From longest increasing subsequences to Whittaker functions and random polymers

Transcription

1 From longest increasing subsequences to Whittaker functions and random polymers Neil O Connell University of Warwick British Mathematical Colloquium, April 2, 2015 Collaborators: I. Corwin, T. Seppäläinen, N. Zygouras Neil O Connell 1 / 35

2 The longest increasing subsequence problem For a permutation σ S n, write L n (σ) = length of longest increasing subsequence in σ E.g. if σ = then L 6 (σ) = 3. Neil O Connell 2 / 35

3 The longest increasing subsequence problem For a permutation σ S n, write L n (σ) = length of longest increasing subsequence in σ E.g. if σ = then L 6 (σ) = 3. Based on Monte-Carlo simulations, Ulam (1961) conjectured that EL n = 1 L n (σ) c n, n. n! σ S n A classical result from combinatorial geometry (Erdős-Szekeres 1935) implies that EL n n 1/2. Neil O Connell 2 / 35

4 The longest increasing subsequence problem Hammersley (1972): The limit c exists, and π/2 c e. Neil O Connell 3 / 35

5 The longest increasing subsequence problem Hammersley (1972): The limit c exists, and π/2 c e. Logan and Shepp (1977): c 2 Neil O Connell 3 / 35

6 The longest increasing subsequence problem Hammersley (1972): The limit c exists, and π/2 c e. Logan and Shepp (1977): c 2 Vershik and Kerov (1977): c = 2 Neil O Connell 3 / 35

7 The longest increasing subsequence problem Hammersley (1972): The limit c exists, and π/2 c e. Logan and Shepp (1977): c 2 Vershik and Kerov (1977): c = 2 Baik, Deift and Johansson (1999): for each x R, 1 n! {σ S n : n 1/6 (L n (σ) 2 n) x} F 2 (x), where F 2 is the Tracy-Widom (GUE) distribution from random matrix theory (Tracy and Widom 1994 limiting distribution of largest eigenvalue of high-dimensional random Hermitian matrix) Neil O Connell 3 / 35

8 The longest increasing subsequence problem Hammersley (1972): The limit c exists, and π/2 c e. Logan and Shepp (1977): c 2 Vershik and Kerov (1977): c = 2 Baik, Deift and Johansson (1999): for each x R, 1 n! {σ S n : n 1/6 (L n (σ) 2 n) x} F 2 (x), where F 2 is the Tracy-Widom (GUE) distribution from random matrix theory (Tracy and Widom 1994 limiting distribution of largest eigenvalue of high-dimensional random Hermitian matrix) How is this possible? Neil O Connell 3 / 35

9 The Robinson-Schensted correspondence From the representation theory of S n, n! = λ n d 2 λ where d λ = number of standard tableaux with shape λ. A standard tableau with shape (4, 3, 1) 8: In other words, S n has the same cardinality as the set of pairs of standard tableaux of size n with the same shape. Neil O Connell 4 / 35

10 The Robinson-Schensted correspondence Robinson (38): A bijection between S n and such pairs σ (P, Q) Schensted (61): L n (σ) = length of longest row of P and Q This yields {σ S n : L n (σ) k} = dλ 2. λ n, λ 1 k Neil O Connell 5 / 35

11 The RSK correspondence Knuth (70): Extends to a bijection between matrices with nonnegative integer entries and pairs of semi-standard tableaux of same shape. A semistandard tableau of shape λ n is a diagram of that shape, filled in with positive integers which are weakly increasing along rows and strictly increasing along columns. A semistandard tableau of shape (5, 3, 1): Neil O Connell 6 / 35

12 Cauchy-Littlewood identity This gives a combinatorial proof of the Cauchy-Littlewood identity (1 x i y j ) 1 = s λ (x)s λ (y), ij λ where s λ are Schur polynomials, defined by s λ (x) = where x = (x 1, x 2,...) and sh P=λ x P, x P = x 1 s in P 1 x 2 s in P Neil O Connell 7 / 35

