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A. Yambartsev

Publications and source records attributed to A. Yambartsev.

At least 19 recordsLinked to original sources
arXiv2025

Moderate, large and super large deviations principles for Poisson process with uniform catastrophes

In this paper, we expand and generalize the findings presented in our previous work on the law of large numbers and the large deviation principle for Poisson processes with uniform catastrophes. We study three distinct scalings: sublinear (moderate deviations), linear (large deviations), and superlinear (superlarge deviations). Across these scales, we establish different yet coherent rate functions.

math.PR↗
arXiv2020

Hawking-Penrose Black Hole Model. Large Emission Regime

In this paper, we propose a stochastic version of the Hawking-Penrose black hole model. We describe the dynamics of the stochastic model as a continuous-time Markov jump process of quanta out and in the black hole. The average of the random process satisfies the deterministic picture accepted in the physical literature. Assuming that the number of quanta is finite the proposed Markov process consists of two components: the number of the quanta in the black hole and the amount of the quanta outside. The stochastic representation allows us to apply large deviation theory to study the asymptotics of probabilities of rare events when the number of quanta grows to infinity. The theory provides explicitly the rate functional for the process. Its infimum over the set of all trajectories leading to a large emission event is attained on the most probable trajectory. This trajectory is a solution of a highly nonlinear Hamiltonian system of equations. Under the condition of the stationarity of the fraction of quanta in the black hole, we found the most probable trajectory corresponding to a large emission event.

math.PR↗
arXiv2019

Local large deviation principle for Wiener process with random resetting

We consider a class of Markov processes with resettings, where at random times, the Markov processes are restarted from a predetermined point or a region. These processes are frequently applied in physics, chemistry, biology, economics, and in population dynamics. In this paper we establish the local large deviation principle (LLDP) for the Wiener processes with random resettings, where the resettings occur at the arrival time of a Poisson process. Here, at each resetting time, a new resetting point is selected at random, according to a conditional distribution.

math.PR↗
arXiv2019

Large deviations in a population dynamics with catastrophes

The large deviation principle on phase space is proved for a class of Markov processes known as random population dynamics with catastrophes. In the paper we study the process which corresponds to the random population dynamics with linear growth and uniform catastrophes, where an eliminating portion of the population is chosen uniformly. The large deviation result provides an optimal trajectory of large fluctuation: it shows how the large fluctuations occur for this class of processes.

math.PR↗
arXiv2019

The local principle of large deviations for compound Poisson process with catastrophes

The continuous time Markov process considered in this paper belongs to a class of population models with linear growth and catastrophes. There, the catastrophes happen at the arrival times of a Poisson process, and at each catastrophe time, a randomly selected portion of the population is eliminated. For this population process, we derive an asymptotic upper bound for the maximum value and prove the local large deviation principle.

math.PR↗
arXiv2018

Large Emission Regime in Mean Field Luminescence

We study a class of random processes on $N$ particles which can be interpreted as stochastic model of luminescence. Each particle can stay in one of two states: Excited state or ground state. Any particle at ground state is excited with a constant rate (pumping). The number of excited particles decreases by means of photon emission through interactions of the particles. We analyse the rare event of flashes, i.e., the emission of a very large number of photons $B$ during a fixed time interval $T$. We employ the theory of large deviations to provide the asymptotics of the probability of such event when the total number of particles $N$ tends to infinity. This theory gives us also the optimal trajectory of scaled process corresponding to this event. The stationary regime of this process we call the large emission regime. In several cases we prove that in the large emission regime a share of excited particles in a system is stable under the changes of the pumping and emission rates.

math.PR↗
arXiv2018

Stochastic Dynamics of Einstein Matter-Radiation Model with Spikes

Motivated by the Einstein classical description of the matter-radiation dynamics we revise a dynamical system producing spikes of the photon emission. Then we study the corresponding stochastic model, which takes into account the randomness of spontaneous and stimulated atomic transitions supporting by a pumping. Our model reduces to Markovian density dependent processes for the inversion coefficient and the photon specific value driven by small but fast jumps. We analyse the model in three limits: the many-component, the mean-field, and the one-component. In the last case it is a positive recurrent Markov chain with sound spikes.

math.PR↗
arXiv2018

Large Fluctuations in two-level systems with stimulated emission

We consider a system of $N$ identical independent Markov processes, each taking values 0 or 1. The system describes a stochastic dynamics of an ensemble of two-level atoms. The atoms are exposed to a photon flux. Under the photon flux action, every atom changes its state with some intensities either from its ground state (state 0) to the excited state (state 1) or from the excited state to the ground state (the stimulated emission). The atom can also change the state from the excited to the ground one spontaneously. We study rare events when the big cumulative emission occurs on the fixed time interval $[0,T]$. To this end we apply the large deviation theory which allows one to make asymptotic analysis as $N\to\infty$.

