. Conclusion

40 In progress A toy model for XXZ spin correlation functions (two-scale problem) Z N = sinh N α c 1 (λ i λ j ) sinh N α c 2 (λ i λ j ) N e N 1+α V (λ i ) dλ i 1 i<j N Open problems Same questions for λ i Z? no Schwinger-Dyson equations... Same (non-perturbative) questions for multi-matrix models? more complicated Schwinger-Dyson equations and convexity issues... Universality from Schwinger-Dyson equations?