arXiv subjects

Tatyana S. Turova

Publications and source records attributed to Tatyana S. Turova.

6 recordsLinked to original sources

arXiv2024

Antiferromagnetic Covariance Structure of Coulomb Chain

We consider a system of particles lined up 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 (2015) 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 show that including interactions beyond the nearest-neighbours ones, leads to qualitatively new features. The order of covariances of distances between pairs of consecutive charges is changed when compared with the former nearest-neighbours case. Furthermore, we discover that the covariances between inter-spacings exhibit the antiferromagnetic property, namely they periodically change sign depending on the parity of number of spacings between them, while their amplitude decays.

math.PR↗

arXiv2015

Critical line of the Ising model on 2-dimensional CDT and its dual

In this paper we study the annealed coupling of an Ising model with 2-dimensional causal dynamical triangulation model. After a short review of previous results, we prove the existence of the so-called critical line and derive its analytical properties and asymptotics. In addition, relations between the model and its dual are investigated.

cond-mat.stat-mech↗

arXiv2009

Diffusion approximation for the components in critical inhomogeneous random graphs of rank 1

Consider the random graph on $n$ vertices $1, ..., n$. Each vertex $i$ is assigned a type $X_i$ with $X_1, ..., X_n$ being independent identically distributed as a nonnegative discrete random variable $X$. We assume that ${\bf E} X^3<\infty$. Given types of all vertices, an edge exists between vertices $i$ and $j$ independent of anything else and with probability $\min \{1, \frac{X_iX_j}{n}(1+\frac{a}{n^{1/3}}) \}$. We study the critical phase, which is known to take place when ${\bf E} X^2=1$. We prove that normalized by $n^{-2/3}$ the asymptotic joint distributions of component sizes of the graph equals the joint distribution of the excursions of a reflecting Brownian motion $B^a(s)$ with diffusion coefficient $\sqrt{{\bf E}X{\bf E}X^3}$ and drift $a-\frac{{\bf E}X^3}{{\bf E}X}s$. This shows that finiteness of ${\bf E}X^3$ is the necessary condition for the diffusion limit. In particular, we conclude that the size of the largest connected component is of order $n^{2/3}$.

math.PR↗

arXiv2008

Asymptotics for the size of the largest component scaled to "log n" in inhomogeneous random graphs

We study the inhomogeneous random graphs in the subcritical case. We derive an exact formula for the size of the largest connected component scaled to $\log n$ where $n$ is the size of the graph. This generalizes the recent result for the "rank 1 case". Here we discover that the same well-known equation for the survival probability, whose positive solution determines the asymptotics of the size of the largest component in the supercritical case, plays the crucial role in the subcritical case as well. But now these are the negative solutions which come into play.

math.PR↗

arXiv2006

Merging percolation on $Z^d$ and classical random graphs: Phase transition

We study a random graph model which is a superposition of the bond percolation model on $Z^d$ with probability $p$ of an edge, and a classical random graph $G(n, c/n)$. We show that this model, being a {\it homogeneous} random graph, has a natural relation to the so-called "rank 1 case" of {\it inhomogeneous} random graphs. This allows us to use the newly developed theory of inhomogeneous random graphs to describe the phase diagram on the set of parameters $c\geq 0$ and $0 \leq p<p_c$, where $p_c=p_c(d)$ is the critical probability for the bond percolation on $Z^d$. The phase transition is similar to the classical random graph, it is of the second order. We also find the scaled size of the largest connected component above the phase transition.

math.PR↗

arXiv2006

Merging percolation and classical random graphs: Phase transition in dimension 1

We study a random graph model which combines properties of the edge percolation model on Z^d and a classical random graph G(n,c/n). We show that this model, being a homogeneous random graph, has a natural relation to the so-called "rank 1 case" of inhomogeneous random graphs. This allows us to use the newly developed theory of inhomogeneous random graphs to describe completely the phase diagram in the case d=1. The phase transition is similar to the classical random graph, it is of the second order. We also find the scaled size of the largest connected component above the phase transition.

math.PR↗