13 Cauchy-Littlewood identity Let (a ij ) (P, Q) under RSK. Then C j = i a ij = j s in P and R i = j a ij = i s in Q. For x = (x 1, x 2,...) and y = (y 1, y 2,...) we have (y i x j ) a ij = ij j x C j j i y R i i = x P y Q. Summing over (a ij ) on the left and (P, Q) with sh P = sh Q on the right gives (1 x i y j ) 1 = ij λ s λ (x)s λ (y). Neil O Connell 8 / 35

14 Tableaux and Gelfand-Tsetlin patterns Semistandard tableaux discrete Gelfand-Tsetlin patterns Neil O Connell 9 / 35

15 The RSK correspondence If (a ij ) N m n, then length of longest row in corresponding tableaux is M = max π (i,j) π a ij (m, n) (1, 1) Neil O Connell 10 / 35

16 Combinatorial interpretation Consider n queues in series: Data: a ij = time required to serve i th customer at j th queue If we start with all customers in first queue, then M is the time taken for all customers to leave the system (Muth 79). Neil O Connell 11 / 35

17 Combinatorial interpretation From the RSK correspondence: If a ij are independent random variables with P(a ij k) = (p i q j ) k then P(M k) = ij (1 p i q j ) λ: λ 1 k s λ (p)s λ (q). cf. Weber (79): The interchangeability of /M/1 queues in series. Johansson (99): As n, m, M Tracy-Widom distribution (and other related asymptotic results) Neil O Connell 12 / 35

18 Surface growth and KPZ universality The queueing system can be thought of as a model for surface growth... Customer Queue Neil O Connell 13 / 35

19 Non-equilibrium statistical mechanics Surface growth and KPZ universality Stochastic interacting particle systems... and belongs to the same universality class as: Integrable systems Random Tilings Research Group Random tiling Burning paper Bacteria colonies 2 KPZ = Kardar-Parisi-Zhang (1986) Neil O Connell 14 / 35

20 Geometric RSK correspondence The RSK mapping can be defined by expressions in the (max, +)-semiring. Replacing these expressions by their (+, ) counterparts, A.N. Kirillov (00) introduced a geometric lifting of RSK correspondence. It is a bi-rational map For n = 2, T : (R >0 ) n n (R >0 ) n n X = (x ij ) (t ij ) = T = T(X). x 21 x 11 x 22 x 12 x 11 x 21 x 12 x 21 /(x 12 + x 21 ) x 11 x 22 (x 12 + x 21 ) x 11 x 12 Neil O Connell 15 / 35

21 Geometric RSK correspondence The analogue of the longest increasing subsequence is the matrix element: t nn = φ Π (n,n) (i,j) φ x ij (n, n) (1, 1) Neil O Connell 16 / 35

22 Geometric RSK correspondence t nm = φ Π (n,m) (i,j) φ x ij (n, m) (1, 1) Neil O Connell 17 / 35

23 Geometric RSK correspondence t n k+1,m k+1... t nm = φ Π (k) (i,j) φ (n,m) x ij (n, m) (1, 1) Neil O Connell 18 / 35

24 Geometric RSK correspondence t n k+1,m k+1... t nm = φ Π (k) (i,j) φ (n,m) x ij (n, m) T(X) = T(X ) (1, 1) Neil O Connell 18 / 35

25 Whittaker functions Whittaker functions were first introduced by Jacquet (67). They play an important role in the theory of automorphic forms and also arise as eigenfunctions of the open quantum Toda chain (Kostant 77) Neil O Connell 19 / 35

26 Whittaker functions Whittaker functions were first introduced by Jacquet (67). They play an important role in the theory of automorphic forms and also arise as eigenfunctions of the open quantum Toda chain (Kostant 77) In the context of GL(n, R), they can be considered as functions Ψ λ (x) on (R >0 ) n, indexed by a (spectral) parameter λ C n Neil O Connell 19 / 35

