Mathematics – Probability
Scientific paper
2011-10-16
Mathematics
Probability
34 pages,6 figures
Scientific paper
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 (TS). 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.
Corwin Ivan
O'Connell Neil
Seppalainen Timo
Zygouras Nikos
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