Theoretical Statistics and Mathematics Unit, ISI Delhi

On the One Dimensional Critical ``Learning from Neighbours'' Model

by Antar Bandyopadhyay, Rahul Roy and Anish Sarkar

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 $\left\{R, B\right\}^{\Zbold}$.
This model was studied by Chatterjee and Xu \cite{ChXu04} in the context of diffusion of technologies in a set-up of
myopic, memoryless agents. In their work they assume different success probabilities of coin tosses according to the colour of the chameleon.
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. As a
corollary we also characterize the set of all translation invariant stationary laws of this
Markov chain. Moreover we show that starting with an i.i.d. colour distribution with
density $p \in [0,1]$ of one colour (say red), the limiting distribution is all red with
probability $\pi\left(\alpha, p\right)$ which is continuous in $p$ and for $p$ ``small" $\pi(p) >
p$. The last result can be interpreted as the model favours the
``underdog".

isid/ms/2009/08 [fulltext]

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