258 research outputs found
Self-regulation in the Bolker-Pacala model
The Markov dynamics is studied of an infinite system of point entities placed
in \mathds{R}^d, in which the constituents disperse and die, also due to
competition. Assuming that the dispersal and competition kernels are continuous
and integrable we show that the evolution of states of this model preserves
their sub-Poissonicity, and hence the local self-regulation (suppression of
clustering) takes place. Upper bounds for the correlation functions of all
orders are also obtained for both long and short dispersals, and for all values
of the intrinsic mortality rate.Comment: arXiv admin note: substantial text overlap with arXiv:1702.0292
Gibbs random fields with unbounded spins on unbounded degree graphs
Gibbs random fields corresponding to systems of real-valued spins (e.g.
systems of interacting anharmonic oscillators) indexed by the vertices of
unbounded degree graphs with a certain summability property are constructed. It
is proven that the set of tempered Gibbs random fields is non-void and weakly
compact, and that they obey uniform exponential integrability estimates. In the
second part of the paper, a class of graphs is described in which the mentioned
summability is obtained as a consequence of a property, by virtue of which
vertices of large degree are located at large distances from each other. The
latter is a stronger version of a metric property, introduced in [Bassalygo, L.
A. and Dobrushin, R. L. (1986). \textrm{Uniqueness of a Gibbs field with a
random potential--an elementary approach.}\textit{Theory Probab. Appl.} {\bf
31} 572--589]
A Phase Transition in a Quenched Amorphous Ferromagnet
Quenched thermodynamic states of an amorphous ferromagnet are studied. The
magnet is a countable collection of point particles chaotically distributed
over , . Each particle bears a real-valued spin with
symmetric a priori distribution; the spin-spin interaction is pair-wise and
attractive. Two spins are supposed to interact if they are neighbors in the
graph defined by a homogeneous Poisson point process. For this model, we prove
that with probability one: (a) quenched thermodynamic states exist; (b) they
are multiple if the particle density (i.e., the intensity of the underlying
point process) and the inverse temperature are big enough; (c) there exist
multiple quenched thermodynamic states which depend on the realizations of the
underlying point process in a measurable way
The statistical dynamics of a spatial logistic model and the related kinetic equation
There is studied an infinite system of point entities in which
reproduce themselves and die, also due to competition. The system's states are
probability measures on the space of configurations of entities. Their
evolution is described by means of a BBGKY-type equation for the corresponding
correlation (moment) functions. It is proved that: (a) these functions evolve
on a bounded time interval and remain sub-Poissonian due to the competition;
(b) in the Vlasov scaling limit they converge to the correlation functions of
the time-dependent Poisson point field the density of which solves the kinetic
equation obtained in the scaling limit from the equation for the correlation
functions. A number of properties of the solutions of the kinetic equation are
also established
- …