Central Limit Theorems for linear statistics for Biorthogonal Ensembles

Transcription

1 Central Limit Theorems for linear statistics for Biorthogonal Ensembles Maurice Duits, Stockholm University Based on joint work with Jonathan Breuer (HUJI) Princeton, April 2, 2014 M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

2 Linear statistic Given a function f : R R and random points x 1,..., x n R the linear statistic X n (f ) is defined as X n (f ) = n f (x j ) j=1 M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

3 Linear statistic Given a function f : R R and random points x 1,..., x n R the linear statistic X n (f ) is defined as X n (f ) = n f (x j ) j=1 In this talk we will be discussing asymptotic behavior as n of the fluctuations of X n (f ) where 1 x 1,..., x n are random from a biorthogonal ensembles with a recurrence (so β = 2) 2 f C 1 (R) with at most polynomial growth at ±. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

4 Central Limit Theorems Typically, there exists a limiting distribution ν of points as n, and we have, almost surely, 1 n X n(f ) = 1 n n f (x j ) f (x)dν(x) j=1 M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

5 Central Limit Theorems Typically, there exists a limiting distribution ν of points as n, and we have, almost surely, 1 n X n(f ) = 1 n n f (x j ) f (x)dν(x) j=1 In several important examples it has been proved that the fluctuations X n (f ) EX n (f ) satisfy a Central Limit Theorem for some σ(f ) 2. X n (f ) EX n (f ) N(0, σ(f ) 2 ), as n, M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

6 Central Limit Theorems Typically, there exists a limiting distribution ν of points as n, and we have, almost surely, 1 n X n(f ) = 1 n n f (x j ) f (x)dν(x) j=1 In several important examples it has been proved that the fluctuations X n (f ) EX n (f ) satisfy a Central Limit Theorem for some σ(f ) 2. X n (f ) EX n (f ) N(0, σ(f ) 2 ), as n, No normalization, since f C 1 (R). In particular, the variance has a limit. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

7 Example: Unitary ensembles Consider the probability measure on R n defined by 1 Z n 1 i<j n (x i x j ) 2 e n n j=1 V (x j ) dx 1 dx n. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

8 Example: Unitary ensembles Consider the probability measure on R n defined by 1 Z n 1 i<j n (x i x j ) 2 e n n j=1 V (x j ) dx 1 dx n. As n there is a limiting density of points µ V, which is the unique minimizer of 1 log x y dν(x)dν(y) + V (x)dν(x) among all probability measures ν on R. Hence 1 n X n(f ) f (x)dµ V (x), almost surely as n. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

9 Example: Unitary ensembles Theorem (Johansson 98) Under certain assumptions on V, in particular S(µ V ) = [γ, δ], we have that for sufficiently smooth f X n (f ) EX n (f ) N(0, in distribution as n, where f k = 1 2π ( δ γ f 2πi 2 0 k f k 2 ) k=1 cos θ + δ + γ ) e ikθ dθ. 2 M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

10 Example: Unitary ensembles Some history Polynomial V, one-cut case and f H 2+ε for some ε > 0 (+additional assumptions) Johansson 98 Techniques were extended in Shcherbina-Kriecherbauer 10 Borot-Guionnet 12 to allow real analytic V. The one-cut assumption is essential. In multi-cut case there is not necessarily a limit to a Gaussian Pastur 06. In the multicut case, full asymptotic expansions of the partition function were recently obtained in Shcherbina 13, Borot-Guionnet 13. These results extend (with some appropriate modifications) to β-ensembles, but in this talk we assume β = 2. Other classical ensembles have been studied: e.g.β-jacobi ensembles Dumitriu-Paquette 12,Borodin-Gorin 13. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

11 Example: Lozenge tilings b c. Tilings of the abc-hexagon with lozenges a M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

12 Example: Lozenge tilings b c. Tilings of the abc-hexagon with lozenges a M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

13 Example: Lozenge tilings b c. Tilings of the abc-hexagon with lozenges a Random tilings: take uniform measure on all possible tilings M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

14 Example: Lozenge tilings M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

15 Gaussian Free Field To each tiling one associates a height function, whose graph is a stepped surface. The pairing of the random height function with a test function gives a linear statistic. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

16 Gaussian Free Field To each tiling one associates a height function, whose graph is a stepped surface. The pairing of the random height function with a test function gives a linear statistic. In many recent works these 2 dimensional fluctuations of the height function turn out ot be described by the Gaussian Free Field (+ complex structure). Borodin, Borodin-Bufetov, Borodin-Gorin, Borodin-Ferrari, D., Kenyon, Kuan, Petrov, M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

