Voter model and coalescent random walks on random graphs

The voter model is a classical Markovian interacting particle system in which each vertex of a graph has a binary opinion that evolves over time depending on the opinion of the neighbors of the vertex. The behaviour of the voter model on infinite regular lattices, as well as on finite lattice boxes, has been studied extensively in the last decades. The analysis relies on the exploitation of a duality relation between the voter model and a system of coalescing random walks. We focus on the voter model on sparse random graphs drawn from the configuration model, both directed and undirected. The project has three lines of research:

1. We analyse the time evolution of the number of discordant edges (i.e., edges between vertices having different opinions). In a recent paper, we completed the analysis in the special case where the graph is random regular (i.e., all vertices have the same degree).
2. We analyse the limiting distribution (when the size of the graph tends to infinity) of the first time at which all the vertices share the same opinion (the so-called consensus time) when the random graph is directed (i.e., sampled according to the directed configuration model).
3. We aim at extending the current results on the consensus time and on the time evolution of the discordant edges on the (undirected) random regular graph to the case where the graph evolves over time (in a Markovian way). More precisely, we want to first consider the case where the graph dynamics is independent of the voter dynamics, and afterwards consider a model with feedback where there is a mutual dependence between the graph dynamics and the voter dynamics.