Graphical representation of certain moment dualities and application to population models with balancing selection

TL;DR

This paper explores duality mechanisms in interacting particle systems, showing how linear transformations can produce moment dualities, with applications to population models with balancing selection and annihilating dual processes.

Contribution

It generalizes existing duality approaches by introducing a linear transformation that yields moment dualities for rescaled processes, including the case q=-1.

Findings

Rescaled dualities of particle systems are achieved through linear transformations.

The case q=-1 explains annihilating dual processes in population models.

Different q-values are analyzed, answering prior open questions.

Abstract

We investigate dual mechanisms for interacting particle systems. Generalizing an approach of Alkemper and Hutzenthaler in the case of coalescing duals, we show that a simple linear transformation leads to a moment duality of suitably rescaled processes. More precisely, we show how dualities of interacting particle systems of the form $H(A,B)=q^{|A\cap B|}, A,B\subset\{0,1\}^N, q\in[-1,1),$ are rescaled to yield moment dualities of rescaled processes. We discuss in particular the case $q=-1,$ which explains why certain population models with balancing selection have an annihilating dual process. We also consider different values of $q,$ and answer a question by Alkemper and Hutzenthaler.