Half-space stationary Kardar-Parisi-Zhang equation

DOI10.1007/S10955-020-02622-ZzbMATH Open1462.82045arXiv2003.03809OpenAlexW3106283304WikidataQ100762347 ScholiaQ100762347MaRDI QIDQ2658884

Author name not available (Why is that?)

Publication date: 25 March 2021

Published in: (Search for Journal in Brave)

Abstract: We study the solution of the Kardar-Parisi-Zhang (KPZ) equation for the stochastic growth of an interface of height $h(x,t)$ on the positive half line, equivalently the free energy of the continuum directed polymer in a half space with a wall at $x=0$. The boundary condition $partia{l}_{x}h(x,t){|}_{x=0}=A$ corresponds to an attractive wall for $A<0$, and leads to the binding of the polymer to the wall below the critical value $A=-1/2$. Here we choose the initial condition $h(x,0)$ to be a Brownian motion in $x>0$ with drift $-(B+1/2)$. When $A+Bo-1$, the solution is stationary, i.e. $h(cdot,t)$ remains at all times a Brownian motion with the same drift, up to a global height shift $h(0,t)$. We show that the distribution of this height shift is invariant under the exchange of parameters $A$ and $B$. For any $A,B>-1/2$, we provide an exact formula characterizing the distribution of $h(0,t)$ at any time $t$, using two methods: the replica Bethe ansatz and a discretization called the log-gamma polymer, for which moment formulae were obtained. We analyze its large time asymptotics for various ranges of parameters $A,B$. In particular, when $(A,B)o(-1/2,-1/2)$, the critical stationary case, the fluctuations of the interface are governed by a universal distribution akin to the Baik-Rains distribution arising in stationary growth on the full-line. It can be expressed in terms of a simple Fredholm determinant, or equivalently in terms of the Painlev'e II transcendent. This provides an analog for the KPZ equation, of some of the results recently obtained by Betea-Ferrari-Occelli in the context of stationary half-space last-passage-percolation. From universality, we expect that limiting distributions found in both models can be shown to coincide.

Full work available at URL: https://arxiv.org/abs/2003.03809

zbMATH Keywords

No records found.

Mathematics Subject Classification ID

No records found.

This page was built for publication: Half-space stationary Kardar-Parisi-Zhang equation

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q2658884)