An exponential limit shape of random $q$-proportion Bulgarian solitaire

We introduce \emph{$p_n$-random $q_n$-proportion Bulgarian solitaire} ($0<p_n,q_n\le 1$), played on $n$ cards distributed in piles. In each pile, a number of cards equal to the proportion $q_n$ of the pile size rounded upward to the nearest integer are candidates to be picked. Each candidate card is picked with probability $p_n$, independently of other candidate cards. This generalizes Popov’s random Bulgarian solitaire, in which there is a single candidate card in each pile. Popov showed that a triangular limit shape is obtained for a fixed $p$ as $n$ tends to infinity. Here we let both $p_n$ and $q_n$ vary with $n$. We show that under the conditions $q_n^2 p_n n/{\log n}\rightarrow \infty$and$p_n q_n \rightarrow 0$as$n\to\infty$,the$p_n$-random$q_n$-proportion Bulgarian solitaire has an exponential limit shape.