Variants of Geometric RSK, Geometric PNG, and the Multipoint Distribution of the Log-Gamma Polymer

Abstract

Abstract. We show that the reformulation of the geometric Robinson-Schensted-Knuth (gRSK) correspondence via local moves, introduced in [OSZ14] can be extended to cases where the input matrix is replaced by more general polygonal, Young-diagram-like, arrays of the form . 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.

Point-to-line last passage percolation and the invariant measure of a system of reflecting Brownian motions

“…We prove (38); equation (37) is more straightforward. Observe that if z ≤ 0 then both sides are zero and for z > 0, (39) tends to zero as ǫ → 0 because for ǫ < z/2 then g analogously to (24) we obtain,…”

“…We want to pass to the limit as ǫ ↓ 0. Equations (37) and (38) show that the right hand side of equation (40) converges. Let x 1 < .…”

“…The point-to-point polymers have been studied in a number of recent papers: the Brownian model in [8,39,41] and the log-gamma polymer in [9,16,42,47] with one motivation being their relationship to the KPZ equation (see [15] for a survey). The point-to-line log-gamma polymer, which corresponds to a flat initial condition for the KPZ equation, has been studied recently by [7,38] using a local version of the geometric RSK correspondence and an expression is given for the Laplace transform of the point-to-line partition function of the log-gamma polymer in terms of Whittaker functions. From Theorem 3 it follows that the Laplace transform of the partition function of the point-to-line Brownian model, which has not been studied previously, is also given by the same expression.…”

Point-to-line polymers and orthogonal Whittaker functions

“…Some of the highlights include the solvability of the Asymmetric Simple Exclusion Process (ASEP) by Tracy and Widom [TW09a,TW09b] via the method of Bethe Ansatz, the construction of Macdonald Processes by Borodin and Corwin [BC14] and various particle processes (q-Totally Asymmetric Simple Exclusion Process, q-Totally Asymmetric Zero Range Process etc.) that fall within this scope, the stochastic six-vertex model [BP16], the Brownian, semi-discrete polymer and its relation to the Quantum Toda hamiltonian as established by O'Connell [O12] and the exactly solvable log-gamma directed polymer, which was introduced by Seppäläinen [Sep12] and analyzed in [COSZ14,OSZ14,BCR13,NZ15].…”

“…In our setting, SO 2n+1 (R)-Whittaker functions emerge through a combinatorial analysis of the log-gamma polymer via the geometric Robinson-Schensted-Knuth correspondence. Using the framework and properties of geometric RSK as in [NZ15], we determine the joint law of all point-to-point partition functions with end point on a line or half line. Subsequently, we are able to derive an integral formula for the Laplace transform of the various point-to-line partition functions after expressing them as a sum of the corresponding point-to-point ones.…”