1. The replicator coalescent
- Author
-
Kyprianou, A. E., Peñaloza, L., and Rogers, T.
- Subjects
Mathematics - Probability ,60J80, 60E10 - Abstract
We consider a stochastic model, called the {\it replicator coalescent}, describing a system of blocks of $k$ different types which undergo pairwise mergers at rates depending on the block types: with rate $C_{i,j}$ blocks of type $i$ and $j$ merge, resulting in a single block of type $i$. The replicator coalescent can be seen as generalisation of Kingman's coalescent death chain in a multi-type setting, although without an underpinning exchangeable partition structure. The name is derived from a remarkable connection we uncover between the instantaneous dynamics of this multi-type coalescent when issued from an arbitrarily large number of blocks, and the so-called {\it replicator equations} from evolutionary game theory. We provide a formal notion of `coming down from infinity' for our model in terms of the its behaviour when issued from infinitely many blocks. Because there are $k$ different types, there are many different `infinities' that the process can come down from. Nonetheless, we show that the exact `infinity' that the process comes down from is irrelevant to the subsequent dynamics as the system is forced through a bottleneck corresponding to an evolutionary stable state of the replicator equations. Thereafter, stochastic effects are felt and the process evolves as a multi-type death chain., Comment: 1 figure
- Published
- 2022