Percolation in deposits for competitive models in (1+1)-dimensions

TL;DR

This study investigates percolation during deposit formation in (1+1)-dimensional models combining ballistic, random, and Family deposition, revealing how deposit height and density scale with substrate size and model type.

Contribution

It introduces and analyzes mixed deposition models, demonstrating crossover behaviors and scaling laws for percolation properties in (1+1)-dimensional systems.

Findings

Mean deposit height scales with substrate length logarithmically.

Scaling law relates deposit density to height at percolation.

Crossover behavior observed in competitive models between different deposition regimes.

Abstract

The percolation behaviour during the deposit formation, when the spanning cluster was formed in the substrate plane, was studied. Two competitive or mixed models of surface layer formation were considered in (1+1)-dimensional geometry. These models are based on the combination of ballistic deposition (BD) and random deposition (RD) models or BD and Family deposition (FD) models. Numerically we find, that for pure RD, FD or BD models the mean height of the percolation deposit $\bar h$ grows with the substrate length $L$ according to the generalized logarithmic law $\bar h\propto (\ln (L))^\gamma$, where $\gamma=1.0$ (RD), $\gamma=0.88\pm 0.020$ (FD) and $\gamma=1.52\pm 0.020$ (BD). For BD model, the scaling law between deposit density $p$ and its mean height $\bar h$ at the point of percolation of type $p-p_\infty \propto \bar h^{-1/\nu_h}$ are observed, where $\nu_h =1.74\pm0.02$ is a scaling coefficient. For competitive models the crossover, %in $h$ versus $L$ corresponding to the RD or FD -like behaviour at small $L$ and the BD-like behaviour at large $L$ are observed.