UC Berkeley

In this thesis we introduce and study two probabilistic models of competition and their applications. The first model is a particular contact process, and is intended to simulate propagation dynamics in real social networks. The second one stems from game theory and is of more theoretical value, as it is used to prove the existence of solutions to a certain non-linear partial differential equation.

The first model consists of two competing first passage percolation processes started from uniformly chosen subsets of a random regular graph on N vertices. The processes are allowed to spread with different rates, start from vertex subsets of different sizes or at different times. We obtain tight results regarding the sizes of the vertex sets occupied by each process, showing that in the generic situation one process will occupy Theta(1)N^α vertices, for some 0 < α < 1. The value of α is calculated in terms of the relative rates of the processes, as well as the sizes of the initial vertex sets and the possible time advantage of one process.

The second model is a version of the stochastic "Tug-of-War" game, played on graphs and smooth domains, with the empty set of terminal states. We prove that, when the running payoff function is shifted by an appropriate constant, the values of the game after n steps converge in the continuous case and the case of finite graphs with loops. Using this we prove the existence of solutions to the infinity Laplace equation with vanishing Neumann boundary condition.