17 Gaussian Free Field To each tiling one associates a height function, whose graph is a stepped surface. The pairing of the random height function with a test function gives a linear statistic. In many recent works these 2 dimensional fluctuations of the height function turn out ot be described by the Gaussian Free Field (+ complex structure). Borodin, Borodin-Bufetov, Borodin-Gorin, Borodin-Ferrari, D., Kenyon, Kuan, Petrov, The example of lozenge tilings of a hexagon has been a special case in Petrov 13. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

18 Gaussian Free Field To each tiling one associates a height function, whose graph is a stepped surface. The pairing of the random height function with a test function gives a linear statistic. In many recent works these 2 dimensional fluctuations of the height function turn out ot be described by the Gaussian Free Field (+ complex structure). Borodin, Borodin-Bufetov, Borodin-Gorin, Borodin-Ferrari, D., Kenyon, Kuan, Petrov, The example of lozenge tilings of a hexagon has been a special case in Petrov 13. For the purpose of this talk, the main interest is in the 1-d fluctuations... M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

19 Example: Lozenge tilings M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

20 Example: Lozenge tilings s M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

21 Example: Lozenge tilings s. The centers x 1..., x n of the red tiles at a vertical line at distance s of the left side, have the jpdf on N n 1 i<j n n (x i x j ) 2 w(x j ), j=1 where w is the Hahn weight (with an appropriate choice of parameters). Johansson 02. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

22 Example: Lozenge tilings s. The centers x 1..., x n of the red tiles at a vertical line at distance s of the left side, have the jpdf on N n 1 i<j n n (x i x j ) 2 w(x j ), j=1 where w is the Hahn weight (with an appropriate choice of parameters). Johansson 02. Fluctuations: X n (f ) EX n (f ) N(0, σ 2 f ) in distribution as n. (1-d section of the GFF+complex structure) M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

23 The previous two models (the Unitary Enembles and vertical sections of lozenge tilings of a hexagon) are both examples of orthogonal polynomial ensembles. We will show that the CLT s for these models are special cases of a more general CLT for such ensembles. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

24 Orthogonal Polynomial Ensembles M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

25 Orthogonal polynomial ensembles Let µ be a Borel measure on R with finite moments x k dµ(x) <. Then the Orthogonal Polynomial Ensemble of size n associated to µ is the probability measure on R n proportional to (x i x j ) 2 dµ(x 1 ) dµ(x n ). 1 i<j n M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

26 Orthogonal polynomial ensembles Let µ be a Borel measure on R with finite moments x k dµ(x) <. Then the Orthogonal Polynomial Ensemble of size n associated to µ is the probability measure on R n proportional to (x i x j ) 2 dµ(x 1 ) dµ(x n ). 1 i<j n We allow for varying measures µ = µ n (but both varying and nonvarying will be discussed) M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

27 Orthogonal polynomial ensembles Let µ be a Borel measure on R with finite moments x k dµ(x) <. Then the Orthogonal Polynomial Ensemble of size n associated to µ is the probability measure on R n proportional to (x i x j ) 2 dµ(x 1 ) dµ(x n ). 1 i<j n We allow for varying measures µ = µ n (but both varying and nonvarying will be discussed) We want to find minimal conditions on the measure µ such that we have for f C 1 (R). X n (f ) EX n (f ) N(0, σ(f ) 2 )) as n, M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

28 Orthogonal polynomials We let p k,n be the orthogonal polynomial of degree k with respect to µ(= µ n ), i.e. Three term recurrence p k,n (x)p l,n (x)dµ(x) = δ kl xp k,n (x) = a k,n p k+1,n (x) + b k,n p k,n (x) + a k 1,n p k 1,n (x) Let J be the Jacobi operator corresponding to µ. b 0,n a 0,n J = J (n) a 0,n b 1,n a 1,n =. a ,n... M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

29 Reproducing kernel The reproducing kernel is defined as n 1 K n (x, y) = p k,n (x)p k,n (y) k=0 M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

30 Reproducing kernel The reproducing kernel is defined as n 1 K n (x, y) = p k,n (x)p k,n (y) k=0 It satisfies the reproducing property K n (x, y)k n (y, z)dµ(y) = K n (x, z). In other words, K n as a operator on L 2 (µ) is an orthogonal projection, i.e. K n = K n and K 2 n = K n, of rank n. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

