arXiv subjects

Tatyana Turova

Publications and source records attributed to Tatyana Turova.

9 recordsLinked to original sources

arXiv2025

Rectangular Gilbert Tessellation

A random planar quadrangulation process is introduced as an approximation for certain cellular automata in terms of random growth of rays from a given set of points. This model turns out to be a particular (rectangular) case of the well-known Gilbert tessellation, which originally models the growth of needle-shaped crystals from the initial random points with a Poisson distribution in a plane. From each point the lines grow on both sides of vertical and horizontal directions until they meet another line. This process results in a rectangular tessellation of the plane. The central and still open question is the distribution of the length of line segments in this tessellation. We derive exponential bounds for the tail of this distribution. The correlations between the segments are proved to decay exponentially with the distance between their initial points. Furthermore, the sign of the correlation is investigated for some instructive examples. In the case when the initial set of points is confined in a box $[0,N]^2$, it is proved that the average number of rays reaching the border of the box has a linear order in $N$.

math.PR↗

arXiv2025

Behavior of the scaling correlation functions under severe subsampling

Scale-invariance is a ubiquitous observation in the dynamics of large distributed complex systems. The computation of its scaling exponents, which provide clues on its origin, is often hampered by the limited available sampling data, making an appropriate mathematical description a challenge. This work investigates the behavior of correlation functions in fractal systems under conditions of severe subsampling. Analytical and numerical results reveal a striking robustness: the correlation functions continue to capture the expected scaling exponents despite substantial data reduction. This behavior is demonstrated numerically for the random 2-D Cantor set and the Sierpinski gasket, both consistent with exact analytical predictions. Similar robustness is observed in 1-D time series both synthetic and experimental, as well as in high-resolution images of a neuronal structure. Overall, these findings are broadly relevant for the structural characterization of biological systems under realistic sampling constraints.

physics.comp-ph↗

arXiv2023

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 $ρ$ between these vertices as $(Nρ)^{-1}$. 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 Erdős-Rényi 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 Erdős-Rényi graph with respect to scaling of the largest component.

math.PR↗

arXiv2016

Phase transition in random distance graphs on the torus

We apply here methods of inhomogeneous random graphs to a class of random distance graphs. This provides an example outside of the rank 1 models which is still solvable as long as the largest connected component is concerned. In particular, we show that some random distance graphs behave exactly as the classical Erdős-Rényi model not only in the supercritical phase (as was already known) but in the subcritical case as well.

math.PR↗

arXiv2016

Phase Transitions in the One-dimensional Coulomb Gas Ensembles

We consider the system of particles on a finite interval with pair-wise nearest neighbours interaction and external force. This model was introduced by Malyshev to study the flow of charged particles on a rigorous mathematical level. It is a simplified version of a 3-dimensional classical Coulomb gas model. We study Gibbs distribution at finite positive temperature extending recent results on the zero temperature case (ground states) with external force. We derive the asymptotic for the mean and for the variances of the distances between the neighbouring charges. We prove that depending on the strength of the external force there are several phase transitions in the local structure of the configuration of the particles in the limit when the number of particles goes to infinity.

math.PR↗

arXiv2016

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.

cond-mat.stat-mech↗

arXiv2015

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 $G_{n,p}$.

math.PR↗

arXiv2012

Bootstrap percolation on the random graph $G_{n,p}$

Bootstrap percolation on the random graph $G_{n,p}$ is a process of spread of "activation" on a given realization 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 $r\geq2$ active neighbors become active as well. We study the size $A^*$ of the final active set. The parameters of the model are, besides $r$ (fixed) and $n$ (tending to $\infty$), the size $a=a(n)$ of the initially active set and the probability $p=p(n)$ of the edges in the graph. We show that the model exhibits a sharp phase transition: depending on the parameters of the model, the final size of activation with a high probability is either $n-o(n)$ or it is $o(n)$. We provide a complete description of the phase diagram on the space of the parameters of the model. In particular, we find the phase transition and compute the asymptotics (in probability) for $A^*$; we also prove a central limit theorem for $A^*$ in some ranges. Furthermore, we provide the asymptotics for the number of steps until the process stops.

math.PR↗

arXiv2011

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 $ρ$ (of neighbouring nodes) which shows that $ρ$ 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.

math.PR↗