Feller property of the multiplicative coalescent with linear deletion

TL;DR

This paper introduces the multiplicative coalescent with linear deletion (MCLD), a process modeling cluster merging and deletion, and proves it is a Feller process, aiding the understanding of scaling limits in percolation models.

Contribution

The paper defines the MCLD process, incorporating linear deletion into the multiplicative coalescent, and establishes its Feller property, advancing the theoretical framework of cluster dynamics.

Findings

MCLD is a well-defined Feller process.

The process models cluster merging and deletion with specific rates.

Results facilitate analysis of scaling limits in percolation models.

Abstract

We modify the definition of Aldous' multiplicative coalescent process and introduce the multiplicative coalescent with linear deletion (MCLD). A state of this process is a square-summable decreasing sequence of cluster sizes. Pairs of clusters merge with a rate equal to the product of their sizes and clusters are deleted with a rate linearly proportional to their size. We prove that the MCLD is a Feller process. This result is a key ingredient in the description of scaling limits of the evolution of component sizes of the mean field frozen percolation model and the so-called rigid representation of such scaling limits.