On the One Dimensional Critical "Learning from Neighbours" Model

TL;DR

This paper analyzes a one-dimensional interacting particle system where chameleons update their colors based on local majority and success probabilities, demonstrating convergence to a single color regardless of initial conditions.

Contribution

It introduces and studies the critical case of the model where success probabilities are equal, proving convergence to a monochromatic state from any initial distribution.

Findings

The system converges to a single color from any translation invariant initial distribution.

Starting with an i.i.d. distribution, the model favors the underdog color in the limit.

No coexistence of colors persists at the critical success probability.

Abstract

We consider a model of a discrete time "interacting particle system" on the integer line where infinitely many changes are allowed at each instance of time. We describe the model using chameleons of two different colours, {\it viz}., red ($R$) and blue ($B$). At each instance of time each chameleon performs an independent but identical coin toss experiment with probability $\alpha$ to decide whether to change its colour or not. If the coin lands head then the creature retains its colour (this is to be interpreted as a "success"), otherwise it observes the colours and coin tosses of its two nearest neighbours and changes its colour only if, among its neighbors and including itself, the proportion of successes of the other colour is larger than the proportion of successes of its own colour. This produces a Markov chain with infinite state space ${R, B}^{\Zbold}$. This model was first studied by Chatterjee and Xu (2004) where different colours had different success probabilities. In this work we consider the "critical" case where the success probability, $\alpha$, is the same irrespective of the colour of the chameleon. We show that starting from any initial translation invariant distribution of colours the Markov chain converges to a limit of a single colour, i.e., even at the critical case there is no "coexistence" of the two colours at the limit. Moreover we show that starting with an i.i.d. colour distribution the limiting distribution gives some advantage to the "underdog".