Variants of geometric RSK, geometric PNG and the multipoint distribution of the log-gamma polymer

TL;DR

This paper extends the geometric RSK correspondence to polygonal arrays and relates it to a geometric PNG model, deriving integral formulas for the log-gamma polymer's joint distribution and connecting it to the Airy process in a scaling limit.

Contribution

It introduces a generalized geometric RSK framework for polygonal arrays and links it to a geometric PNG model, providing new integral formulas and asymptotic results.

Findings

Derived integral formulas for the joint distribution of log-gamma polymer partition functions.

Extended geometric RSK to polygonal arrays and related it to geometric PNG.

Showed convergence to the Airy process in a specific scaling limit.

Abstract

We show that the reformulation of the geometric Robinson-Schensted-Knuth (gRSK) correspondence via local moves, introduced in \cite{OSZ14} can be extended to cases where the input matrix is replaced by more general polygonal, Young-diagram-like, arrays of the form $\polygon$. We also show that a rearrangement of the sequence of the local moves gives rise to a geometric version of the polynuclear growth model (PNG). These reformulations are used to obtain integral formulae for the Laplace transform of the joint distribution of the point-to-point partition functions of the log-gamma polymer at different space-time points. In the case of two points at equal time $N$ and space at distance of order $N^{2/3}$, we show formally that the joint law of the partition functions, scaled by $N^{1/3}$, converges to the two-point function of the Airy process