On global fluctuations of non-colliding processes Maurice Duits March 21, Warwick 1.Motivation Some examples of non-colliding processes or non-intersecting paths in the class of determinantal point processes Global fluctuations via multi-time linear statistics Choose a number of horizontal sections ⌧m for m = 1, . . . , N and consider the linear statistic N X n X X n (f ) = f (⌧m , xj (⌧m )) m=1 j=1 In this talk we will be interested in the behavior of the fluctuations of Xn (f ) as n ! 1 We will discuss N=1 separately. 2. Some example of results for linear statistics Linear statistics and the GUE For a smooth function f consider the linear statistic Xn (f ) = n X f (xj ) j=1 where, for now, the points x1 , . . . , xn are the eigenvalues of a (properly scaled) GUE matrix. Then, as n ! 1 , the global distribution is given by 1 1 EXn (f ) ! n ⇡ Z p 2 p 2 p f (x) 2 x2 dx We will be mostly interested in the limiting behavior of the fluctuations Xn (f ) EXn (f ) Linear statistics and the GUE Central Limit Theorem: For the fluctuations we have as n ! 1 , for sufficiently smooth functions f , EXn (f ) ! N (0, (f )2 ) Xn (f ) where 2 f and = k=1 1 fk = ⇡ 1 X Johansson ’98,…. Z ⇡ k|fk |2 p f ( 2 cos ✓) cos k✓d✓ 0 No normalization! Only for sufficiently smooth functions. One dimensional section of a 2d Gaussian Free Field. Linear statistics and the GUE Dyson (1962). Consider a Hermitian matrix for which the entries (up to symmetry) evolve according to independent Ornstein-Uhlenbeck processes. Then the ordered eigenvalues form a non-colliding process The invariant measure is the eigenvalue distribution for the GUE. What about multi-time fluctuations in the stationary situation? Stationary DBM Choose a number of horizontal sections ⌧m for m = 1, . . . , N Xn (f ) = N X n X f (⌧m , xj (⌧m )) m=1 j=1 Multi-time Global CLT: As n ! 1, for sufficiently smooth functions f , EXn (f ) ! N (0, (f )2 ) Xn (f ) where (f )2 = 1 X N X e |⌧m1 ⌧m2 |k (m1 ) (m2 ) fk , kfk m1 ,m2 =1 k=1 and (m) fk Borodin, D. 1 = ⇡ Z ⇡ f (m, 0 p 2 cos ✓) cos k✓d✓ 3. Biorthogonal Ensembles Single time fluctuations Biorthogonal Ensembles n R Consider a probability measure on defined by 1 det ( Zn n n j (xk ))j,k=1 dµ(x1 ) · · · dµ(xn ) j (xk ))j,k=1 det ( where µ is a Borel measure functions with respect to µ j and j are some square integrable By taking linear combination of the the rows in the determinants and the fact that the above is probability measure, we can assume, without loss of generality, that Z j (x) k (x)dµ(x) This also implies that Zn = n! = jk Recurrence Matrices Our main assumption will be that the functions j and j are part of large system of biorthogonal functions { j }j and { j }j , Z j (x) k (x)dµ(x) = jk for which there exists a banded matrix J such that 0 B x@ 0 (x) C 1 (x)A .. . This means that the x 1 j (x) j 0 B =J@ 0 (x) 1 C 1 (x)A .. . satisfy a finite term recurrence = X |k|⇢ Jj,j+k j+k (x) Orthogonal polynomials and Jacobi matrices An example of such a system are the Orthogonal Polynomial Ensembles. In that case j (x) = j (x) = pj (x) are the orthonormal polynomials. That is, the polynomials of degree j with positive leading coefficients satisfying Z pj (x)pk (x)dµ(x) = jk There exists {ak }k ⇢ (0, 1) and {bk }k ⇢ R such that xpk (x) = ak+1 pk+1 (x) + bk+1 pk (x) + ak pk 1 (x) The recurrence matrix, or Jacobi operator, is symmetric tridiagonal 0 b1 Ba1 B J =B0 @ .. . a1 b2 a2 .. . 0 a2 b3 .. . 