arXiv subjects

Stanislav Volkov

Publications and source records attributed to Stanislav Volkov.

45 records · Page 3Linked to original sources

arXiv2009

Stability of a growth process generated by monomer filling with nearest-neighbour cooperative effects

We study stability of a growth process generated by sequential adsorption of particles on a one-dimensional lattice torus, that is, the process formed by the numbers of adsorbed particles at lattice sites, called heights. Here the stability of process, loosely speaking, means that its components grow at approximately the same rate. To assess stability quantitatively, we investigate the stochastic process formed by differences of heights. The model can be regarded as a variant of a Polya urn scheme with local geometric interaction.

math.PR↗

arXiv2009

Forest fires on $\Z_+$ with ignition only at 0

We consider a version of the forest fire model on graph $G$, where each vertex of a graph becomes occupied with rate one. A fixed vertex $v_0$ is hit by lightning with the same rate, and when this occurs, the whole cluster of occupied vertices containing $v_0$ is burnt out. We show that when $G=Z_{+}$, the times between consecutive burnouts at vertex $n$, divided by $\log n$, converge weakly as $n\to\infty$ to a random variable which distribution is $1-ρ(x)$ where $ρ(x)$ is the Dickman function. We also show that on transitive graphs with a non-trivial site percolation threshold and one infinite cluster at most, the distributions of the time till the first burnout of {\it any} vertex have exponential tails. Finally, we give an elementary proof of an interesting limit: $\lim_{n\to\infty} \sum_{k=1}^n {n \choose k} (-1)^k \log k -\log\log n=γ$.

math.PR↗

arXiv2007

Random environment on coloured trees

In this paper, we study a regular rooted coloured tree with random labels assigned to its edges, where the distribution of the label assigned to an edge depends on the colours of its endpoints. We obtain some new results relevant to this model and also show how our model generalizes many other probabilistic models, including random walk in random environment on trees, recursive distributional equations and multi-type branching random walk on $\mathbb{R}$.

math.PR↗

arXiv2007

A note on random walks in a hypercube

We study a simple random walk on an n-dimensional hypercube. For any starting position we find the probability of hitting vertex a before hitting vertex b, whenever a and b share the same edge. This generalizes the model in Doyle, P., and Snell, J., "Random Walks and Electric Networks", Mathematical Association of America, 1984 (see Exercise 1.3.7 there).

math.PR↗

arXiv2007

Urn-related random walk with drift $ρx^α / t^β$

We study a one-dimensional random walk whose expected drift depends both on time and the position of a particle. We establish a non-trivial phase transition for the recurrence vs. transience of the walk, and show some interesting applications to Friedman's urn, as well as showing the connection with Lamperti's walk with asymptotically zero drift.

math.PR↗

arXiv2002

Packet delay in models of data networks

We investigate individual packet delay in a model of data networks with table-free, partial table and full table routing. We present analytical estimation for the average packet delay in a network with small partial routing table. Dependence of the delay on the size of the network and on the size of the partial routing table is examined numerically. Consequences for network scalability are discussed.

cs.NI↗

arXiv1999

Loss of tension in an infinite membrane with holes distributed by Poisson law

If one randomly punches holes in an infinite tensed membrane, when does the tension cease to exist? This problem was introduced by R. Connelly in connection with applications of rigidity theory to natural sciences. We outline a mathematical theory of tension based on graph rigidity theory and introduce several probabilistic models for this problem. We show that if the ``centers'' of the holes are distributed in R^2 according to Poisson law with parameter λ>0, and the distribution of sizes of the holes is independent of the distribution of their centers, the tension vanishes on all of R^2 for any value of λ. In fact, it follows from a more general result on the behavior of iterative convex hulls of connected subsets of R^d, when the initial configuration of subsets is distributed according to Poisson law and the sizes of the elements of the original configuration are independent of this Poisson distribution. For the latter problem we establish the existence of a critical threshold in terms of the number of iterative convex hull operations required for covering all of R^d. The processes described in the paper are somewhat related to bootstrap and rigidity percolation models.

math.PR↗

arXiv1997

Markov chains in a field of traps

A general criterion is given for when a Markov chain trapped with probability p(x) in state x will be almost surely trapped. The quenched (state x is a trap forever with probability p(x)) and annealed (state x traps with probability p(x) on each visit) problems are shown to be equivalent.

math.PR↗

arXiv1997

Vertex-reinfoced random walk on Z visits finitely many states

Vertex-reinforced random walk is defined in Pemantle's (1988) thesis; it is a random walk that is biased to visit sites it has already visited a lot. We show that this reinforcement scheme, in contrast to the scheme of edge-reinforcement, causes random walk on a line to get trapped in a finite set.

math.PR↗