31 Reproducing kernel The reproducing kernel is defined as n 1 K n (x, y) = p k,n (x)p k,n (y) k=0 It satisfies the reproducing property K n (x, y)k n (y, z)dµ(y) = K n (x, z). In other words, K n as a operator on L 2 (µ) is an orthogonal projection, i.e. K n = K n and K 2 n = K n, of rank n. Christoffel-Darboux formula: p n,n (x)p n 1,n (y) p n,n (y)p n 1,n (x) K n (x, y) = a n,n. x y M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

32 Determinantal point process The Orthogonal Polynomial Ensemble of size n associated to a Borel measure µ is a determinantal point process on R with kernel K n and reference measure µ. This means that ) E [exp tx n (f )] = det (I + (e tf 1)K n L 2 (µ) where the right-hand side is the Fredholm determinant for operator with integral kernel (e tf (x) 1)K n (x, y) acting on L 2 (µ). M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

33 A first question We recall that we ask whether we have a Central Limit Theorem where X n (f ) EX n (f )? N(0, σ(f ) 2 ) X n (f ) = n f (x j ), where f is sufficiently smooth and x 1,..., x n are from the OPE of size n for a Borel measure µ(= µ n ) on R. Question: can we expect this to hold for general measures µ? Do we have the correct normalization or does the normalization depend on regularity properties of µ? j=1 M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

34 Concentration inequality Theorem (Breuer-D 13) For a bounded function f we have [ E e t(xn(f ) EXn(f )] exp(a t 2 Var X n (f )), t 1 3 f where A is a universal constant. And hence ( ( ε 2 )) P ( X n (f ) EX n (f ) ε) 2 exp min 4A Var X n (f ), ε 6 f M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

35 Concentration inequality Theorem (Breuer-D 13) For a bounded function f we have [ E e t(xn(f ) EXn(f )] exp(a t 2 Var X n (f )), t 1 3 f where A is a universal constant. And hence ( ( ε 2 )) P ( X n (f ) EX n (f ) ε) 2 exp min 4A Var X n (f ), ε 6 f The theorem holds for any determinantal process that has a kernel K n that is an orthogonal projection, i.e. K n = K 2 n and K n = K n. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

36 Concentration inequality There is always the crude bound VarX n (f ) = n f 2. With the concentration inequality this implies 1 n 1/2+ε (X n(f ) EX n (f )) 0, almost surely as n for any bounded f and ε > 0. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

37 Concentration inequality There is always the crude bound VarX n (f ) = n f 2. With the concentration inequality this implies 1 n 1/2+ε (X n(f ) EX n (f )) 0, almost surely as n for any bounded f and ε > 0. In particular, if 1 n EX n(f ) has limit, then 1 n X n(f ) converges to the same limit almost surely. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

38 Concentration inequality There is always the crude bound VarX n (f ) = n f 2. With the concentration inequality this implies 1 n 1/2+ε (X n(f ) EX n (f )) 0, almost surely as n for any bounded f and ε > 0. In particular, if 1 n EX n(f ) has limit, then 1 n X n(f ) converges to the same limit almost surely. No conditions on µ! This holds for general DPP with orthogonal projection as a kernel. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

39 Concentration inequality There is always the crude bound VarX n (f ) = n f 2. With the concentration inequality this implies 1 n 1/2+ε (X n(f ) EX n (f )) 0, almost surely as n for any bounded f and ε > 0. In particular, if 1 n EX n(f ) has limit, then 1 n X n(f ) converges to the same limit almost surely. No conditions on µ! This holds for general DPP with orthogonal projection as a kernel. The bound on the variance can be improved significantly for C 1 functions. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

40 Computing the variance Compute the variance of a linear statistic VarX n (f ) M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

41 Computing the variance Compute the variance of a linear statistic VarX n (f ) = f (x) 2 K n (x, x)dµ(x) f (x)f (y)k n (x, y) 2 dµ(x)dµ(y) M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

42 Computing the variance Compute the variance of a linear statistic VarX n (f ) = f (x) 2 K n (x, x)dµ(x) f (x)f (y)k n (x, y) 2 dµ(x)dµ(y) = 1 (f (x) f (y)) 2 K n (x, y) 2 dµ(x)dµ(y) 2 M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

43 Computing the variance Compute the variance of a linear statistic VarX n (f ) = f (x) 2 K n (x, x)dµ(x) f (x)f (y)k n (x, y) 2 dµ(x)dµ(y) = 1 (f (x) f (y)) 2 K n (x, y) 2 dµ(x)dµ(y) 2 ( ) f (x) f (y) 2 (x y) 2 K n (x, y) 2 dµ(x)dµ(y) = 1 2 x y M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

