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

## TL;DR

This paper introduces a generalized random Bulgarian solitaire model with variable parameters and proves that, under certain conditions, it exhibits an exponential limit shape as the number of cards grows large.

## Contribution

It extends Popov's model by allowing both the proportion and probability parameters to vary with the number of cards, establishing conditions for an exponential limit shape.

## Key findings

- Under specified conditions, the model converges to an exponential limit shape.
- The results generalize previous fixed-parameter models to variable-parameter settings.
- The paper provides rigorous proof of the limit shape behavior.

## 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.