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- Title
Convergence in the p-Contest.
- Authors
Kennerberg, Philip; Volkov, Stanislav
- Abstract
We study asymptotic properties of the following Markov system of 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 , is removed and replaced by an independent 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 and p > 1 . In particular, when p < 1 and ζ ∼ 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 ζ satisfies certain mild conditions (e.g. 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 . 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 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.
- Subjects
LYAPUNOV functions; BEAUTY contests; EVIDENCE; MASS transfer coefficients
- Publication
Journal of Statistical Physics, 2020, Vol 178, Issue 5, p1096
- ISSN
0022-4715
- Publication type
Article
- DOI
10.1007/s10955-020-02491-6