  This dissertation develops, for several families of statistical mechanical
and random growth models, techniques for analyzing infinite-volume asymptotics.
In the statistical mechanical setting, we focus on the low-temperature phases
of spin glasses and directed polymers, wherein the ensembles exhibit
localization which is physically phenomenological. We quantify this behavior in
several ways and establish connections to properties of the limiting free
energy. We also consider two popular zero-temperature polymer models, namely
first- and last-passage percolation. For these random growth models, we
investigate the order of fluctuations in their growth rates, which are
analogous to free energy.
