Convergence in the $p$-contest

We study asymptotic properties of the following Markov system of $N \geq 3$ points in~$[0,1]$. At each time step, the point farthest from the current centre of mass, multiplied by a constant $p>0$, is removed and replaced by an independent $\zeta$-distributed point; the problem, inspired by variants of the Bak--Sneppen model of evolution and called a $p$-contest, was posed in [Grinfeld, M, Knight, P.A., and Wade, A.R. Rank-driven Markov processes, J. Stat. Phys. 146 (2012)]. We obtain various criteria for the convergences of the system, both for $p<1$ and $p>1$. In particular, when $p<1$ and $\zeta\sim U[0,1]$, we show that the limiting configuration converges to zero. When $p>1$, we show that the configuration must converge to either zero or one, and we present an example where both outcomes are possible. Finally, when $p>1$, $N=3$ and $\zeta$ satisfies certain conditions (e.g.~$\zeta\sim U[0,1]$), we prove that the configuration can only converge to one a.s. Our paper substantially extends the results of [Grinfeld, M., Volkov, S., and Wade, A.R. Convergence in a multidimensional randomized Keynesian beauty contest. Adv. in Appl. Probab. 47 (2015)] and [Kennerberg, P., and Volkov, S. Jante's law process. Adv. in Appl. Probab. 50 (2018)] where it was assumed that $p=1$. Unlike the previous models, one can no longer use the Lyapunov function based just on the radius of gyration; when $0<p<1$ one has to find a much finer tuned function which turns out to be a supermartingale; the proof of this fact constitutes an unwieldy, albeit necessary, part of the paper.