Tropical combinatorics and Whittaker functions

Abstract

We establish a fundamental connection between the geometric RSK correspondence and GL(N, R)-Whittaker functions, analogous to the well known relationship between the RSK correspondence and Schur functions. This gives rise to a natural family of measures associated with GL(N, R)-Whittaker functions which are the analogues in this setting of the Schur measures on integer partitions. The corresponding analogue of the Cauchy-Littlewood identity can be seen as a generalisation of an integral identity for GL(N, R)-Whittaker functions due to Bump and Stade. As an application, we obtain an explicit integral formula for the Laplace transform of the law of the partition function associated with a one-dimensional directed polymer model with log-gamma weights recently introduced by one of the authors (TS).

The Strict-Weak Lattice Polymer

“…(Subsequent to this paper, a generalization of this model called the Beta Polymer was discovered and analyzed in [2]). The strict-weak model introduced here differs from the earlier studied log-gamma polymer [9,10,15] in the definition of the admissible polymer paths. The strict-weak model uses gamma-distributed weights on the edges (or vertices, depending on the formulation chosen) while the log-gamma polymer uses inverse gamma weights.…”

“…As explained in [14], the analog of the smallest part for the pure alpha Whittaker process is related to the strict-weak polymer free energy. Methods coming from Whittaker processes [10] provide a route to write down a Laplace transform formula for the strict-weak polymer partition function which can be turned (using identities similar to those of [9]) into the Fredholm determinant formula present herein. This is the approach taken in [14].…”

Observables of Macdonald processes

“…Using A.N. Kirillov's [71,85] tropical RSK correspondence, Corwin, O'Connell, Seppäläinen and Zygouras [39] introduce a different Whittaker process to describe the free energy hierarchy for this polymer (see Section 5.3 for more on this). Presently we call this the α-Whittaker process (see Section 4.2) and it arises as the limit of the Macdonald process when ρ is a pure alpha specialization (some α i > 0, β i = 0 for all i ≥ 1 and γ = 0).…”

“…Analogously to the O'Connell-Yor semi-discrete polymer, the hierarchy f (n) evolves as a Markov chain in n with state space in R N (N +1)/2 . An explicit construction of this Markov chain as a function of the weights d is given in [39] (appealing to the recursive formulation of the tropical RSK correspondence which is given in [85]). We will not restate the kernel of this Markov chain on R N (N +1)/2 , but only remark that it is not the same as the limiting Markov chain which corresponds to the scaling limit of the dynamics given in Proposition 2.3.5.…”

Airy Processes and Variational Problems

“…Versions of this conjecture have been proved for two related models in the point-to-point case: the continuum random polymer in [ACQ11] (building on results of [TW08a; TW08b; TW09]) and the semi-discrete polymer of O'Connell and Yor in [BC11; BCF12] (see also [O'C12]). In the setting of discrete directed random polymers, [COSZ11] showed that if the weights are chosen so that −w i,j is distributed as the logarithm of a Gamma random variable with parameter θ i +θ j (for some fixed θ i 's andθ j 's) then the model is exactly solvable in the sense explained above. This was later used in [BCR13] to prove that the asymptotic fluctuations of the free energy of the point-to-point polymer (at least for low enough temperature) have the conjectured Tracy-Widom GUE distribution.…”