A proof of Andrews' conjecture on Partitions with no short sequences

Abstract

Holroyd, Liggett, and Romik introduced the following probability model. Let $C_1,C_2,...$ be independent events with probabilities ${\P }_s(C_n)=1-e^{-ns}$ under a probability measure ${\P }_s$ with $0<s<1$. Let $A_k$ be the event that there is no sequence of $k$ consecutive $C_i$ that do not occur. We given an asymptotic for ${\P }_s(A_k)$ with a relative error term that goes to 0 as $s\to 0$. This establishes a conjecture of Andrews.