We introduce a new family of directed, multi-state bootstrap percolation models that naturally occur as the “convolution” of classical bootstrap percolation models as well as generalized $k$-cross models studied by Gravner, Holroyd, Liggett, and the first two authors. We prove bounds for the probability of indefinite growth by relating the percolation process to sequences of random variables that characterize the percolation growth combinatorics. The corresponding stochastic processes are of independent interest, and we prove a general bound for the limiting density of their probability distributions. We prove these bounds using new results for the convexity and monotonicity of linear operators; these results are of independent interest and the techniques also apply to other stochastic processes with “forbidden patterns”. In the simplest case of the new multi-state percolation models, we prove a stronger result that gives the precise asymptotic behavior for the limiting probability density of the corresponding stochastic process. This follows from the surprising appearance of Ramanujan’s mock theta functions, whose cuspidal asymptotics are closely connected to the limiting probabilities.