44 Computing the variance Compute the variance of a linear statistic VarX n (f ) = f (x) 2 K n (x, x)dµ(x) f (x)f (y)k n (x, y) 2 dµ(x)dµ(y) = 1 (f (x) f (y)) 2 K n (x, y) 2 dµ(x)dµ(y) 2 = 1 ( ) f (x) f (y) 2 (x y) 2 K n (x, y) 2 dµ(x)dµ(y) 2 x y ( ) = a2 n,n f (x) f (y) 2 (p n (x)p n 1(y) p n (y)p n 1(x)) 2 dµ(x)dµ(y) 2 x y M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

45 Lipschitz functions If f is a Lipschitz function, i.e. f L = sup x,y f (x) f (y) x y <, then VarX n (f ) ( ) = a2 n,n f (x) f (y) 2 (p n (x)p n 1(y) p n (y)p n 1(x)) 2 dµ(x)dµ(y) 2 x y M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

46 Lipschitz functions If f is a Lipschitz function, i.e. f L = sup x,y f (x) f (y) x y <, then VarX n (f ) ( ) = a2 n,n f (x) f (y) 2 (p n (x)p n 1(y) p n (y)p n 1(x)) 2 dµ(x)dµ(y) 2 x y a2 n,n f 2 L 2 (p n (x)p n 1 (y) p n (y)p n 1 (x)) 2 dµ(x)dµ(y) = f 2 La 2 n,n M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

47 Lipschitz functions If f is a Lipschitz function, i.e. f L = sup x,y f (x) f (y) x y <, then VarX n (f ) ( ) = a2 n,n f (x) f (y) 2 (p n (x)p n 1(y) p n (y)p n 1(x)) 2 dµ(x)dµ(y) 2 x y a2 n,n f 2 L 2 (p n (x)p n 1 (y) p n (y)p n 1 (x)) 2 dµ(x)dµ(y) = f 2 La 2 n,n Hence if we have a n,n = O(1) as n, then the variance remains bounded. Pastur 06 M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

48 Lipschitz functions If f is a Lipschitz function, i.e. f L = sup x,y f (x) f (y) x y <, then VarX n (f ) ( ) = a2 n,n f (x) f (y) 2 (p n (x)p n 1(y) p n (y)p n 1(x)) 2 dµ(x)dµ(y) 2 x y a2 n,n f 2 L 2 (p n (x)p n 1 (y) p n (y)p n 1 (x)) 2 dµ(x)dµ(y) = f 2 La 2 n,n Hence if we have a n,n = O(1) as n, then the variance remains bounded. Pastur 06 Conclusion: if f is Lipschitz and a n,n = O(1) as n, then X n (f ) EX n (f ) 1, and we have exponential tails uniformly in n. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

49 Polynomial test functions Recall the formula for the variance: VarX n (f ) ( ) = a2 n,n f (x) f (y) 2 (p n (x)p n 1(y) p n (y)p n 1(x)) 2 dµ(x)dµ(y) 2 x y M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

50 Polynomial test functions Recall the formula for the variance: VarX n (f ) ( ) = a2 n,n f (x) f (y) 2 (p n (x)p n 1(y) p n (y)p n 1(x)) 2 dµ(x)dµ(y) 2 x y If is a polynomial, the variance can be written in terms of finitely many recurrence coefficients a n+k,n and b n+k,n. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

51 Polynomial test functions Recall the formula for the variance: VarX n (f ) ( ) = a2 n,n f (x) f (y) 2 (p n (x)p n 1(y) p n (y)p n 1(x)) 2 dµ(x)dµ(y) 2 x y If is a polynomial, the variance can be written in terms of finitely many recurrence coefficients a n+k,n and b n+k,n. In particular, For k Z we have that a n+k,n and b n+k,n have a limit as n = VarX n (f ) has a limit for any polynomial f M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

52 CLT for Orthogonal Polynomial Ensembles Theorem (Breuer-D 13) Assume that for any k Z we have a n+k,n a b n+k,n b as n. Then, if f is a polynomial with real coefficients, ( ) X n (f ) EX n (f ) N 0, k ^f k 2 k=1 where ^f k = 1 2π 2π 0 f (2a cos θ + b)e iθk dθ. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

53 CLT for Orthogonal Polynomial Ensembles Theorem (Breuer-D 13) Assume that there exists a subsequence {n j } j such that for any k Z we have a nj +k,n j a b nj +k,n j b as j. Then, if f is a polynomial with real coefficients, ( ) X nj (f ) EX nj (f ) N 0, k ^f k 2 k=1 where ^f k = 1 2π 2π 0 f (2a cos θ + b)e iθk dθ. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

