Large fluctuations of the KPZ equation in a half-space

TL;DR

This paper develops a unifying method to analyze the short-time height distribution of the KPZ equation in a half-space, revealing large deviation behaviors and novel boundary condition solutions.

Contribution

It introduces a new analytical framework for KPZ in half-space, including Fredholm Pfaffian and determinant formulas for boundary conditions.

Findings

Height distribution follows a large deviation form with explicit rate functions.

The left tail amplitude in half-space is half of that in full space.

Left tails remain valid at all times, indicating persistent large deviation behavior.

Abstract

We investigate the short-time regime of the KPZ equation in $1+1$ dimensions and develop a unifying method to obtain the height distribution in this regime, valid whenever an exact solution exists in the form of a Fredholm Pfaffian or determinant. These include the droplet and stationary initial conditions in full space, previously obtained by a different method. The novel results concern the droplet initial condition in a half space for several Neumann boundary conditions: hard wall, symmetric, and critical. In all cases, the height probability distribution takes the large deviation form $P(H,t) \sim \exp( - \Phi(H)/\sqrt{t})$ for small time. We obtain the rate function $\Phi(H)$ analytically for the above cases. It has a Gaussian form in the center with asymmetric tails, $|H|^{5/2}$ on the negative side, and $H^{3/2}$ on the positive side. The amplitude of the left tail for the half-space is found to be half the one of the full space. As in the full space case, we find that these left tails remain valid at all times. In addition, we present here (i) a new Fredholm Pfaffian formula for the solution of the hard wall boundary condition and (ii) two Fredholm determinant representations for the solutions of the hard wall and the symmetric boundary respectively.