A random variant of the game of plates and olives

TL;DR

This paper studies a random variant of the game of plates and olives, showing that the number of olives grows linearly and concentrates around its expected value as the game progresses.

Contribution

It introduces a stochastic version of the game and proves linear growth and concentration results for the number of olives.

Findings

Number of olives is linear with high probability.

Number of olives concentrates around its expectation.

The process's behavior is well-understood asymptotically.

Abstract

The game of plates and olives was originally formulated by Nicolaescu and encodes the evolution of the topology of the sublevel sets of Morse functions. We consider a random variant of this game. The process starts with an empty table. There are four different types of moves: (1) add a new plate to the table, (2) combine two plates and their olives onto one plate, removing the second plate from the table, (3) add an olive to a plate, and (4) remove an olive from a plate. We show that with high probability the number of olives is linear as the total number of moves goes to infinity. Furthermore, we prove that the number of olives is concentrated around its expectation.