54 Central Limit Theorems for C 1 functions Theorem (Breuer-D 13) Suppose there is a compact E R such that for k N we have x k K n (x, x)dµ(x) = o(1/n), R\E as n. Then the CLT holds for any f C 1 (R) such that f (x) C(1 + x k ) for some C > 0 and k N. Corollary If µ is non-varying and has compact support S(µ), then the CLT holds for any f C 1 (S(µ)). M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

55 Examples 1a and 1b Unitary Ensembles. Absolutely continuous measure with S(µ V ) = [γ, δ]. dµ n (x) = e nv (x) dx. Lozenge tiling of the hexagon. Discrete measure: dµ = N x=0 ( α + x x ) ( ) β + N x δ N x x. This called the Hahn weight and the orthogonal polynomials are the Hahn polynomials. In both cases one can prove that, with given parameters, the recurrence coefficients have limits and the CLT follows from the general CLT. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

56 Example 2 The following result for the non-varying situation follows by combining the CLT with a theorem by Denissov and Rakhmanov. Theorem Fix dµ = w(x)dx + dµ sing, and assume that S ess (µ) = [γ, δ] and w(x) > 0 Lebesgue a.e. on [γ, δ]. Then for any f C 1 (R) we have X n (f ) EX n (f ) N(0, k f k 2 ) k=1 where ^f k = 1 2πi 2π 0 f ( δ γ 2 cos θ + δ + γ ) e ikθ dθ. 2 M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

57 Example 3: different CLT s for different subsequences! Partition N into blocks A 1, B 2, A 2, B 2,... such that and set b n = 0 and A j = 2 j2, B j = 3 j2. a n = { 1, n A j 1 2, n B j Then the spectral measure µ for J is supported on [ 2, 2] and is singular with respect to the Lebesgue measure. For the OPE for µ one can show that 1 n X n(f ) f (x)dµ eq (x) where µ eq is the equilibrium measure for the interval [ 2, 2]. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

58 Example 3: different CLT s for different subsequences! Hence the exists subsequence n j such that where X nj (f ) EX nj (f ) N(0, ^f k = 1 2π 2π 0 k ^f k 2 ) k=1 f (cos θ)e ikθ dθ. There also exists subsequences n j such that X n j (f ) EX n j (f ) N(0, k f k 2 ) k=1 where f k = 1 2π 2π 0 f (2 cos θ)e ikθ dθ. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

59 Example 4 Example 3 can be modified such that the spectral measure is absolutely continuous on [ 1, 1] and purely singular on [ 2, 2] \ ( 1, 1) and b n = 0 and the a n are such that for each a [ 1 2, 1] there exists a subsequence {n j (a)} j such that k Z a nj (a)+k a, as j. Hence for each a there is an CLT for a special subsequence and each CLT is different. We stil have 1 lim n n X n(f ) = f (x)dµ eq (x) where µ eq is the equilibrium measure for the interval [ 2, 2]. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

60 Right Limits M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

61 Some notation Let J be the Jacobi operator corresponding to µ. b 0,n a 0,n J = J (n) a 0,n b 1,n a 1,n =. a ,n... Tri-diagonal semi-infinite. Since we allow varying measures, we allow J to depend on n and write J = J (n). Given a Laurent polynomial s(z) = p j= q s jz j, the Laurent matrix/operator L(s) and Toeplitz matrix/operator T (s) are defined as (L(s)) k,l = s k l, k, l Z. (T (s)) k,l = s k l, k, l N. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

62 Right limits Definition A two-sided infinte matrix J R is call a right limit of {J (n) } n iff there exists a subsequence {n j } j such that ( J R) = lim J (n j ) k,l j n j +k,n j +l, k, l Z. Example: there exists a subsequence n j such that for all k we have a nj +k,n j a and b nj +k,n j b if and only if L(az + b + a/z) is a right limit of {J (n) } n. Right limits have proved to be very useful when studying the spectra of Jacobi operators Last-Simon 99, Last-Simon 06, Remling 11 M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

63 Laurent matrices as right limits In this way, we can reformulate the CLT for OPE: Theorem (Breuer-D 13) Suppose that L(s) is a right limit of J (n) with subsequence {n j } j. Then, for any polynomial f with real coefficients, ( ) X nj (f ) EX nj (f ) N 0, k ^f k 2, k=1 where ^f k = 1 f (s(z)) dz 2πi z =1 z k+1. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

