Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-24T17:54:22.906Z Has data issue: false hasContentIssue false
  • English
  • Français

Bit flipping and time to recover

Part of: Theory of computing Markov processes

Published online by Cambridge University Press:  24 October 2016

Anton Muratov  and
Sergei Zuyev
Anton Muratov*
Affiliation:
KTH Royal Institute of Technology
Sergei Zuyev*
Affiliation:
Chalmers University of Technology
*
*Postal address: KTH Royal Institute of Technology, EES, Osquldas v. 10, 100 44 Stockholm, Sweden. Email address: [email protected]
** Postal address: Chalmers University of Technology, MV, 412 96 Gothenburg, Sweden. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We call `bits' a sequence of devices indexed by positive integers, where every device can be in two states: 0 (idle) and 1 (active). Start from the `ground state' of the system when all bits are in 0-state. In our first binary flipping (BF) model the evolution of the system behaves as follows. At each time step choose one bit from a given distribution P on the positive integers independently of anything else, then flip the state of this bit to the opposite state. In our second damaged bits (DB) model a `damaged' state is added: each selected idling bit changes to active, but selecting an active bit changes its state to damaged in which it then stays forever. In both models we analyse the recurrence of the system's ground state when no bits are active. We present sufficient conditions for both the BF and DB models to show recurrent or transient behaviour, depending on the properties of the distribution P. We provide a bound for fractional moments of the return time to the ground state for the BF model, and prove a central limit theorem for the number of active bits for both models.

Keywords

Binary systembit flippingrandom walk on a groupMarkov chain recurrencecritical behaviour

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 68Q87: Probability in computer science (algorithm analysis, random structures, phase transitions, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 3 , September 2016 , pp. 650 - 666
Copyright
Copyright © Applied Probability Trust 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Aspandiiarov, S.,Iasnogorodski, R. and Menshikov, M. (1996).Passage-time moments for nonnegative stochastic processes and an application to reflected random walks in a quadrant.Ann. Prob. 24,932–960.Google Scholar
[2] Athreya, K. B. and Karlin, S. (1968).Embedding of urn schemes into continuous time Markov branching process and related limit theorems.Ann. Math. Statist. 39,1801–1817.CrossRefGoogle Scholar
[3] Balogh, J. and Pemantle, R. (2007).The Klee–Minty random edge chain moves with linear speed.Random Structures Algorithms 30,464–483.CrossRefGoogle Scholar
[4] Benjamini, I.,Häggström, O. Peres, Y. and Steif, J. E. (2003).Which properties of a random sequence are dynamically sensitive?.Ann. Prob. 31,1–34.Google Scholar
[5] Borovkov, A. A. (1998). In Probability Theory,Gordon and Breach,Amsterdam.Google Scholar
[6] Lyons, R.,Pemantle, R. and Peres, Y. (1996).Random walks on the lamplighter group.Ann. Prob. 24,1993–2006.Google Scholar
[7] Steif, J. E.(2009).A survey of dynamical percolation. In Fractal Geometry and Stochastics IV(Progr. Prob. 61),Birkhäuser,Basel, pp.145–174.Google Scholar
[8] Woess, W. (2000).Random Walks on Infinite Graphs and Groups(Camb. Tracts Math. 138).IEEE,Cambridge University Press.CrossRefGoogle Scholar