Self-similar behavior of the exchange-driven growth model with product kernel

TL;DR

This paper rigorously analyzes the self-similar behavior and coarsening rates of the exchange-driven growth model with a product kernel, establishing convergence to known profiles and connecting discrete and continuous models.

Contribution

It provides a rigorous mathematical proof of the coarsening rates and self-similar convergence for the exchange-driven growth model with product kernels, including new inequalities and scaling limits.

Findings

Established coarsening rates for the model.

Proved convergence to self-similar profiles.

Linked the discrete model to a continuous heat equation.

Abstract

We study the self-similar behavior of the exchange-driven growth model, which describes a process in which pairs of clusters, consisting of an integer number of monomers, interact through the exchange of a single monomer. The rate of exchange is given by an interaction kernel $K(k,l)$ which depends on the sizes $k$ and $l$ of the two interacting clusters and is assumed to be of product form $(k\,l)^\lambda$ for $\lambda \in [0,2)$. We rigorously establish the coarsening rates and convergence to the self-similar profile found by Ben-Naim and Krapivsky [7]. For the explicit kernel, the evolution is linked to a discrete weighted heat equation on the positive integers by a nonlinear time-change. For this equation, we establish a new weighted Nash inequality that yields scaling-invariant decay and continuity estimates. Together with a replacement identity that links the discrete operator to its continuous analog, we derive a discrete-to-continuum scaling limit for the weighted heat equation. Reverting the time-change under the use of additional moment estimates, the analysis of the linear equation yields coarsening rates and self-similar convergence of the exchange-driven growth model.