27 Whittaker functions Whittaker functions were first introduced by Jacquet (67). They play an important role in the theory of automorphic forms and also arise as eigenfunctions of the open quantum Toda chain (Kostant 77) In the context of GL(n, R), they can be considered as functions Ψ λ (x) on (R >0 ) n, indexed by a (spectral) parameter λ C n The following Gauss-Givental representation for Ψ λ is due to Givental (97), Joe-Kim (03), Gerasimov-Kharchev-Lebedev-Oblezin (06) Neil O Connell 19 / 35

28 Whittaker functions A triangle P with shape x (R >0 ) n is an array of positive real numbers: P = z 22 z 11 z 21 z nn z n1 with bottom row z n = x. Denote by (x) the set of triangles with shape x. Neil O Connell 20 / 35

29 Whittaker functions Let P = z 22 z 11 z 21 z nn z n1 Define P λ = R λ1 1 ( R2 R 1 ) λ2 ( ) λn Rn, λ C n, R k = R n 1 k i=1 z ki Neil O Connell 21 / 35

30 Whittaker functions Let P = z 22 z 11 z 21 z nn z n1 Define P λ = R λ1 1 ( R2 R 1 ) λ2 ( ) λn Rn, λ C n, R k = R n 1 z 11 z 21 k i=1 z ki F(P) = a b z a z b z33 z 22 z 32 z 31 Neil O Connell 21 / 35

31 Whittaker functions For λ C n and x (R >0 ) n, define Ψ λ (x) = where dp = 1 i k<n dz ki/z ki. For n = 2, (x) P λ e F(P) dp, Ψ (ν/2, ν/2) (x) = 2K ν ( 2 x 2 /x 1 ). These are called GL(n)-Whittaker functions. As we shall see, they are the analogue of the Schur polynomials in the geometric setting. Neil O Connell 22 / 35

32 Geometric RSK correspondence Recall X = (x ij ) (t ij ) = T = t 31 t 21 t 32 t 11 t 22 t 33 t 12 t 23 t 13 = pair of triangles of same shape (t nn,..., t 11 ). t nn = φ Π (n,n) (i,j) φ x ij (n, n) (1, 1) Neil O Connell 23 / 35

33 Whittaker measures Let a, b R n with a i + b j > 0 and define P(dX) = ij Γ(a i + b j ) 1 x a i b j 1 ij e 1/x ij dx ij. Theorem (Corwin-O C-Seppäläinen-Zygouras 14) Under P, the law of the shape of the output under geometric RSK is given by the Whittaker measure on R n + defined by µ a,b (dx) = ij Γ(a i + b j ) 1 e 1/xn Ψ a (x)ψ b (x) n i=1 dx i x i. Neil O Connell 24 / 35

34 Application to random polymers Corollary Suppose a i > 0 for each i and b j < 0 for each j. Then Ee stnn = s n i=1 (b i λ i ) Γ(λ i b j ) ιr m ij ij Γ(a i + λ j ) Γ(a i + b j ) s n(λ)dλ, where s n (λ) = 1 (2πι) n n! Γ(λ i λ j ) 1. i j Neil O Connell 25 / 35

35 Combinatorial approach Recall: X = (x ij ) (t ij ) = T(X) = (P, Q). The following is a refinement of the previous theorem. Theorem (O C-Seppäläinen-Zygouras 14) The map (log x ij ) (log t ij ) has Jacobian ±1 For ν, λ C n, The following identity holds: ij ij x ν i+λ j ij = P λ Q ν 1 = 1 + F(P) + F(Q) x ij t 11 This theorem (a) explains the appearance of Whittaker functions and (b) extends to models with symmetry. Neil O Connell 26 / 35

36 Analogue of the Cauchy-Littlewood identity It follows that ij x νi λj ij e dx 1/xij ij = P λ Q ν e 1/t11 F(P) F(Q) x ij ij dt ij t ij. Integrating both sides gives, for R(ν i + λ j ) > 0: Corollary (Stade 02) Γ(ν i + λ j ) = ij R n + e 1/xn Ψ ν (x)ψ λ (x) n i=1 dx i x i. This is equivalent to a Whittaker integral identity which was conjectured by Bump (89) and proved by Stade (02). The integral is associated with Archimedean L-factors of automorphic L-functions on GL(n, R) GL(n, R). Neil O Connell 27 / 35

