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.

Let X be a translation invariant point process on the Euclidean space E and let D, a subset of E, be a bounded open set whose boundary has zero Lebesgue measure. We ask what does the point configuration O obtained by taking the points of X outside D tell us about the point configuration I obtained from the points of X inside D? We focus mainly on translation invariant point processes on the plane. We show that for the Ginibre ensemble, O determines the number of points in I. We refer to this kind of behaviour as ``rigidity''. For the translation-invariant zero process of a planar Gaussian Analytic Function, we show that O determines the number as well as the centre of mass of the points in I. Further, in both models we prove that the outside says ``nothing more'' about the inside, in the sense that the conditional distribution of the inside points, given the outside, is mutually absolutely continuous with respect to the Lebesgue measure on its supporting submanifold. We further show that the conditional density (of the inside points given the outside) is, roughly speaking, comparable to a squared Vandermonde density. In particular, this shows that even under spatial conditioning, the points exhibit repulsion which is quadratic in their mutual separation. We apply these results to the study of continuum percolation on these point processes, and establish the existence of a non-trivial critical radius and the uniqueness of infinite cluster in the supercritical regime. En route, we obtain new estimates on hole probabilities for zeroes of the planar Gaussian Analytic Function. Finally, we apply these ideas to prove completeness properties of random exponential functions originating from "rigid" determinantal point processes. We conclude by establishing miscellaneous other results on determinantal point processes. These include answers to two questions of Lyons and Steif on certain models of stationary determinantal processes on the integers, one involving insertion and deletion tolerance, and the other regarding the recovery of the driving function from the process.

In this thesis we study the mixing times of Markov chains, e.g., the

rate of convergence of Markov chains to stationary measures. We

focus on Glauber dynamics for the (classical) Ising model as well as

random walks on random graphs.

We first provide a complete picture for the evolution of the mixing

times and spectral gaps for the mean-field Ising model. In

particular, we pin down the scaling window, and prove a cutoff

phenomenon at high temperatures, as well as confirm the power law at

criticality. We then move to the critical Ising model at Bethe

lattice (regular trees), where the criticality corresponds to the

reconstruction threshold. We establish that the mixing time and the

spectral gap are polynomial in the surface area, which is the height

of the tree in this special case. Afterwards, we show that the

mixing time of Glauber dynamics for the (ferromagnetic) Ising model

on an arbitrary $n$-vertex graph at any temperature has a lower

bound of $n \log n/4$, confirming a folklore theorem in the special

case of Ising model.

In the second part, we study the random walk on the largest

component of the near-supcritical Erd\"os-R {e}nyi graph. Using a

complete characterization of the structure for the

near-supercritical random graph, as well as various techniques to

bound the mixing times in terms of spectral profile, we obtain the

correct order for the mixing time in this regime, which demonstrates

a smooth interpolation between the critical and the supercritical

regime.