64 A general limit theorem Theorem Let J R be a right limit of J with subsequence {n j } j. Define { ( ) JM R (J R ) kl, k, l = M,..., M, =. kl 0, otherwise Let P be{ the projection on the negative coefficients x k k < 0 (P x) k =. Then for any polynomial f we have 0 k 0. lim E [ exp t(x nj (f ) EX nj (f )) ] j ) = lim M et Tr P f (JR M ) det (I + P (e tf (JR M ) I )P In particular, both limits exist. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

65 A general limit theorem If J R is a Laurent matrix, the right-hand side can be computed giving the previous CLT. It is an interesting open problem to find the limiting value at the right-hand side if J R is, for example, two-periodic. This should of course match with the formulas for the multi-cut case by Shcherbina 13, Borot-Guionnet 13 M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

66 Biorthogonal ensembles M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

67 Biorthogonal ensembles Let µ be a Borel measure on R and {φ j },{ψ j } two families of linearly indendent functions. Then the corresponding biorthogonal ensemble of size n is defined as the probability measure on R n proportional to det (φ k 1 (x j )) n j,k=1 det (ψ k 1(x j )) n j,k=1 dµ(x 1) dµ(x n ) M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

68 Biorthogonal ensembles Let µ be a Borel measure on R and {φ j },{ψ j } two families of linearly indendent functions. Then the corresponding biorthogonal ensemble of size n is defined as the probability measure on R n proportional to det (φ k 1 (x j )) n j,k=1 det (ψ k 1(x j )) n j,k=1 dµ(x 1) dµ(x n ) By taking linear combination of the rows we can assume that φ k (x)ψ l (x)dµ(x) = δ kl. R M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

69 Biorthogonal ensembles Let µ be a Borel measure on R and {φ j },{ψ j } two families of linearly indendent functions. Then the corresponding biorthogonal ensemble of size n is defined as the probability measure on R n proportional to det (φ k 1 (x j )) n j,k=1 det (ψ k 1(x j )) n j,k=1 dµ(x 1) dµ(x n ) By taking linear combination of the rows we can assume that φ k (x)ψ l (x)dµ(x) = δ kl. R The biorthogonal ensembles is a determinantal point process on R with kernel n 1 K n (x, y) = φ j (x)ψ j (y) and reference measure µ. k=0 M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

70 Biorthogonal ensembles with a recurrence We will assume that there exists a banded one-sided infinite matrix J such that φ 0 (x) φ 0 (x) φ 1 (x) x φ 2 (x) = J φ 1 (x) φ 2 (x).. Since J is banded this is a recurrence. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

71 Biorthogonal ensembles with a recurrence We will assume that there exists a banded one-sided infinite matrix J such that φ 0 (x) φ 0 (x) φ 1 (x) x φ 2 (x) = J φ 1 (x) φ 2 (x).. Since J is banded this is a recurrence. We will allow that J varies with n, i.e. J = J (n), in which case we assume that the number of bands is independent of n. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

72 Biorthogonal ensembles with a recurrence We will assume that there exists a banded one-sided infinite matrix J such that φ 0 (x) φ 0 (x) φ 1 (x) x φ 2 (x) = J φ 1 (x) φ 2 (x).. Since J is banded this is a recurrence. We will allow that J varies with n, i.e. J = J (n), in which case we assume that the number of bands is independent of n. There many interesting biorthogonal ensembles that admit such a recurrence. In particular when we are concerned Multiple Orthogonal Polynomials Ensembles. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

73 Example: unitary ensembles with external source Consider the eigenvalues of a n n Hermitian matrix randomly chosen from 1 Z n e c Tr(M2 AM) dm where c > 0 and A is a diagonal matrix with eigenvalues a 1,..., a m of multiplicities k 1,..., k m. Brezin-Hikami 97. Alternatively, consider n non-intersecting brownian paths, with k j paths conditioned to start at 0 a time t = 0 and end at b j at time t = 1. Location of paths at time t has the same distribution as the eigenvalues of M with a j = 2tb j and c = 1/t(1 t). M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

74 The eigenvalues form a biorthogonal ensemble constructed out of multiple Hermite polynomials with m weights. These polynomials satisfy an m + 2-term recurrence. Bleher-Kuijlaars 04 M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

75 Example: two matrix model Consider the eigenvalues of two n n Hermitian matrices (M 1, M 2 ) randomly chosen from 1 Z n e Tr(V (M 1)+W (M 2 ) τm 1 M 2 ) dm 1 dm 2 where V and W are polynomials such that the integral converges. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

