Induced gelation in a two-site spatial coagulation model

A two-site spatial coagulation model is considered. Particles of masses $m$ and $n$ at the same site form a new particle of mass $m+n$ at rate $mn$. Independently, particles jump to the other site at a constant rate. The limit (for increasing particle numbers) of this model is expected to be nondeterministic after the gelation time, namely, one or two giant particles randomly jump between the two sites. Moreover, a new effect of induced gelation is observed--the gelation happening at the site with the larger initial number of monomers immediately induces gelation at the other site. Induced gelation is shown to be of logarithmic order. The limiting behavior of the model is derived rigorously up to the gelation time, while the expected post-gelation behavior is illustrated by a numerical simulation.Comment: Published at http://dx.doi.org/10.1214/105051605000000755 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

An Ergodic Theorem for the Quantum Relative Entropy

We prove the ergodic version of the quantum Stein's lemma which was conjectured by Hiai and Petz. The result provides an operational and statistical interpretation of the quantum relative entropy as a statistical measure of distinguishability, and contains as a special case the quantum version of the Shannon-McMillan theorem for ergodic states. A version of the quantum relative Asymptotic Equipartition Property (AEP) is given.Comment: 19 pages, no figure

Level Crossing Probabilities II: Polygonal Recurrence of Multidimensional Random Walks

In part I (math.PR/0406392) we proved for an arbitrary one-dimensional random walk with independent increments that the probability of crossing a level at a given time n is of the maximal order square root of n. In higher dimensions we call a random walk 'polygonally recurrent' (resp. transient) if a.s. infinitely many (resp. finitely many) of the straight lines between two consecutive sites hit a given bounded set. The above estimate implies that three-dimensional random walks with independent components are polygonally transient. Similarly a directionally reinforced random walk on Z^3 in the sense of Mauldin, Monticino and v.Weizsaecker [1] is transient. On the other hand we construct an example of a transient but polygonally recurrent random walk with independent components on Z^2.Comment: 23 pages, errors and typos corrected, references adde

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 Structure of Spatial Branching Processes

The paper is a contribution to the theory of branching processes with discrete time and a general phase space in the sense of [2]. We characterize the class of regular, i.e. in a sense sufficiently random, branching processes (Φk) k∈Z by almost sure properties of their realizations without making any assumptions about stationarity or existence of moments. This enables us to classify the clans of (Φk) into the regular part and the completely non-regular part. It turns out that the completely non-regular branching processes are built up from single-line processes, whereas the regular ones are mixtures of left-tail trivial processes with a Poisson family structure