math.PR↗
arXiv2017

Large Fluctuations of Radiation in Stochastically Activated Two-Level Systems

We study the large fluctuations of emitted radiations in the system of $N$ non-interacting two-level atoms. Two methods are used to calculate the probability of the large fluctuations and the time dependence of the excitation and emission. The first method is based on the large deviation principle for Markov processes. The second one uses an analogue of the quantum formalism for classical probability problems. Particularly we prove that in a large fluctuation limit approximately a half of the atoms are excited. This fact is independent on the fraction of the excited atoms in the stationary state.

cond-mat.stat-mech↗
arXiv2015

Random walks in a queueing network environment

We propose a class of models of random walks in a random environment where an exact solution can be given for a stationary distribution. The tool is the detailed balance equations.

math.PR↗
arXiv2014

Dynamics of Tectonic Plates

We suggest a model that describes a mutual dynamic of tectonic plates. The dynamic is a sort of stick-slip one which is modeled by a Markov random process. The process defines a microlevel of the dynamic. A macrolevel is obtained by a scaling limit which leads to a system of integro-differential equations which determines a kind of mean field systems. Conditions when Gutenberg-Richter empirical law are presented on the mean field level. These conditions are rather universal and do not depend on features of resistant forces.

math.PR↗
arXiv2013

Large deviations for excursions of non-homogeneous Markov processes

In this paper, the large deviations on trajectory level for ergodic Markov processes are studied. These processes take values in the non-negative quadrant of the two dimension lattice and are concentrated on step-wise functions. The rates of jumps towards the axes (jump down) depend on the position of the process -- the higher the position, the greater the rate. The rates of jumps going in the same direction as the axes (jump up) are constants. Therefore the processes are ergodic. The large deviations are studied under equal scalings of both space and time. The scaled versions of the processes converge to 0. The main result is that the probabilities of long excursions out of 0 tend to 0 exponentially fast with an exponent proportional to the square of the scaling parameter. A proportionality coefficient is an integral of a linear combination of path components. A rate function of the large deviation principle is calculated for continuous functions only.

math.PR↗
arXiv2013

Bounds on the critical line via transfer matrix methods for an Ising model coupled to causal dynamical triangulations

We introduce a transfer matrix formalism for the (annealed) Ising model coupled to two-dimensional causal dynamical triangulations. Using the Krein-Rutman theory of positivity preserving operators we study several properties of the emerging transfer matrix. In particular, we determine regions in the quadrant of parameters beta, mu >0 where the infinite-volume free energy converges, yielding results on the convergence and asymptotic properties of the partition function and the Gibbs measure.

math-ph↗
arXiv2013

A Mermin--Wagner theorem on Lorentzian triangulations with quantum spins

We consider infinite random casual Lorentzian triangulations emerging in quantum gravity for critical values of parameters. With each vertex of the triangulation we associate a Hilbert space representing a bosonic particle moving in accordance with standard laws of Quantum Mechanics. The particles interact via two-body potentials decaying with the graph distance. A Mermin--Wagner type theorem is proven for infinite-volume reduced density matrices related to solutions to DLR equations in the Feynman--Kac (FK) representation.

math-ph↗
arXiv2013

Growth of uniform infinite causal triangulations

We introduce a growth process which samples sections of uniform infinite causal triangulations by elementary moves in which a single triangle is added. A relation to a random walk on the integer half line is shown. This relation is used to estimate the geodesic distance of a given triangle to the rooted boundary in terms of the time of the growth process and to determine from this the fractal dimension. Furthermore, convergence of the boundary process to a diffusion process is shown leading to an interesting duality relation between the growth process and a corresponding branching process.

math-ph↗
arXiv2012

A Mermin--Wagner theorem for Gibbs states on Lorentzian triangulations

We establish a Mermin--Wagner type theorem for Gibbs states on infinite random Lorentzian triangulations (LT) arising in models of quantum gravity. Such a triangulation is naturally related to the distribution $\sf P$ of a critical Galton--Watson tree, conditional upon non-extinction. At the vertices of the triangles we place classical spins taking values in a torus $M$ of dimension $d$, with a given group action of a torus ${\tt G}$ of dimension $d'\leq d$. In the main body of the paper we assume that the spins interact via a two-body nearest-neighbor potential $U(x,y)$ invariant under the action of ${\tt G}$. We analyze quenched Gibbs measures generated by $U$ and prove that, for $\sf P$-almost all Lorentzian triangulations, every such Gibbs measure is ${\tt G}$-invariant, which means the absence of spontaneous continuous symmetry-breaking.

math-ph↗
arXiv2011

A note on weak convergence results for uniform infinite causal triangulations

We discuss uniform infinite causal triangulations and equivalence to the size biased branching process measure - the critical Galton-Watson branching process distribution conditioned on non-extinction. Using known results from the theory of branching processes, this relation is used to prove weak convergence of the joint length-area process of a uniform infinite causal triangulations to a limiting diffusion. The diffusion equation enables us to determine the physical Hamiltonian and Green's function from the Feynman-Kac procedure, providing us with a mathematical rigorous proof of certain scaling limits of causal dynamical triangulations.

math-ph↗