The Limit Shape of a Stochastic Bulgarian Solitaire

TL;DR

This paper analyzes a stochastic variant of Bulgarian solitaire, demonstrating that as the number of cards grows, the process converges to an exponential limit shape with quantifiable deviation bounds.

Contribution

It introduces a stochastic model of Bulgarian solitaire and proves convergence to an exponential limit shape with deviation probability bounds.

Findings

Convergence to an exponential limit shape as card number increases

Quantitative bounds on deviation probability from the limit shape

Connection between the number of rounds and deviation from the limit shape

Abstract

We consider a stochastic version of Bulgarian solitaire: A number of cards are distributed in piles; in every round a new pile is formed by cards from the old piles, and each card is picked independently with a fixed probability. This game corresponds to a multi-square birth-and-death process on Young diagrams of integer partitions. We prove that this process converges in a strong sense to an exponential limit shape as the number of cards tends to infinity. Furthermore, we bound the probability of deviation from the limit shape and relate this to the number of rounds played in the solitaire.