Exact formulas for random growth with half-flat initial data

Abstract

We obtain exact formulas for moments and generating functions of the height function of the asymmetric simple exclusion process at one spatial point, starting from special initial data in which every positive even site is initially occupied. These complement earlier formulas of E. Lee [J. Stat. Phys. 140 (2010) 635-647] but, unlike those formulas, ours are suitable in principle for asymptotics. We also explain how our formulas are related to divergent series formulas for half-flat KPZ of Le Doussal and Calabrese [J. Stat. Mech. 2012 (2012) P06001], which we also recover using the methods of this paper. These generating functions are given as a series without any apparent Fredholm determinant or Pfaffian structure. In the long time limit, formal asymptotics show that the fluctuations are given by the Airy$_{2\to1}$ marginals.Comment: Published at http://dx.doi.org/10.1214/15-AAP1099 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

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

“…Mathematically rigorous formulae for joint moments for related particle models, q-Push(T)ASEP's, have been obtained, see for example [CP15]. In principle these can degenerate to formulae for joint moments of polymer partition functions Z (m,n) for various points (m, n) which would, however, need to share the same coordinate n. Finally, we close this review of related results by mentioning the work of Ortmann-Quastel-Remenik [OQR15a], [OQR15b]. There they obtain formulae for the one point distribution (as opposed to our treatment of two-point distributions) of the asymmetric exclusion process with flat and half-flat initial data.…”

A Pfaffian Representation for Flat ASEP

“…Of course, this means that we are getting formulas for the exponential moments E flat OEe mh.t;0/ of flat KPZ. One can also think of this in terms of the solution of the delta Bose gas with flat initial data, which is the solution v.tI E x/ of the following system of equations, where we write [5,31] for more details):…”

“…In [31] we obtained an explicit formula for the moments of the SHE started with half-flat initial condition by taking a weakly asymmetric limit of the half-flat ASEP moment formula (Theorem 2.4 above). The same formula was obtained in [31] directly for the delta Bose gas with a slightly more general initial condition using the same method that led in that paper to the moment formulas for half-flat ASEP.…”

“…The first formulas were for the narrow wedge and half-Brownian initial conditions [1,12], which were obtained by taking limits of earlier formulas for ASEP with step and step-Bernoulli initial conditions due to Tracy and Widom [39,40]. The remaining key initial data, flat, half-flat, and Brownian, are more challenging and had to wait several years ( [7,31] and the present work). Meanwhile there was a parallel effort in the physics community working directly from KPZ, using the method of replicas to derive rigorous formulas for the moments, and then writing divergent series for appropriate generating functions for the one-point distributions of KPZ, which could then be formally manipulated into convergent Fredholm series [11,16,25,26].…”

“…In a prequel article [31], we derived a formula for a generating function in the half-flat case, meaning that initially only even positive sites are occupied. Because such data is left-finite, meaning that there is a leftmost particle, the duality methods of [8] apply, and with suitable guessing, inspired, in part, by the formulas of [28], one is able to obtain an exact solution to the duality equations, which provide formulas for the exponential moments of the ASEP height function and can then be turned into a certain generating function.…”