Exact formulas for random growth with half-flat initial data

TL;DR

This paper derives exact formulas for the height function moments of the ASEP with specific initial data, enabling asymptotic analysis and connecting to KPZ universality class fluctuations.

Contribution

It provides new exact formulas suitable for asymptotic analysis, extending previous results and linking to divergent series formulas for half-flat KPZ.

Findings

Formulas are suitable for asymptotics.

Fluctuations converge to Airy_{2→1} marginals.

Connections to divergent series formulas for KPZ.

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.