A multiplicative coalescent with asynchronous multiple mergers

TL;DR

This paper introduces a new Markov coalescent process modeling the merging of blocks based on asynchronous sampling, analyzes its asymptotic behavior, and applies it to study random graph component sizes and related coagulation equations.

Contribution

It defines a novel multiplicative coalescent with asynchronous multiple mergers, analyzes its asymptotic distribution, and connects it to BGW processes and coagulation equations.

Findings

Asymptotic distribution of coalescent time determined

Bound established between block size and BGW process

Size estimates for largest blocks in various regimes

Abstract

We define a Markov process on the partitions of $[n]=\{1,\ldots,n\}$ by drawing a sample in $[n]$ at each time of a Poisson process, by merging blocks that contain one of these points and by leaving all other blocks unchanged. This coalescent process appears in the study of the connected components of random graph processes in which connected subgraphs are added over time with probabilities that depend only on their size. First, we determine the asymptotic distribution of the coalescent time. Then, we define a Bienayme-Galton-Watson (BGW) process such that its total population size dominates the block size of an element. We compute a bound for the distance between the total population size distribution and the block size distribution at a time proportional to $n$. As a first application of this result, we establish the coagulation equations associated with this coalescent process. As a second application, we estimate the size of the largest block in the subcritical and supercritical regimes as well as in the critical window.