Geometric RSK, Whittaker functions and random polymers Neil - PowerPoint PPT Presentation

Geometric RSK, Whittaker functions and random polymers Neil O’Connell University of Warwick / Trinity College Dublin School and Workshop on Random Interacting Systems Bath, June 25, 2014 Collaborators: I. Corwin, T. Seppäläinen, N. Zygouras Neil O’Connell 1 / 62

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. Neil O’Connell 2 / 62

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. Based on Monte-Carlo simulations, Ulam (1961) conjectured that L n ( σ ) ∼ c √ n , EL n = 1 � n → ∞ . n ! σ ∈ S n A classical result from combinatorial geometry (Erd˝ os-Szekeres 1935) √ implies that EL n ≥ n − 1 / 2. Neil O’Connell 2 / 62

The longest increasing subsequence problem Hammersley (1972): The limit c exists, and π/ 2 ≤ c ≤ e . Neil O’Connell 3 / 62

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 / 62

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 / 62

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 , n ! |{ σ ∈ S n : n − 1 / 6 ( L n ( σ ) − 2 √ n ) ≤ x }| → F 2 ( x ) , 1 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 / 62

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 , n ! |{ σ ∈ S n : n − 1 / 6 ( L n ( σ ) − 2 √ n ) ≤ x }| → F 2 ( x ) , 1 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 / 62

The Robinson-Schensted correspondence From the representation theory of S n , � d 2 n ! = λ λ ⊢ n 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 4 / 62

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 � d 2 |{ σ ∈ S n : L n ( σ ) ≤ k }| = λ . λ ⊢ n , λ 1 ≤ k Neil O’Connell 5 / 62

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 6 / 62

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 � x P , s λ ( x ) = sh P = λ where x = ( x 1 , x 2 , . . . ) and x P = x ♯ 1 ′ s in P x ♯ 2 ′ s in P . . . . 1 2 Neil O’Connell 7 / 62

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 = x C j � � � y R i i = x P y Q . j ij j i 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 = � � s λ ( x ) s λ ( y ) . ij λ Neil O’Connell 8 / 62

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 9 / 62

The RSK correspondence If ( a ij ) ∈ N m × n , then length of longest row in corresponding tableaux is ( m , n ) � M = max π ( i , j ) ∈ π a ij ( 1 , 1 ) Neil O’Connell 10 / 62

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 / 62

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 ) = ( 1 − p i q j ) s λ ( p ) s λ ( q ) . ij λ : λ 1 ≤ k 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 / 62

Surface growth and KPZ universality The queueing system can be thought of as a model for surface growth . . . Customer 5 4 3 2 1 1 2 3 4 Queue Neil O’Connell 13 / 62

Surface growth and KPZ universality . . . and belongs to the same universality class as: Random tiling Burning paper Bacteria colonies KPZ = Kardar-Parisi-Zhang (1986) Neil O’Connell 14 / 62

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 T : ( R > 0 ) n × n → ( R > 0 ) n × n X = ( x ij ) �→ ( t ij ) = T = T ( X ) . For n = 2, x 21 x 11 x 21 �→ x 12 x 21 / ( x 12 + x 21 ) x 11 x 22 ( x 12 + x 21 ) x 11 x 22 x 12 x 11 x 12 Neil O’Connell 15 / 62

Geometric RSK correspondence The analogue of the ‘longest increasing subsequence’ is the matrix element: � � t nn = x ij φ ∈ Π ( n , n ) ( i , j ) ∈ φ ( n , n ) ( 1 , 1 ) Neil O’Connell 16 / 62

Geometric RSK correspondence � � t nm = x ij φ ∈ Π ( n , m ) ( i , j ) ∈ φ ( n , m ) ( 1 , 1 ) Neil O’Connell 17 / 62

Geometric RSK correspondence � � t n − k + 1 , m − k + 1 . . . t nm = x ij φ ∈ Π ( k ) ( i , j ) ∈ φ ( n , m ) ( n , m ) ( 1 , 1 ) Neil O’Connell 18 / 62

Geometric RSK correspondence � � t n − k + 1 , m − k + 1 . . . t nm = x ij φ ∈ Π ( k ) ( i , j ) ∈ φ ( n , m ) ( n , m ) T ( X ) ′ = T ( X ′ ) ( 1 , 1 ) Neil O’Connell 18 / 62

Whittaker functions A triangle P with shape x ∈ ( R > 0 ) n is an array of positive real numbers: z 11 z 22 z 21 P = z nn z n 1 with bottom row z n · = x . Denote by ∆( x ) the set of triangles with shape x . Neil O’Connell 19 / 62

Whittaker functions Let z 11 z 22 z 21 P = z nn z n 1 Define � R n � λ 2 � λ n k � R 2 P λ = R λ 1 � λ ∈ C n , · · · , R k = z ki 1 R 1 R n − 1 i = 1 Neil O’Connell 20 / 62

Whittaker functions Let z 11 z 22 z 21 P = z nn z n 1 Define � R n � λ 2 � λ n k � R 2 P λ = R λ 1 � λ ∈ C n , · · · , R k = z ki 1 R 1 R n − 1 i = 1 z 11 z a z 22 z 21 � F ( P ) = z b z 33 z 32 z 31 a → b Neil O’Connell 20 / 62

Whittaker functions For λ ∈ C n and x ∈ ( R > 0 ) n , define � P − λ e −F ( P ) dP , Ψ λ ( x ) = ∆( x ) where dP = � 1 ≤ i ≤ k < n dz ki / z ki . For n = 2, � � � Ψ ( ν/ 2 , − ν/ 2 ) ( x ) = 2 K ν 2 π 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 21 / 62

Geometric RSK correspondence Recall t 31 t 21 t 32 X = ( x ij ) �→ ( t ij ) = T = t 11 t 22 t 33 t 12 t 23 t 13 = pair of triangles of same shape ( t nn , . . . , t 11 ) . ( n , n ) t nn = � � ( i , j ) ∈ φ x ij φ ∈ Π ( n , n ) ( 1 , 1 ) Neil O’Connell 22 / 62

Whittaker measures Let a , b ∈ R n with a i + b j > 0 and define Γ( a i + b j ) − 1 x − a i − b j − 1 � e − 1 / x ij dx ij . P ( dX ) = ij 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 n dx i � Γ( a i + b j ) − 1 e − 1 / x n Ψ a ( x )Ψ b ( x ) � µ a , b ( dx ) = . x i ij i = 1 Neil O’Connell 23 / 62

Application to random polymers Corollary Suppose a i > 0 for each i and b j < 0 for each j. Then � Γ( a i + λ j ) � n E e − st nn = i = 1 ( b i − λ i ) � � Γ( λ i − b j ) Γ( a i + b j ) s n ( λ ) d λ, ι R m s ij ij where 1 � Γ( λ i − λ j ) − 1 . s n ( λ ) = ( 2 πι ) n n ! i � = j Neil O’Connell 24 / 62