# When fast and slow interfaces grow together: connection to the half-space problem of the Kardar-Parisi-Zhang class

## TL;DR

This study investigates height fluctuations of 1+1 dimensional KPZ interfaces with different growth speeds on each side, revealing boundary fluctuation distributions linked to random matrix theory and crossover behaviors.

## Contribution

It demonstrates that boundary fluctuations in KPZ interfaces with asymmetric growth are described by the half-space KPZ problem and identifies the GSE Tracy-Widom distribution as the boundary fluctuation distribution.

## Key findings

- Boundary fluctuations follow the GSE Tracy-Widom distribution.
- Crossover occurs when growth speed difference is small.
- Simulation results match theoretical half-space KPZ predictions.

## Abstract

We study height fluctuations of interfaces in the $(1+1)$-dimensional Kardar-Parisi-Zhang (KPZ) class, growing at different speeds in the left half and the right half of space. Carrying out simulations of the discrete polynuclear growth model with two different growth rates, combined with the standard setting for the droplet, flat, and stationary geometries, we find that the fluctuation properties at and near the boundary are described by the KPZ half-space problem developed in the theoretical literature. In particular, in the droplet case, the distribution at the boundary is given by the largest-eigenvalue distribution of random matrices in the Gaussian symplectic ensemble, often called the GSE Tracy-Widom distribution. We also characterize crossover from the full-space statistics to the half-space one, which arises when the difference between the two growth speeds is small.