Geometric RSK, Whittaker functions and random polymers

Geometric RSK, Whittaker functions and random polymers Neil O Connell University of Warwick Advances in Probability: Integrability, Universality and Beyond Oxford, September 29, 2014 Collaborators: I. Corwin, T. Seppäläinen, N. Zygouras Neil O Connell 1 / 64

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 = 154263 then L 6 (σ) = 3. Ulam (1960) What is typical behaviour of L n (σ)? Long history... 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 2 / 64

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: 1 3 5 6 2 4 8 7 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 3 / 64

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 4 / 64

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): 1 2 2 5 7 3 3 8 4 Neil O Connell 5 / 64

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 2.... Neil O Connell 6 / 64

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 7 / 64

Tableaux and Gelfand-Tsetlin patterns Semistandard tableaux discrete Gelfand-Tsetlin patterns 1 1 1 2 2 3 2 2 3 3 3 3 0 1 2 3 4 5 6 Neil O Connell 8 / 64

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 9 / 64

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 10 / 64

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 11 / 64

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 12 / 64

Geometric RSK correspondence t nm = φ Π (n,m) (i,j) φ x ij (n, m) (1, 1) Neil O Connell 13 / 64

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 14 / 64

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 14 / 64

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 15 / 64

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 16 / 64

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 17 / 64

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 17 / 64

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. They are the analogue of Schur polynomials in the geometric setting. Neil O Connell 18 / 64

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 19 / 64

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 20 / 64

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 If a i + b j = θ for all i, j, this is the log-gamma polymer model introduced by Seppäläinen (2012). Neil O Connell 21 / 64

Application to random polymers Using the above integral formula, Borodin, Corwin and Remenik (2013) have shown that for θ < θ (for technical reasons) log t nn c(θ)n d(θ)n 1/3 dist F 2. The constant c(θ) = 2Ψ(θ/2) and bound on fluctuation exponent χ < 1/3 were established earlier by Seppäläinen (2012). Neil O Connell 22 / 64

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 23 / 64

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 24 / 64

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 This description is a re-formulation of Noumi and Yamada s (2004) geometric row insertion algorithm. Neil O Connell 25 / 64

Local moves The basic move is: b a d c Neil O Connell 26 / 64

Local moves The basic move is: b bc ab + ac bd + cd c Neil O Connell 27 / 64

Local moves This can be applied at any position in the matrix: c b f a e i d h g Neil O Connell 28 / 64

Local moves This can be applied at any position in the matrix: c b f a e i d h g Neil O Connell 29 / 64

Local moves This can be applied at any position in the matrix: c ce bc + be cf + ef a e i d h g Neil O Connell 30 / 64

Local moves This can be applied at any position in the matrix: c b f a e i d h g Neil O Connell 31 / 64

Local moves This can be applied at any position in the matrix: c b f a e i eg de + dg eh + gh g Neil O Connell 32 / 64

Local moves This can be applied at any position in the matrix: c b f a e i d h g Neil O Connell 33 / 64

Local moves This can be applied at any position in the matrix: c b f a e i d h dg Neil O Connell 34 / 64

Local moves This can be applied at any position in the matrix: c b f a e i d h g Neil O Connell 35 / 64

Local moves This can be applied at any position in the matrix: c ab f a e i d h g Neil O Connell 36 / 64

Local moves This can be applied at any position in the matrix: c b f a e i d h g Neil O Connell 37 / 64

Local moves This can be applied at any position in the matrix: c b f a e i d h g Neil O Connell 38 / 64

Local moves Start with: x 31 x 21 x 32 x 11 x 22 x 33 x 12 x 23 x 13 Neil O Connell 39 / 64

Local moves Apply the local moves in the following order: Neil O Connell 40 / 64

Local moves Apply the local moves in the following order: Neil O Connell 41 / 64

Local moves Apply the local moves in the following order: Neil O Connell 42 / 64

Local moves Apply the local moves in the following order: Neil O Connell 43 / 64

Local moves Apply the local moves in the following order: Neil O Connell 44 / 64

Local moves Apply the local moves in the following order: Neil O Connell 45 / 64

Local moves Apply the local moves in the following order: Neil O Connell 46 / 64

Local moves Apply the local moves in the following order: Neil O Connell 47 / 64

Local moves Apply the local moves in the following order: Neil O Connell 48 / 64

Local moves Apply the local moves in the following order: Neil O Connell 49 / 64

Local moves Apply the local moves in the following order: Neil O Connell 50 / 64

Local moves Apply the local moves in the following order: Neil O Connell 51 / 64

Local moves Apply the local moves in the following order: Neil O Connell 52 / 64

Local moves Apply the local moves in the following order: Neil O Connell 53 / 64

Local moves To arrive at: t 31 t 21 t 32 t 11 t 22 t 33 t 12 t 23 t 13 Neil O Connell 54 / 64