76 Example: two matrix model Consider the eigenvalues of two n n Hermitian matrices (M 1, M 2 ) randomly chosen from 1 Z n e Tr(V (M 1)+W (M 2 ) τm 1 M 2 ) dm 1 dm 2 where V and W are polynomials such that the integral converges. In Eynard-Mehta 98 it was shown that the eigenvalue correlations have determinantal struture with kernels constructed out of the biorthogonal polynomials p j and q k of degree j and k such that p j (x)q k (y)e V (x) W (y)+τxy dxdy = δ jk. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

77 Example: two matrix model The eigenvalues of M 1, when averaged over M 2, form a biorthogonal ensembles with φ j (x) = p j 1 (x) ψ j (x) = q j 1 (y)e (V (x)+w (y) τxy)) dy. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

78 Example: two matrix model The eigenvalues of M 1, when averaged over M 2, form a biorthogonal ensembles with φ j (x) = p j 1 (x) ψ j (x) = q j 1 (y)e (V (x)+w (y) τxy)) dy. The polynomials p j satisfy a d W + 1-term recurrence, where d W is the degree of W. Bertola-Eynard-Harnad 02 M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

79 Example: two matrix model The eigenvalues of M 1, when averaged over M 2, form a biorthogonal ensembles with φ j (x) = p j 1 (x) ψ j (x) = q j 1 (y)e (V (x)+w (y) τxy)) dy. The polynomials p j satisfy a d W + 1-term recurrence, where d W is the degree of W. Bertola-Eynard-Harnad 02 In D-Kuijlaars 08, D-Kuijlaars-Mo 12, D-Geudens-Kuijlaars 11, D-Geudens 13 the asymptotics have been computed for W (y) = y 4 /4 + αy 2 /2 and V even, using Riemann-Hilbert methods, from which we can compute the recurrence coefficients. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

80 CLT for biorthogonal ensembles Theorem (Breuer-D 13) Assume that L(s) is a right limit of J (n) with subsequence {n j } j for some s(z) = p j= q s jz j. Then, if f is a polynomial with real coefficients, X nj (f ) EX nj (f ) N ( 0, ) k ^f k 2, k=1 as j, where ^f k = 1 f (s(z)) dz 2πi z =1 z k+1. Note: for orthogonal polynomial ensembles p = q = 1. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

81 CLT for biorthogonal ensembles Theorem (Breuer-D 13) Assume that L(s) is a right limit of J (n) with subsequence {n j } j for some s(z) = p j= q s jz j. Then, if f is a polynomial with real coefficients, X nj (f ) EX nj (f ) N ( 0, ) k ^f k 2, k=1 as j, where ^f k = 1 f (s(z)) dz 2πi z =1 z k+1. Note: for orthogonal polynomial ensembles p = q = 1. In case K n K n then we have no concentration inequality and hence extension to f C 1 (R), without more a priori knowledge. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

82 A general limit theorem Theorem Let J R be a right limit of J with subsequence {n j } j. Define { ( ) JM R (J R ) kl, k, l = M,..., M, =. kl 0, otherwise Let P be{ the projection on the negative coefficients x k k < 0 (P x) k =. Then for any polynomial f we have 0 k 0. lim E [exp t(x n(f ) EX n (f ))] j In particular, both limits exist. = lim M et Tr P f (JR M ) det (I + P (e tf (JR M ) I )P ) M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

83 Some words about the proofs M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

84 Cumulant expansion Expand the moment-generating function E [exp tx n (f )]) = det(1 + (e tf 1)K n ) ) = exp Tr log (1 + (e tf 1)K n ( ) = exp t m C m (n) (f ) m=1 where C (n) m (f ) = m ( 1) j j=1 The are called the cumulants. j l 1 + +l j =m,l i 1 Tr f l 1 K n f l j K n l 1! l j! M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

85 For a Central Limit Theorem one needs to show that { lim C m (n) 2σ(f ) 2, m = 2 (f ) n 0, m 3. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

86 For a Central Limit Theorem one needs to show that { lim C m (n) 2σ(f ) 2, m = 2 (f ) n 0, m 3. A first problem: each term in the sum C (n) m (f ) = m ( 1) j j=1 j l 1 + +l j =m,l i 1 Tr f l 1 K n f l j K n l 1! l j! grows linearly with n. Clearly, some very effective cancelation must come into play! M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

87 By using the identity x = log (1 + (e x 1)) and expanding the right-hand side we obtain if m 2. m ( 1) j j=1 j l 1 + +l j =m,l i 1 1 l 1! l j! = 0 M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

