Branching systems and spatial fragmentations

The largest part of this thesis is concerned with the study of a fragmentation process in which rectangles break up into progressively smaller pieces at rates that depend on their shape. Long, thin rectangles are more likely to break quickly, and are also more likely to split along their longest side. We are interested in the evolution of the system at large times: how many fragments are there of different shapes and sizes, and how did they reach that state? We give an almost sure growth rate along paths by studying an equivalent branching random walk. Our analysis is highly technical due to the spatial dependence of the rates and the fact that we work under weaker assumptions than the usual large deviations regime for random walks. In the second part of the thesis we focus on a different, but related problem: estimating the probability that the paths of a random walk stay close to a given function. We prove a small deviation result about the unscaled paths of either a compound Poisson process, or a random walk in discrete time. Our proof strategy involves a Brownian motion approximation on smaller time intervals, which allows us to take advantage of the sharpest estimates currently available on the probability that a Brownian motion lies in a tube about a given function.