A model of discrete interacting updates

TL;DR

This paper analyzes a stochastic process involving counters that are randomly updated based on their states, establishing recurrence and bounds for the average update rate, and discussing the complexity and open problems in the model.

Contribution

It introduces a novel model of interacting counters, proves recurrence and bounds for the average update rate, and discusses the complexity and open questions of the problem.

Findings

Distances between counters form a positive recurrent Markov chain

Existence of a stationary average update rate V(N)

Provided bounds for V(N) as N approaches infinity

Abstract

We consider $N$ counters taking integer values which are subject to the following dynamics. At every time, a pair of distinct counters is chosen uniformly at random and their states are updated according to the following rule. If the states are different, then the smaller one is increased by $1$, while if the states are the same, both of them are increased by $1$. We show that, for a fixed $N$, the distances between consecutive ordered counters form a positive recurrent Markov chain and there exists the speed $V(N)$ defined as the average number of counters updated per time step in the stationary regime. We provide non-trivial upper and lower bounds for $V(N)$ as $N\to \infty$. Despite the simple formulation of the problem, its analysis seems to be highly complicated. We also provide a list of open problems and discuss various methods one may want to use, and obstacles one encounters.