Linear and superlinear spread for continuous-time frog model

Consider a stochastic growth model on \mathbb{Z} ^\rm{d}. Start with some active particle at the origin and sleeping particles elsewhere. The initial number of particles at x \in \mathbb{Z} ^\rm{d} is $\eta(x)$, where $\eta (x)$ are independent random variables distributed according to $\mu$. Active particles perform a simple continuous-time random walk while sleeping particles stay put until the first arrival of an active particle to their location. Upon the arrival all sleeping particles at the site activate at once and start moving according to their own simple random walks. The aim of this paper is to give conditions on $\mu$ under which the spread of the process is linear or faster than linear. The main technique is comparison with other stochastic growth models.Comment: Added references related to the percolation model in Section

Universally Typical Sets for Ergodic Sources of Multidimensional Data

We lift important results about universally typical sets, typically sampled sets, and empirical entropy estimation in the theory of samplings of discrete ergodic information sources from the usual one-dimensional discrete-time setting to a multidimensional lattice setting. We use techniques of packings and coverings with multidimensional windows to construct sequences of multidimensional array sets which in the limit build the generated samples of any ergodic source of entropy rate below an $h_0$ with probability one and whose cardinality grows at most at exponential rate $h_0$.Comment: 15 pages, 1 figure. To appear in Kybernetika. This replacement corrects typos and slightly strengthens the main theore

On the complexity of polygonal billiards

We show that the complexity of the billiard in a typical polygon grows cubically and the number of saddle connections grows quadratically along certain subsequences. It is known that the set of points whose first n-bounces hits the same sequence of sides as the orbit of an aperiodic phase point z converges to z. We establishe a polynomial lower bound estimate on this convergence rate for almost every z. This yields an upper bound on the upper metric complexity and upper slow entropy of polygonal billiards. We also prove significant deviations from the expected convergence behavior. Finally we extend these results to higher dimensions as well as to arbitrary invariant measures

Passive Supporters of Terrorism and Phase Transitions

We discuss some social contagion processes to describe the formation and spread of radical opinions. The dynamics of opinion spread involves local threshold processes as well as mean field effects. We calculate and observe phase transitions in the dynamical variables resulting in a rapidly increasing number of passive supporters. This strongly indicates that military solutions are inappropriate.Comment: references added concerning previous work of S. Gala

Shape theorem for a one-dimensional growing particle system with a bounded number of occupants per site

We consider a one-dimensional discrete-space birth process with a bounded number of particle per site. Under the assumptions of the finite range of interaction, translation invariance, and non-degeneracy, we prove a shape theorem. We also derive a limiting estimate and an exponential estimate on the fluctuations of the position of the rightmost particle.Comment: 17 pages; a short section devoted to numerical simulations is added; other small changes and improvement

The continuous-time frog model can spread arbitrarily fast

The aim of the paper is to demonstrate that the continuous-time frog model can spreadarbitrary fast. The set of sites visited by an active particle can become infinite in a finitetime

Spatial growth processes with long range dispersion: microscopics, mesoscopics, and discrepancy in spread rate

We consider the speed of propagation of a {continuous-time continuous-space} branching random walk with the additional restriction that the birth rate at any spatial point cannot exceed $1$. The dispersion kernel is taken to have density that decays polynomially as $|x|^{- 2\alpha}$, $x \to \infty$. We show that if $\alpha > 2$, then the system spreads at a linear speed, {while for $\alpha \in (\frac 12 ,2]$ the spread is faster than linear}. We also consider the mesoscopic equation corresponding to the microscopic stochastic system. We show that in contrast to the microscopic process, the solution to the mesoscopic equation spreads exponentially fast for every $\alpha > \frac 12$.Comment: v2 update: A new result is added covering the case $alpha < 2$ for the microscopic model. Further remarks and heuristic comments are added, including connections to other models. Many minor changes are mad