Percolation, Probability, and Partitions
I will describe recent work (joint with Kathrin Bringmann) on improved
estimates for metastability thresholds for a certain family of
generalized bootstrap percolation processes on the square lattice,
which were first studied by Holroyd, Liggett, and Romik. These
probability estimates are notable for their direct connection to the
asymptotics for partitions without sequences and also to cuspidal
expansions for hypergeometric q-series; some of the cases involve
modular forms as well as Ramanujan's famous mock theta functions.
One key technical result is also of independent interest in
combinatorial probability. Consider a sequence on independent events
X_1, ..., X_n that each occur with probability u_i. We prove very
tight lower and upper bounds for the probability that the sequence
contains no k-gaps, which means that there are no k successive events
which do not occur. These bounds are an interpolated version of the
entropy for the corresponding Markov process when all of the
probabilities u_i are identical.