Aspects of Spatial Random Processes

Abstract: This thesis consists of five papers dealing with various aspects of spatial random processes. In the first three papers the main focus is on a special class of such processes, namely measures of maximal entropy for subshifts of finite type. These are shown to be closely related to Gibbs measures, and the question of whether a given subshift of finite type has a unique measure of maximal entropy is also discussed. In the fourth paper the theory of subshifts of finite type is exploited in order to derive properties of uniform spanning trees for a certain class of infinite graphs. The fifth paper deals with stationary first passage percolation, the main result being a necessary and sufficient condition for a compact set B in R^d to arise as the asymptotic shape for some stationary measure on the passage times.