Title: An exponential limit shape of random $q$-proportion Bulgarian solitaire,
Abstract: 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.