37 Local moves Proof of second theorem uses new description of the grsk map T as a composition of a sequence of local moves applied to the input matrix x 31 x 21 x 32 x 11 x 22 x 33 x 12 x 23 x 13 Neil O Connell 28 / 35

38 Local moves The basic move is: b a d c Neil O Connell 29 / 35

39 Local moves The basic move is: b bc ab + ac bd + cd c Neil O Connell 30 / 35

40 Combinatorial approach Recall: X = (x ij ) (t ij ) = T(X) = (P, Q). The following is a refinement of the previous theorem. Theorem (O C-Seppäläinen-Zygouras 14) The map (log x ij ) (log t ij ) has Jacobian ±1 For ν, λ C n, The following identity holds: ij ij x ν i+λ j ij = P λ Q ν 1 = 1 + F(P) + F(Q) x ij t 11 This theorem (a) explains the appearance of Whittaker functions and (b) extends to models with symmetry. Neil O Connell 31 / 35

41 Integrable systems point of view Quantum Toda chain n 1 H = 2 i=1 e x i+1 x i Hψ λ = ( λ 2 ) i ψλ ψ λ (x) = Ψ λ (e x ) Associated diffusion process in R n with generator i L λ = 1 2 ψ λ(x) 1( H λ 2 i ) ψλ (x) = log ψ λ. Neil O Connell 32 / 35

42 Geometric RSK (in continuous time) Let η : [0, ) R n be smooth (or Brownian) with η(0) = 0. Define b(t) in upper triangular matrices by ḃ = ɛ( η)b, b(0) = Id., where Consider the principal minors ] k = det [b ij λ λ ɛ(λ) = λ n λ n 1 i k, n k+1 j n, and define x i = log( i / i 1 ), where 0 = 1. 1 k n Neil O Connell 33 / 35

43 Geometric RSK (in continuous time) The main results described above are in fact discrete-time versions of: Theorem (O C 12) If η(t) = B(t) + λt then x(t) is a diffusion with generator L λ. Neil O Connell 34 / 35

44 Geometric RSK (in continuous time) The main results described above are in fact discrete-time versions of: Theorem (O C 12) If η(t) = B(t) + λt then x(t) is a diffusion with generator L λ. What happens if we remove the noise? Neil O Connell 34 / 35

45 Geometric RSK (in continuous time) The main results described above are in fact discrete-time versions of: Theorem (O C 12) If η(t) = B(t) + λt then x(t) is a diffusion with generator L λ. What happens if we remove the noise? Theorem (O C 13) If η(t) = λt then x(t) is a solution to the classical Toda flow with constants of motion (Lax matrix eigenvalues) λ 1,..., λ n. Neil O Connell 34 / 35

46 Geometric RSK (in continuous time) The main results described above are in fact discrete-time versions of: Theorem (O C 12) If η(t) = B(t) + λt then x(t) is a diffusion with generator L λ. What happens if we remove the noise? Theorem (O C 13) If η(t) = λt then x(t) is a solution to the classical Toda flow with constants of motion (Lax matrix eigenvalues) λ 1,..., λ n. This phenomena extends to hyperbolic Calogero-Moser systems. In all cases, the quantum systems (in imaginary time) are obtained by adding noise to the constants of motion in particular representations of the classical systems. Neil O Connell 34 / 35

47 References N. O Connell. Stochastic Bäcklund transformations. arxiv: To appear in Seminaires de Probabilités, special volume for Marc Yor. N. O Connell. Geometric RSK and the Toda lattice. Illinois J. Math., N. O Connell, T. Seppäläinen, N. Zygouras. Geometric RSK correspondence, Whittaker functions and symmetrized random polymers. Inventiones Math., I. Corwin, N. O Connell, T. Seppäläinen, N. Zygouras. Tropical Combinatorics and Whittaker functions. Duke Math. J., N. O Connell. Directed polymers and the quantum Toda lattice. Ann. Probab. 40 (2012) Neil O Connell 35 / 35