Ivan Corwin: Tropical combinatorics and Whittaker functions
Abstract
The Robinson-Schensted-Knuth (RSK) correspondence is a combinatorial mapping which plays a
fundamental role in the theory of Young tableaux, symmetric functions, ultra-discrete
integrable systems and representation theory. It is also the basic structure that lies
behind the `solvability' of a particular family of combinatorial models in probability and
statistical physics which include longest increasing subsequence problems, directed last
passage percolation in 1+1 dimensions, the totally asymmetric exclusion process, queues in
series and discrete models for surface growth. There is a geometric version of the RSK
correspondence introduced by A.N. Kirillov, known as the `tropical RSK correspondence'. We
show that, with a particular family of product measures on its domain, the tropical RSK
correspondence is closely related to GL(N,R)-Whittaker functions and yields analogues in
this setting of the Schur measures and Schur processes on integer partitions.
As an application, we give an explicit integral formula for the generating function of the
partition function of a family of lattice one-dimensional directed polymer models with
log-gamma weights recently introduced by one of the authors. This positive temperature
extension of K. Johansson's work on last passage percolation offers an approach towards
rigorously computing statistics of the Kardar-Parisi-Zhang non-linear stochastic partial
differential equation.
This is based on joint work with Neil O'Connell, Timo Seppalainen and Nikos Zygouras.