Voter model on k-partite graphs. Asymptotic behavior conditional on non-absorption

Thanks to its incredible ability to model real-life problems, the research field of interacting particle systems has become one of the main application of stochastic processes. We treated the case of the voter model, where the set of particles represents a group of people each of whom holds one of two different opinions (0 and 1) on a political issue. The state space of the process is defined by [image: \{0,1\}^V], where [image: V] represents the vertex set of a complete [image: k]-partite graph with parameters [image: n,m_1,\dots,m_{k-1}]. Since the graph is finite, consensus, i.e. states where opinions are all equal, occurs almost surely. The aim of the thesis is to investigate the asymptotic behavior of the quasi-stationary distributions, for the process conditioned to never reach consensus, as [image: n] tends to infinity. In particular, we want to find out if the lack of consensus is due to a minority number of dissenters, or if the opinions are relatively balanced.