## Journal of Applied Probability

- J. Appl. Probab.
- Volume 53, Number 3 (2016), 650-666.

### >Bit flipping and time to recover

Anton Muratov and Sergei Zuyev

#### 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.

#### Article information

**Source**

J. Appl. Probab. Volume 53, Number 3 (2016), 650-666.

**Dates**

First available in Project Euclid: 13 October 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.jap/1476370768

**Mathematical Reviews number (MathSciNet)**

MR3570086

**Zentralblatt MATH identifier**

1351.60104

**Subjects**

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.) [See also 68W20, 68W40]

**Keywords**

Binary system bit flipping random walk on a group Markov chain recurrence critical behaviour

#### Citation

Muratov, Anton; Zuyev, Sergei. >Bit flipping and time to recover. J. Appl. Probab. 53 (2016), no. 3, 650--666.https://projecteuclid.org/euclid.jap/1476370768