Convergence in the p-Contest

Abstract

AbstractWe study asymptotic properties of the following Markov system of $$N \ge 3$$N≥3 points in [0, 1]. At each time step, the point farthest from the current centre of mass, multiplied by a constant $$p>0$$p>0, is removed and replaced by an independent $$\zeta $$ζ-distributed point; the problem, inspired by variants of the Bak–Sneppen model of evolution and called a p-contest, was posed in Grinfeld et al. (J Stat Phys 146, 378–407, 2012). We obtain various criteria for the convergences of the system, both for $$p<1$$p<1 and $$p>1$$p>1. In particular, when $$p<1$$p<1 and $$\zeta \sim U[0,1]$$ζ∼U[0,1], we show that the limiting configuration converges to zero. When $$p>1$$p>1, we show that the configuration must converge to either zero or one, and we present an example where both outcomes are possible. Finally, when $$p>1$$p>1, $$N=3$$N=3 and $$\zeta $$ζ satisfies certain mild conditions (e.g. $$\zeta \sim U[0,1]$$ζ∼U[0,1]), we prove that the configuration converges to one a.s. Our paper substantially extends the results of Grinfeld et al. (Adv Appl Probab 47:57–82, 2015) and Kennerberg and Volkov (Adv Appl Probab 50:414–439, 2018) where it was assumed that $$p=1$$p=1. Unlike the previous models, one can no longer use the Lyapunov function based just on the radius of gyration; when $$0<p<1$$0<p<1 one has to find a more finely tuned function which turns out to be a supermartingale; the proof of this fact constitutes an unwieldy, albeit necessary, part of the paper.