arXiv subjects

George Kordzakhia

Publications and source records attributed to George Kordzakhia.

4 recordsLinked to original sources

arXiv2018

Phase Transition for the Chase-Escape Model on 2D Lattices

Chase-Escape is a simple stochastic model that describes a predator-prey interaction. In this model, there are two types of particles, red and blue. Red particles colonize adjacent empty sites at an exponential rate $λ_{R}$, whereas blue particles take over adjacent red sites at exponential rate $λ_{B}$, but can never colonize empty sites directly. Numerical simulations suggest that there is a critical value $p_{c}$ for the relative growth rate $p:=λ_{R}/λ_{B}$. When $p p_{c}$ mutual survival occurs with positive probability. In particular, $p_{c} \approx 0.50$ for the square lattice case ($\mathbb Z^{2}$). Our simulations provide a plausible explanation for the critical value. Near the critical value, the set of occupied sites exhibits a fractal nature, and the hole sizes approximately follow a power-law distribution.

cond-mat.dis-nn↗

arXiv2006

Ergodicity and Mixing Properties of the Northeast Model

The Northeast Model is a spin system on the two-dimensional integer lattice that evolves according to the following rule: Whenever a site's southerly and westerly nearest neighbors have spin $1$, it may reset its own spin by tossing a $p$-coin; at all other times, its spin remains frozen. It is proved that the northeast model has a phase transition at $p_{c}=1-β_{c}$, where $β_{c}$ is the critical parameter for oriented percolation. For $p 2/3$ it is shown that there is exponential decay of correlations.

math.PR↗

arXiv2005

A two-species competition model on Z^d

We consider a two-type stochastic competition model on the integer lattice Z^d. The model describes the space evolution of two ``species'' competing for territory along their boundaries. Each site of the space may contain only one representative (also referred to as a particle) of either type. The spread mechanism for both species is the same: each particle produces offspring independently of other particles and can place them only at the neighboring sites that are either unoccupied, or occupied by particles of the opposite type. In the second case, the old particle is killed by the newborn. The rate of birth for each particle is equal to the number of neighboring sites available for expansion. The main problem we address concerns the possibility of the long-term coexistence of the two species. We have shown that if we start the process with finitely many representatives of each type, then, under the assumption that the limit set in the corresponding first passage percolation model is uniformly curved, there is positive probability of coexistence.

math.PR↗

arXiv2005

An Oriented Competition model on Z_{+}^2

We consider a two-type oriented competition model on the first quadrant of the two-dimensional integer lattice. Each vertex of the space may contain only one particle of either Red type or Blue type. A vertex flips to the color of a randomly chosen southwest nearest neighbor at exponential rate 2. At time zero there is one Red particle located at (1,0) and one Blue particle located at (0,1). The main result is a partial shape theorem: Denote by R(t) and B(t) the red and blue regions at time t. Then (i) eventually the upper half of the unit square contains no points of B(t)=t, and the lower half no points of R(t)=t; and (ii) with positive probability there are angular sectors rooted at (1,1) that are eventually either red or blue. The second result is contingent on the uniform curvature of the boundary of the corresponding Richardson shape.

math.PR↗