Determination of Finite Size Effect in Lattice Models From the Local Height Difference Distribution

TL;DR

This paper introduces a method to analyze finite size effects in lattice growth models by examining local height difference distributions, applying it to various models including KPZ, and proposing a new model with rapid convergence to KPZ dynamics.

Contribution

It develops a novel approach using local height difference distributions to identify finite size effects and introduces a new model that quickly reaches KPZ behavior.

Findings

Invariance of height difference distribution indicates finite size effects.

The new model converges rapidly to KPZ dynamics.

Predicted roughness constant closely matches known KPZ values.

Abstract

Growth of interfaces during vapor deposition is analyzed on a discrete lattice. It leads to finding distribution of local heights, measurable for any lattice model. Invariance in the change of this distribution in time is used to determine the finite size effects in various models The analysis is applied to the discrete linear growth equation and Kardar-Parisi-Zhang (KPZ) equation. A new model is devised that shows early convergence to the KPZ dynamics. Various known conservative and non conservative models are tested on a one dimensional substrate by comparing the growth results with the exact KPZ and linear growth equation results. The comparison helps in establishing the condition that helps in determining the presence of finite size effect for the given model. The new model is used in 2+1 dimensions to predict close to the true value of roughness constant for KPZ equation.