Symmetric input matrix Symmetry properties of grsk: T(X ) = T(X) X (P, Q) X (Q, P). X = X P = Q Theorem (O C-Seppäläinen-Zygouras 14) The restriction of T to symmetric matrices is volume-preserving. Neil O Connell 56 / 64

Symmetric input matrix The analogue of the Cauchy-Littlewood identity in this setting is: Corollary Suppose s > 0 and Rλ i > 0 for each i. Then (R >0 ) n e sx 1 Ψ n λ (x) n i=1 dx i x i = s n i=1 λ i i Γ(λ i ) i<j Γ(λ i + λ j ). This is equivalent to a Whittaker integral identity which was conjectured by Bump-Friedberg (90) and proved by Stade (01). Neil O Connell 57 / 64

Symmetric input matrix Corollary Let α i > 0 for each i and define P α (dx) = Z 1 α i x α i ii i<j x α i α j ij e 1 2 i 1 x ii i<j 1 x ij i j dx ij x ij. Then P α (sh P dx) = c 1 α e 1/2xn Ψ n α(x) i dx i x i, where c α = i Γ(α i ) i<j Γ(α i + α j ). Neil O Connell 58 / 64

Application to symmetrised random polymer (reflecting boundary conditions) Formally, this yields the integral formula: E α e stnn = s i λ Γ(λ i i ) Γ(α i + λ j ) Γ(α i ) i,j i<j i for appropriate vertical contours which stay to the right of zero. Γ(λ i + λ j ) Γ(α i + α j ) s n(λ)dλ Neil O Connell 59 / 64

Semi-discrete polymer and integrable systems Discrete KPZ equation (cf. TASEP) dx 1 = db 1, dx i = db i e X i X i 1 dt, i 2. Polymer interpretation: for appropriate initial conditions, X n (t) = log partition function of semi-discrete random polymer. n n 1 3 2 1 t 1 t 2 t 3 t n 2 t n 1 t O C-Yor 01 (combines ideas from queueing theory and mathematical finance) Neil O Connell 60 / 64

Connection to quantum Toda / Whittaker functions 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 ) i Theorem (O C 12) If (B 1,..., B n ) is a BM with drift λ then, for appropriate initial conditions, the process X n (t), t > 0 has the same law as the first coordinate of a diffusion process in R n with generator L λ = 1 2 ψ λ(x) 1( H λ 2 i ) ψλ (x) = 1 2 + log ψ λ. Yields (determinantal) formulae for law of X n (t). Generalises earlier result of Matsumoto-Yor 99, which corresponds to n = 2. Neil O Connell 61 / 64

Geometric RSK (in continuous time) Biane, Bougerol, O C 05 (cf. Kirillov 00): 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 λ 1 1 0... 0 0 λ 2 1... 0 ɛ(λ) =...... λ n 1 1 0... λ n Neil O Connell 62 / 64

Geometric RSK (in continuous time) Consider the principal minors ] k = det [b ij 1 i k, n k+1 j n, 1 k n and define x i = log( i / i 1 ), where 0 = 1. Theorem (O C 12, 14) (a) If η(t) = B(t) + λt then x(t) is a diffusion with generator L λ. (b) 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. The above construction extends naturally to rational and 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 63 / 64

References N. O Connell. Stochastic Bäcklund transformations. arxiv:1403.5709. To appear in Seminaires de Probabilités, special volume for Marc Yor. N. O Connell. Geometric RSK and the Toda lattice. Illinois J. Math., 2014. N. O Connell, T. Seppäläinen, N. Zygouras. Geometric RSK correspondence, Whittaker functions and symmetrized random polymers. Inventiones Math., 2014. I. Corwin, N. O Connell, T. Seppäläinen, N. Zygouras. Tropical Combinatorics and Whittaker functions. Duke Math. J., 2014. N. O Connell. Directed polymers and the quantum Toda lattice. Ann. Probab. 40 (2012) 437 458. Neil O Connell 64 / 64