Coalescence on the real line

Abstract:

We study a geometrically constrained coalescence model derived from spin systems. Given two probability distributions $\mathbb {P}_R$ and $\mathbb {P}_B$ on the positive reals with finite means, colour the real line alternately with red and blue intervals so that the lengths of the red intervals have distribution $\mathbb {P}_R$, the lengths of the blue intervals have distribution $\mathbb {P}_B$, and distinct intervals have independent lengths. Now, iteratively update this colouring of the line by coalescing intervals: change the colour of any interval that is surrounded by longer intervals so that these three consecutive intervals subsequently form a single monochromatic interval. We say that a colour (either red or blue) wins if every point of the line is eventually of that colour. Holroyd, in 2010, asked the following question: under what natural conditions on the initial distributions is one of the colours almost surely guaranteed to win? It turns out that the answer to this question can be quite counter-intuitive due to the non-monotone dynamics of the model. In this paper, we investigate various notions of “advantage” one of the colours might initially possess, and in the course of doing so, we determine which of the two colours emerges victorious for various non-trivial pairs of initial distributions.