1 ... . . .C C . . .C A .. . Relevant part of the recurrence matrix (1) Limit shape: If is a polynomial then 1 lim (EXn (f ) n!1 n Tr Pn f (J )Pn ) = 0 This means that the expected value of the moments of the empirical measure only depend on coefficients in an upper right block of size. J = ( (n + O(1)) ⇥ (n + O(1)) ) Relevant parts of the recurrence matrix (2) Fluctuations: The moments of the fluctuations E [(Xn (f ) m EXn (f )) ] Ẽ h⇣ Xn (f ) ẼXn (f ) of the moments of the empirical measure only depend on O(1) size block around n,n entry More precisly, The m-th moment only depends on the entries Jn+j,n+k , |j|, |k| M. where M depends on m and the degree f of but not on n . J = ( ) ⌘m i !0 Universality of fluctuations Theorem (Breuer-D, to appear in JAMS) Given two biorthogonal ensembles, with recurrence matrices J and J˜ such that for all k and ` we have ⇣ ⌘ ˜n+k,n+` = 0, lim J J n+k,n+` n!1 Then, for any polynomial f (x) and for each m , as n ! 1 , E [(Xn (f ) m EXn (f )) ] Question: What is the typical behavior in the red blocks? Ẽ h⇣ Xn (f ) J = ẼXn (f ) ( ⌘m i !0 ) Central Limit Theorem Theorem (Breuer-D, to appear in JAMS) Suppose that we have a biorthogonal ensemble with a recurrence matrix satisfying lim Jn+k,n+` = ak n!1 ` Then for any polynomial f (x) as n ! 1 , Xn (f ) where EXn (f ) ! N 0, 1 X kfk f k=1 k ! 0 1 I X 1 j A dz @ fk = f aj z k+1 2⇡i |z|=1 z j ´In the symmetric case j = j , we also formulate general and weak conditions to allow C 1 functions of polynomial growth. Example GUE The eigenvalues of a GUE matrix have the 1 ⇣ det (pj n! n (x )) 1 k j,k=1 n ⌘2 Y e nx2j dxj j=1 where pj (x) are rescaled and normalized Hermite polynomials, i.e. Z nx2 pj (x)pk (x)e dx = jk The (rescaled) Hermite polynomials have there recurrence r r k+1 k xpk (x) = pk+1 (x) + pk 1 (x) 2n 2n Hence we find the following recurrence matrix and the relevant limit 0 p0 B 1 1 B J = p1 B B J =p 2n nB B @ p 1 p0 2 p 2 0 p 3 p 3 .. . .. . 1 C C C C .. C .C A (J )n+k,n+l = ( p1 2 0 |k l| = 1 otherwise Remarks on the generality This theorem gives a criterium for which the model has Gaussian limiting fluctuation for the linear statistics. Work is still needed to show that the criteria is practical. For models involving classical functions / orthogonal polynomials, the criteria is often satisfied and can be simply looked up on any standard reference on special functions. Often, the question of finding leading term in the asymptotics for the recurrence coefficients is an intrinsically easier question that finding the full asymptotic description for the correlation kernel. In certain cases there are general result ensuring that the criterium is fulfilled for a remarkably wide class of models. The criterium fails in multi-cut cases since the recurrence coefficients have quasi-periodic behavior. The fluctuations are not always gaussian. Pastur ’06, Borot-Guionnet ’13, Shcherbina ’13 Example: OPRL As the first consider j = j = pj 1 the OP’s for µ Y 1 2 (xi (det pj (xk )) dµ(x1 ) · · · dµ(xn ) ⇠ n! 1i<jn xj )2 dµ(x1 ) · · · dµ(xn ) Recurrence xpk (x) = ak+1 pk+1 (x) + bk+1 pk (x) + ak pk 1 (x) Jacobi operator 0 b1 Ba1 B J =B0 @ .. . a1 b2 a2 .. . 0 a2 b3 .. . 1 ... . . .C C . . .C A .. . We allow varying measures so all objects may depend on n Example: OPRL For non-varying measures we have the: Denissov-Rakhmanov Theorem: If µ is a compactly supported measure for which the essential part of the support is a single interval [b 2a, b + 2a] and the absolutely continuous part has a density that is strictly positive a.e. on the essential part. Then an ! a bn ! b For varying measure related to Unitary Ensembles, we have limits of the recurrence coefficients for one-cut situations. Also many discrete examples from tiling models fall in this class. DBM with random initial condition Take as the initial configuration the eigenvalues of a matrix take randomly from a unitary ensemble with a convex potential