Coalescence on the real line

27 January 2015
Bhargav Narayanan

Given two probability distributions $P_R$ and $P_B$ 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 $P_R$, the lengths of the blue intervals have distribution $P_B$, 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. Say that a colour (either red or blue) `wins' if every point of the line is eventually of that colour. I will attempt to answer the following question: under what natural conditions on the distributions is one of the colours almost surely guaranteed to win?

  • Combinatorial Theory Seminar