Preprint C1446/2019
Super-diffusive Limit of an Exclusion Process and Convergence From Coalescence to the Kingman’s Coalescent
Enrique Idael Chavez Sarmiento

This thesis has three parts. In the first part we consider a one-dimensional, weakly asymmetric, boundary driven exclusion process on a one-dimensional discrete interval, in a super-diffusive time scale. In this part we derive an equation which describes the evolution of the density of the particles until certain point. In the second part we consider a process associated to the simple random walk in a discrete torus. We place a particle at each site of the torus and let them evolve as independent, nearest-neighbor, symmetric, continuous-time random walks. Each time two particles meet, they  coalesce into one. We prove that, in a convenient scale of time, the sequence of total number of particles of these processes, when the size of the torus grows, converges to the total number of partitions in Kingman's coalescent. Finally, in the third part of this thesis, we consider a process similar to the previous one. We consider a finite number of i.i.d. irreducible and transitive Markov chains in continuous time, over a finite state space. Each time two chains meet, they stay together. This mechanism induces a process in the set of partitions of the first natural numbers. Starting from the invariant measure, we find conditions under which a sequence of these processes, in an appropriate scale of time, converges to the Kingman's coalescent that starts with finite equivalence classes. In particular, we prove this convergence in the reversible case under a condition that involves the relaxation time.