V 1 e n n Tr V (M ) dM Then run DBM and consider the distribution of the points at time t . These have the same distribution as the eigenvalues of t e MV + p 1 e 2t M GUE (Recall talk by Kuijlaars on Monday) In this case, we have multiple orthogonal polynomials with wider band recurrence matrices but we still find a Toeplitz structure in the limit with relative ease, see Breuer-D. DBM with two initial points Now consider DBM with two starting points ↵1 and ↵2 . We assume that the number of paths n is even and that half of the points start at each point. (Equivalent picture of brownian bridges) ↵1 ↵2 Multiple Hermite polynomials The locations at time t form a biorthogonal ensembles constructed by Multiple Hermite Polynomials (Johansson ’02, Bleher-Kuijlaars ’04) 1 Pk1 ,k2 (x) = p 2⇡i Z 1 Qk1 ,k2 (x) = p 2⇡2⇡i (z k1 a1 ) (z k2 a2 ) e 1 2 (z x)2 dz. iR I (w a1 ) k1 (w a2 ) k2 e 1 2 (w x)2 dz. a1 ,a2 The Pk1 ,k2 is a polynomial of degree k1 + k2. Then, up to a rescaling and re-parametrization, the points at time t form a biorthogonal ensemble with { n } j j=1 = {P0,0 , P1,0 , P1,1 , P2,1 , . . .} { n } j j=1 = {Q1,0 , Q1,1 , Q2,1 , Q2,2 , . . .} Recurrence relations for Multiple Hermite Polynomials One can show that xPk1 ,k2 (x) = Pk1 +1,k2 (x) + a1 Pk1 ,k2 (x) + k1 Pk1 1,k2 (x) + k2 Pk1 ,k2 1 (x) We can exchange indices according to the rule Pk1 +1,k2 (x) = Pk1 ,k2 +1 (x) + (a2 a1 )Pk1 ,k2 (x). Then we have the following recurrence relation xPk1 ,k2 (x) = Pk1 +1,k2 (x) + a1 Pk1 ,k2 (x) + (k1 + k2 )Pk1 ,k2 xPk1 ,k2 (x) = Pk1 ,k2 +1 (x) + a2 Pk1 ,k2 (x) + (k1 + k2 )Pk1 1 (x) 1,k2 (x) (a2 (a1 a1 )k1 Pk1 a2 )k1 Pk1 1,k2 1 (x). 1,k2 1 (x). Limit of the recurrence matrix After some computation by plugging in the recurrences we find lim n!1 where and ⇣ ⌘ (J (n) )n+k,n+l 0 A0 B A1 B J = J B @ A1 = (e t A 1 A0 A1 t e ) ✓ (J)n+k,n+l = 0. A 1 A0 .. . ✓1 2 (a1 ◆ A .. 1 .. . a2 ) . C C C A 1 1 2 (a2 0 a1 1 A0 = e + (et 0 a2 ✓ ◆ t 0 0 A 1=e 1 0 t 1 ✓ 0 0 e ) 1 0 t a1 ) ◆ ◆ A more general Central Limit Theorem The CLT of Breuer-D. can be generalized into Theorem Suppose there exists and such that ⇣ ⌘ (n) lim (J )n+k,n+l (J)n+k,n+l = 0. n!1 and that for a polynomial we can decompose p(J) = p(J)+ + p(J) such that 1) p(J)+ is lower triangular, 2) p(J) is upper triangular, 3) [p(J)+ , p(J) ] is trace class Then, as n ! 1, we have Xn (p) D EXn (p) ! N (0, Tr[p(J)+ , p(J) ]) , Some comments If J is a Toeplitz matrix, the conditions are satisfied for every polynomial and therefore the aforementioned theorem in Breuer-D is a direct consequence of this theorem. This follows by simple identities for Toeplitz matrices. If J is a block Toeplitz matrix, then this generically fails, although it may be true for some polynomials (e.g. in the symmetric two-cut case for unitary ensemble, even polynomials still satisfy a CLT) However, in the case of multiple Hermite polynomials, the condition is true for every polynomial! This is because of a very particular structure in the blocks. It is an interesting problem for the theory of multiple orthogonal polynomials to characterize conditions under which this happens. CLT for DBM with two starting points Theorem [D. upcoming] Let f be a polynomial. Then EXn (f ) ! N N 0, 0, Xn (f ) where 1 fk = 2⇡i and A1 = (et I e t) ✓ ✓1 2 (a1 ◆ a2 ) 0 kTrTr(f(fk kff kk)) k=1 p (zA1 + A0 + A2 /z) a1 1 A0 = e + (et 0 a2 ✓ ◆ t 0 0 A 1=e 1 0 t 1 X X !! 