Bootstrap percolation on a graph with random and local connections

Let $G_{n,p}^1$ be a superposition of the random graph $G_{n,p}$ and a one-dimensional lattice: the $n$ vertices are set to be on a ring with fixed edges between the consecutive vertices, and with random independent edges given with probability $p$ between any pair of vertices. Bootstrap percolation on a random graph is a process of spread of "activation" on a given realisation of the graph with a given number of initially active nodes. At each step those vertices which have not been active but have at least $r \geq 2$ active neighbours become active as well. We study the size of the final active set in the limit when $n\rightarrow \infty$. The parameters of the model are $n$, the size $A_0=A_0(n)$ of the initially active set and the probability $p=p(n)$ of the edges in the graph. Bootstrap percolation process on $G_{n,p}$ was studied earlier. Here we show that the addition of $n$ local connections to the graph $G_{n,p}$ leads to a more narrow critical window for the phase transition, preserving however, the critical scaling of parameters known for the model on $G_{n,p}$. We discover a range of parameters which yields percolation on $G_{n,p}^1$ but not on $G_{n,p}$.Comment: 38 pages, 2 figure

The Ising model on the random planar causal triangulation: bounds on the critical line and magnetization properties

We investigate a Gibbs (annealed) probability measure defined on Ising spin configurations on causal triangulations of the plane. We study the region where such measure can be defined and provide bounds on the boundary of this region (critical line). We prove that for any finite random triangulation the magnetization of the central spin is sensitive of the boundary conditions. Furthermore, we show that in the infinite volume limit, the magnetization of the central spin vanishes for values of the temperature high enough.Comment: 28 pages, 2 figures, 1 section adde

Covariance Structure of Coulomb Multiparticle System

We consider a system of particles on a finite interval with Coulomb 3-dimensional interactions between close neighbours, i.e. only a few other neighbours apart. This model was introduced by Malyshev [Probl. Inf. Transm. 51 (2015) 31-36] to study the flow of charged particles. Notably even the nearest-neighbours interactions case, the only one studied previously, was proved to exhibit multiple phase transitions depending on the strength of the external force when the number of particles goes to infinity. Here we include as well interactions beyond the nearest-neighbours ones. Surprisingly but this leads to qualitatively new features even when the external force is zero. The order of the covariances of distances between pairs of consecutive charges is changed when compared with the former nearest-neighbours case, and moreover the covariances exhibit periodicity in sign: the interspacings are positively correlated if the number of interspacings between them is odd, otherwise, they are negatively correlated. In the course of the proof we derive Gaussian approximation for the limit distribution for dependent variables described by a Gibbs distribution.Comment: 43 page

A dynamic network in a dynamic population: asymptotic properties

We derive asymptotic properties for a stochastic dynamic network model in a stochastic dynamic population. In the model, nodes give birth to new nodes until they die, each node being equipped with a social index given at birth. During the life of a node it creates edges to other nodes, nodes with high social index at higher rate, and edges disappear randomly in time. For this model we derive criterion for when a giant connected component exists after the process has evolved for a long period of time, assuming the node population grows to infinity. We also obtain an explicit expression for the degree correlation $\rho$ (of neighbouring nodes) which shows that $\rho$ is always positive irrespective of parameter values in one of the two treated submodels, and may be either positive or negative in the other model, depending on the parameters

Scaling of Components in Critical Geometric Random Graphs on 2-dim Torus

We consider random graphs on the set of $N^2$ vertices placed on the discrete $2$-dimensional torus. The edges between pairs of vertices are independent, and their probabilities decay with the distance $\rho$ between these vertices as $(N\rho)^{-1}$. This is an example of an inhomogeneous random graph which is not of rank 1. The reported previously results on the sub- and super-critical cases of this model exhibit great similarity to the classical Erd\H{o}s-R\'{e}nyi graphs. Here we study the critical phase. A diffusion approximation for the size of the largest connected component rescaled with $(N^2)^{-2/3}$ is derived. This completes the proof that in all regimes the model is within the same class as Erd\H{o}s-R\'{e}nyi graph with respect to scaling of the largest component.Comment: 65 page