Convergence in the p-Contest
Kennerberg, Philip LU and Volkov, Stas LU
(2020)
In Journal of Statistical Physics
178(5).
p.1096-1125
- 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.... (More)
- 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. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/3e7886a6-c4b1-4f26-a668-cac3debc4e4e
- author
- Kennerberg, Philip
LU
and Volkov, Stas
LU
- organization
- publishing date
- 2020-03
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- Keynesian beauty contest, Jante’s law, Rank-driven process
- in
- Journal of Statistical Physics
- volume
- 178
- issue
- 5
- pages
- 30 pages
- publisher
- Springer
- external identifiers
-
- scopus:85078219476
- ISSN
- 1572-9613
- DOI
- 10.1007/s10955-020-02491-6
- language
- English
- LU publication?
- yes
- id
- 3e7886a6-c4b1-4f26-a668-cac3debc4e4e
- date added to LUP
- 2020-01-28 19:23:15
- date last changed
- 2022-04-18 20:10:29
@article{3e7886a6-c4b1-4f26-a668-cac3debc4e4e,
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.}},
author = {{Kennerberg, Philip and Volkov, Stas}},
issn = {{1572-9613}},
keywords = {{Keynesian beauty contest; Jante’s law; Rank-driven process}},
language = {{eng}},
number = {{5}},
pages = {{1096--1125}},
publisher = {{Springer}},
series = {{Journal of Statistical Physics}},
title = {{Convergence in the p-Contest}},
url = {{http://dx.doi.org/10.1007/s10955-020-02491-6}},
doi = {{10.1007/s10955-020-02491-6}},
volume = {{178}},
year = {{2020}},
}