1 1 2 (a2 dz z k+1 ◆ a1 ) ✓ ◆ 0 0 t e ) 1 0 4. Extended Biorthogonal Ensembles Multi-time fluctuations Extended Biorthogonal Ensembles Now consider a probability measure on (Rn )N of the form N 1 Y 1 n n det ( (x )) det (T (x , x )) j,1 1,k m m,i m+1,j j,k=1 i,j=1 Zn m=1 N Y n Y n ⇥ det ( j,N (xN,k ))j,k=1 dµm (xm,j ), m=1 j=1 We assume that the operators Tm : L2 (µm ) ! L2 (µm+1 ) defined by Tm f (y) = are bounded. Z f (x)Tm (x, y)dµm (x), Recurrence matrices We assume that the functions are biorthogonal in the sense Z j,N (x)TN 1 TN Then define the functions j,m = Tm 1 2 · · · T1 j,m · · · T1 k,1 (x)dµN (x) and j,1 , = jk , j,m j,m = Tm⇤ · · · TN⇤ 1 j,N , which are biorthogonal as well Z j,m (x) k,m (x)dµ(x) = jk The marginals for fixed m gives a biorthogonal ensemble with respect to the functions j,m and j,m . Recurrence matrices We assume that that Z j,1 and j,N (x)TN k,N 1 TN 2 can be embedded into families such · · · T1 k,1 (x)dµN (x) = jk , and that there exists band matrices Jm 0 B B xB @ 0,m (x) 1 0 C B B C = Jm B 2,m (x)A @ .. . 1,m (x)C 0,m (x) 1 C C. 2,m (x)A .. . 1,m (x)C The matrices may depend on n , except for the bandwidths. Universality Consider linear statistics Xn (f ) = N X n X f (m, xj,m ) m=1 j=1 Theorem (D. arXiv) Suppose we have two different extended biorthogonal ensembles such that for each m ⇣ ⌘ lim (Jm )n+k,n+l (J˜m )n+k,n+l = 0 n!1 Then for any f (x, m) such that it is a polynomial in h E (Xn (f ) as n ! 1 EXn (f )) k i e E ⇣ Xn (f ) x and k 2 N e n (f ) EX ⌘k ! 0, Central Limit Theorem Theorem (D. arXiv) Suppose that for each m, k, ` lim (Jm )n+k,n+l = n!1 (m) ak l , Then for any f (m, x) that is a polynomial in x, as n ! 1 , Xn (f ) N EXn (f ) ! 0, 2 N X N X 1 X (m1 ) (m2 ) f k kfk m1 =1 m2 =m1 +1 k=1 where (m) fk 1 = 2⇡i I f (m, |z|=1 + N X 1 X m=1 k=1 X j (m) (m) f k kfk dz (m) j aj z ) k+1 . z ! , Example: Stationary DBM In stationary DBM we have 1 det ( Zn n j,1 (x1,k ))j,k=1 det Ptm+1 n j,N (xN,k ))j,k=1 ⌧1 (j 1) =e ⌧N (j = e j,N j,1 n i,j=1 N Y n Y dµ (xm,j ), dxmm,j m=1 j=1 where tm (xm,i , xm+1,j ) m=1 ⇥ det ( N Y1 Hj 1) Hj p 1( nx) p 1 ( nx) e nx2 where Hj stands for the j-th Hermite polynomial and 1 Pt (x, y) = p e 2⇡n 1 n e 2t (e t x y)2 is the transition probability for the OU process. Example: Dyson’s Brownian motion (2) Then j,m (x) = Tm 1 · · · T1 j,1 (x) j,m =e ⌧m (j 1) Hj 1( p nx) and hence we obtain the recurrence matrices )jk = e (J(Jmm)jk ⌧m (j k) (JHermite )jk Multi-time Global CLT: As n ! 1, for sufficiently smooth functions f EXn (f ) ! N (0, (f )2 ) Xn (f ) where (f )2 = N X 1 X e |⌧m1 ⌧m2 |k (m1 ) (m2 ) fk , kfk m1 ,m2 =1 k=1 and (m) fk 1 = ⇡ Z ⇡ f (m, 0 p 2 cos ✓) cos k✓d✓ Further comments This construction can be generalized in a straightforward way. The Hermite functions are eigenfunctions for the generator of the OrnsteinUhlenbeck processes. Similar statements are true other process for which the generator has the classical orthogonal polynomials as eigenfunctions. Such as Laguerre and Jacobi polynomials in the continuous case and in the discrete case one can turn to birth/death processes, Meixner, Charlier, Krawtchouk. It can also be applied to tiling models, such as lozenge tilings of a regular hexagon. One can rigorously show that the CLT explains a convergence of the fluctuation of the associated random height function to the Gaussian Free Field. By an extension we can also handle certain cases involving multiple orthogonal polynomials: Multi-time CLT for DBM with two starting points Theorem [D. upcoming] Let f (⌧, x) be a polynomial in x . Then Xn (f ) N 0, 2 N X EXn (f ) ! N N X 1 X Tr (fk f k=1 ⇣ (m1 ) (m2 ) f k k Tr fk m1 =1 m2 =m1 +1 k=1 0, X ⌘ + k) ! N X 1 X ⇣ (m) (m) f k k Tr fk m=1 k=1 ⌘ ! where (m) fk 1 = 2⇡i I f |z|=1 and (m) A1 (m) A0 A (m) 1 = (e ⌧m ⇣ (m) ⌧m , A1 z e ✓ ) 1 a2 ✓ ◆ 0 0 = e ⌧m 1 0 =e ⌧m a1 0 ⌧m ✓1 2 (a1 ◆ + (m) A0 a2 ) 0 + (e⌧m + (m) A 1z 1 1 1 2 (a2 ◆ a1 ) ✓ ◆ 0 0 e ⌧m ) 1 0 ⌘ dz z k+1 Thank you for you attention!