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We generalize and improve results of Andrews, Gravner, Holroyd, Liggett, andRomik on metastability thresholds for generalized two-dimensional bootstrappercolation 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 unifiedand improved treatment of existing results for bootstrap, modified bootstrap,and Frobose percolation. Furthermore, we prove improved asymptotic bounds forthe generating functions of partitions without k-gaps, which are also relatedto 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 ofindependent events with monotonically increasing probabilities contains no"k-gap" patterns, which interpolates the general Markov chain solution thatarises in the case that all of the probabilities are equal.

Improved bounds on metastability thresholds and probabilities for generalized bootstrap percolation