Universal Behavior of the Coefficients of the Continuous Equation in Competitive Growth Models

TL;DR

This paper demonstrates that the coefficients in the continuous equations of competitive growth models depend quadratically on the probability parameter, revealing a universal behavior linked to different average time rates, supported by analytical derivations and numerical verification.

Contribution

It analytically derives the continuous equations for competitive growth models and uncovers their universal quadratic dependence on the probability parameter p.

Findings

Coefficients are quadratic in p across models.

Analytical derivation of continuous equations from microscopic rules.

Numerical results confirm the analytical and scaling predictions.

Abstract

The competitive growth models involving only one kind of particles (CGM), are a mixture of two processes one with probability $p$ and the other with probability $1-p$. The $p-$dependance produce crossovers between two different regimes. We demonstrate that the coefficients of the continuous equation, describing their universality classes, are quadratic in $p$ (or $1-p$). We show that the origin of such dependance is the existence of two different average time rates. Thus, the quadratic $p-$dependance is an universal behavior of all the CGM. We derive analytically the continuous equations for two CGM, in 1+1 dimensions, from the microscopic rules using a regularization procedure. We propose generalized scalings that reproduce the scaling behavior in each regime. In order to verify the analytic results and the scalings, we perform numerical integrations of the derived analytical equations. The results are in excellent agreement with those of the microscopic CGM presented here and with the proposed scalings.