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.