arXiv subjects

Anton Muratov

Publications and source records attributed to Anton Muratov.

3 recordsLinked to original sources

arXiv2014

Bit flipping and time to recover

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 is the following: at each time step choose one bit from a given distribution $\mathcal{P}$ on the integers independently of anything else, then flip the state of this bit to the opposite. 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 BF and DB models to show recurrent or transient behaviour, depending on the properties of $\mathcal{P}$. 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.

math.PR↗

arXiv2014

Neighbour-dependent point shifts and random exchange models: invariance and attractors

Consider a stationary renewal point process on the real line and divide each of the segments it defines in a proportion given by \iid realisations of a fixed distribution $G$ supported by [0,1]. We ask ourselves for which interpoint distribution $F$ and which division distributions $G$, the division points is again a renewal process with the same $F$? An evident case is that of degenerate $F$ and $G$. Interestingly, the only other possibility is when $F$ is Gamma and $G$ is Beta with related parameters. In particular, the division points of a Poisson process is again Poisson, if the division distribution is Beta: B$(r,1-r)$ for some $0<r<1$. We show a similar behaviour of random exchange models when a countable number of `agents' exchange randomly distributed parts of their `masses' with neighbours. More generally, a Dirichlet distribution arises in these models as a fixed point distribution preserving independence of the masses at each step. We also show that for each $G$ there is a unique attractor, a distribution of the infinite sequence of masses, which is a fixed point of the random exchange and to which iterations of a non-equilibrium configuration of masses converge weakly. In particular, iteratively applying B$(r,1-r)$-divisions to a realisation of any renewal process with finite second moment of $F$ yields a Poisson process of the same intensity in the limit.

math.PR↗

arXiv2012

LISA: Locally Interacting Sequential Adsorbtion

We study a class of dynamically constructed point processes in which at every step a new point (particle) is added to the current configuration with a distribution depending on the local structure around a uniformly chosen particle. This class covers, in particular, generalised Polya urn scheme, Dubbins--Freedman random measures and cooperative sequential adsorption models studied previously. Specifically, we address models where the distribution of a newly added particle is determined by the distance to the closest particle from the chosen one. We address boundedness of the processes and convergence properties of the corresponding sample measure. We show that in general the limiting measure is random when exists and that this is the case for a wide class of almost surely bounded processes.

math.PR↗