88 By using the identity x = log (1 + (e x 1)) and expanding the right-hand side we obtain if m 2. m ( 1) j j=1 j l 1 + +l j =m,l i 1 1 l 1! l j! = 0 Hence C (n) m (f ) = m ( 1) j j=1 j l 1 + +l j =m,l i 1 Tr f l 1 K n f l j K n Tr f m K n l 1! l j! for m 2. Now each term in the double sum can be proved to be bounded and this captures a first cancelation. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

89 Concentration inequality Using Kn 2 = K n and Kn = K n one can show that Tr f l 1 K n f l j K n Tr f m K n jm 2 f m 2 [f, K n ] 2 2, m 2 where [f, K n ] 2 is the Hilbert-Schmidt norm for the commutator of K n anf f (viewed as a multiplication operator). M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

90 Concentration inequality Using Kn 2 = K n and Kn = K n one can show that Tr f l 1 K n f l j K n Tr f m K n jm 2 f m 2 [f, K n ] 2 2, m 2 where [f, K n ] 2 is the Hilbert-Schmidt norm for the commutator of K n anf f (viewed as a multiplication operator). If K n = K n then [f, K n ] 2 2 = 2VarX n (f ) and the mentioned concentration inequality Breuer-D 13 follows after a further arranging of terms. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

91 Concentration inequality Using Kn 2 = K n and Kn = K n one can show that Tr f l 1 K n f l j K n Tr f m K n jm 2 f m 2 [f, K n ] 2 2, m 2 where [f, K n ] 2 is the Hilbert-Schmidt norm for the commutator of K n anf f (viewed as a multiplication operator). If K n = K n then [f, K n ] 2 2 = 2VarX n (f ) and the mentioned concentration inequality Breuer-D 13 follows after a further arranging of terms. For a CLT we need to capture another cancelation. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

92 Use orthogonality to rewrite C (n) m (f ) = = m ( 1) j j=1 j m ( 1) j j=1 j l 1 + +l j =m,l i 1 l 1 + +l j =m,l i 1 Tr f l 1 K n f l j K n Tr f m K n l 1! l j! Tr f (J) l 1 P n f (J) l j P n Tr f (J) m P n l 1! l j! M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

93 Use orthogonality to rewrite C (n) m (f ) = = m ( 1) j j=1 j m ( 1) j j=1 j l 1 + +l j =m,l i 1 l 1 + +l j =m,l i 1 Tr f l 1 K n f l j K n Tr f m K n l 1! l j! Tr f (J) l 1 P n f (J) l j P n Tr f (J) m P n l 1! l j! By using the band structure of J (and hence of f (J) for polynomial f ) it is not hard to prove that Tr f (J) l 1 P n f (J) l j P n Tr f (J) m P n only depends on a finite and fixed number of recurrence coefficients. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

94 Use orthogonality to rewrite C (n) m (f ) = = m ( 1) j j=1 j m ( 1) j j=1 j l 1 + +l j =m,l i 1 l 1 + +l j =m,l i 1 Tr f l 1 K n f l j K n Tr f m K n l 1! l j! Tr f (J) l 1 P n f (J) l j P n Tr f (J) m P n l 1! l j! By using the band structure of J (and hence of f (J) for polynomial f ) it is not hard to prove that Tr f (J) l 1 P n f (J) l j P n Tr f (J) m P n only depends on a finite and fixed number of recurrence coefficients. This does not prove the desired asymptotic behavior, but it does show that only a relatively small part of J matters in the fluctuation. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

95 Comparison/Universality principle If J 1 and J 2 have the same right limit J R along the same subsequence {n j }, then the leading term in the asympotic behavior of the fluctuations for the linear statistics for polynomial f along that subsequence is the same. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

96 Comparison/Universality principle If J 1 and J 2 have the same right limit J R along the same subsequence {n j }, then the leading term in the asympotic behavior of the fluctuations for the linear statistics for polynomial f along that subsequence is the same. Hence if J has a Laurent matrix L(s) as a right limit, then we can assume that J = T (s) from the start! Here T (s) is the Toeplitz operator with symbol s. Hence we only need to compute the CLT for the special Toeplitz case and the rest follows by the comparison principle. M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

97 Lemma (Breuer-D 13) Let s(z) = p j= q s jz j be a Laurent polynomial. Then ( ( ) lim det I + P n (e T (s) I )P n e Tr PnT (s) 1 = exp n 2 ) ks k s k k=1 Proof is based on Ehrhardt s generalization of the Helton-Howe-Pincus formula: if [A, B] is trace class, then det e A e A+B e B = exp 1 Tr[A, B]. 2 (This beautiful formula essentially captures the final cancelations in the cumulant expansion) M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53

98 Thank you for your attention! M. Duits (SU) CLT s for linear statistics Princeton, April 2, / 53