Improved bounds on metastability thresholds and probabilities for generalized bootstrap percolation

We generalize and improve results of Andrews, Gravner, Holroyd, Liggett, and Romik on metastability thresholds for generalized two-dimensional bootstrap percolation models, and answer several of their open problems and conjectures. Specifically, we prove slow convergence and localization bounds for Holroyd, Liggett, and Romik's k-percolation models, and in the process provide a unified and improved treatment of existing results for bootstrap, modified bootstrap, and Frobose percolation. Furthermore, we prove improved asymptotic bounds for the generating functions of partitions without k-gaps, which are also related to certain infinite probability processes relevant to these percolation models. One of our key technical probability results is also of independent interest. We prove new upper and lower bounds for the probability that a sequence of independent events with monotonically increasing probabilities contains no "k-gap" patterns, which interpolates the general Markov chain solution that arises in the case that all of the probabilities are equal.