arXiv subjects

Stanislav Volkov

Publications and source records attributed to Stanislav Volkov.

At least 19 recordsLinked to original sources

arXiv2026

First to reach $n$ game

We consider a game with two players, consisting of a number of rounds, where the first player to win $n$ rounds becomes the overall winner. Who wins each individual round is governed by a certain urn having two types of balls (type 1 and type 2). At each round, we randomly pick a ball from the urn, and its type determines which of the two players wins. We study the game under three regimes. In the first and the third regimes, a ball is taken without replacement, whilst in the second regime, it is returned to the urn with one more ball of the same colour. We study the properties of the random variables equal to the properly defined overall net profits of the players, and the results are drastically different in all three regimes.

math.PR↗

arXiv2026

The Loss of Tension in an Infinite Membrane with Holes of Decaying Spatial Density

What is the effect of randomly removing material from an infinite stretched membrane? Under what conditions can the membrane still sustain tension? This problem was introduced by Robert Connelly in connection with applications of rigidity theory in the natural sciences, and was later studied in M. V. Menshikov, K. A. Rybnikov, and S. E. Volkov, "The loss of tension in an infinite membrane with holes distributed according to a Poisson law" (2002); a discrete version was also considered in Robert Connelly, Konstantin Rybnikov, and Stanislav Volkov, "Percolation and the Loss of Tension in an Infinite Triangular Lattice" (2001). We study a mathematical framework based on a non-homogeneous Poisson point process whose intensity $λ$ tends to zero at infinity. The hole shapes are i.i.d.\ and independent of their locations. We show that if the intensity does not decay too quickly, then tension is still lost throughout the whole plane, as in the homogeneous model studied in 2002. Conversely, we give sufficient conditions under which complete loss of tension does not occur. Thus, both destruction and non-destruction regimes are possible even when the intensity tends to zero, indicating a phase transition in the model. The processes studied here are closely related to bootstrap percolation.

math.PR↗

arXiv2026

Gaussian consensus processes and their Lyapunov exponents

We introduce a simple dynamic model of opinion formation, in which a finite population of individuals hold vector-valued opinions. At each time step, each individual's opinion moves towards the mean opinion but is then perturbed independently by a centred multivariate Gaussian random variable, with covariance proportional to the covariance matrix of the opinions of the population. We establish precise necessary and sufficient conditions on the parameters of the model, under which all opinions converge to a common limiting value. Asymptotically perfect correlation emerges between opinions on different topics. Our results are rigorous and based on properties of the partial products of an i.i.d. sequence of random matrices. Each matrix is a fixed linear combination of the identity matrix and a real Ginibre matrix. We derive an analytic expression for the maximal Lyapunov exponent of this product sequence. We also analyze a continuous-time analogue of our model.

math.PR↗

arXiv2026

Two-dimensional Rademacher walk

We study a generalisation of the one-dimensional Rademacher random walk introduced in Bhattacharya and Volkov (2023) to $\mathbb{Z}^2$ (for $d\ge 3$, the Rademacher random walk is always transient, as follows from Theorem 8.8 in Englander and Volkov (2025)). This walk is defined as the sum of a sequence of independent steps, where each step goes in one of the four possible directions with equal probability, and the size of the $n$th step is $a_n$ where $\{a_n\}$ is a given sequence of positive integers. We establish some general conditions under which the walk is recurrent or transient.

math.PR↗

arXiv2025

On extinction and survival of the Bak-Sneppen model on arbitrary graphs

We study the discrete Bak-Sneppen model introduced by Barbay and Kenyon (2001) "On the discrete Bak-Sneppen model of self-organized criticality". We extend their results as well as the non-triviality result of Meester and Znamenskiy (2002) for a finite segment of $\mathbb{Z}^1$ with the periodic boundary condition to a large class of graphs, by using coupling between the Bak-Sneppen model and the oriented percolation in a quadrant. This allows us to avoid dealing with the so-called avalanches, thus simplifying many arguments.

math.PR↗

arXiv2025

Forest Fire Model on $\mathbb{Z}_{+}$ with Delays

We consider a generalization of the forest fire model on $\mathbb{Z}_+$ with ignition at zero only, studied in [arXiv:0907.1821]. Unlike that model, we allow delays in the spread of the fires as well as the non-zero burning time of individual ``trees''. We obtain some general properties for this model, which cover, among others, the phenomena of an ``infinite fire'', not present in the original model.

math.PR↗

arXiv2024

Transience of continuous-time conservative random walks

We consider two continuous-time generalizations of conservative random walks introduced in [J.Englander and S.Volkov (2022)], an orthogonal and a spherically-symmetrical one; the latter model is known as {\em random flights}. For both models, we show the transience of the walks when $d\ge 2$ and the rate of changing of direction follows power law $t^{-α}$, $0<α\le 1$, or the law $(\ln t)^{-β}$ where $β>2$.

math.PR↗

arXiv2023

The limit point in the Jante's law process has an absolutely continuous distribution

We study a stochastic model of consensus formation, introduced in 2015 by Grinfeld, Volkov and Wade, who called it a multidimensional randomized Keynesian beauty contest. The model was generalized by Kennerberg and Volkov, who called their generalization the Jante's law process. We consider a version of the model where the space of possible opinions is a convex body $\mathcal{B}$ in $\mathbb{R}^d$. $N$ individuals in a population each hold a (multidimensional) opinion in $\mathcal{B}$. Repeatedly, the individual whose opinion is furthest from the center of mass of the $N$ current opinions chooses a new opinion, sampled uniformly at random from $\mathcal{B}$. Kennerberg and Volkov showed that the set of opinions that are not furthest from the center of mass converges to a random limit point. We show that the distribution of the limit opinion is continuous, thus proving the conjecture made after Proposition 3.2 in Grinfeld et al.

math.PR↗

arXiv2022

Recurrence and transience of Rademacher series

We introduce the notion of {\bf a}-walk $S(n)=a_1 X_1+\dots+a_n X_n$, based on a sequence of positive numbers ${\bf a}=(a_1,a_2,\dots)$ and a Rademacher sequence $X_1,X_2,\dots$. We study recurrence/transience (properly defined) of such walks for various sequences of ${\bf a}$. In particular, we establish the classification in the cases where $a_k=\lfloor k^β\rfloor$, $β>0$, as well as in the case $a_k=\lceil \log_γk \rceil$ or $a_k=\log_γk$ for $γ>1$.

math.PR↗

arXiv2022

Conservative Random Walk

Recently, in ["The coin-turning walk and its scaling limit", Electronic Journal of Probability, 25 (2020)], the ``coin-turning walk'' was introduced on ${\mathbb Z}$. It is a non-Markovian process where the steps form a (possibly) time-inhomogeneous Markov chain. In this article, we follow up the investigation by introducing analogous processes in ${\mathbb Z}^d$, $d\ge 2$: at time $n$ the direction of the process is ``updated'' with probability $p_n$; otherwise the next step repeats the previous one. We study some of the fundamental properties of these walks, such as transience/recurrence and scaling limits. Our results complement previous ones in the literature about ``correlated'' (or ``Newtonian'') and ``persistent'' random walks.

math.PR↗

arXiv2022

Generalizations of forest fires with ignition at origin

We study generalizations of the Forest Fire model introduced in [van den Berg, J., and Járai, A. A. "On the asymptotic density in a one-dimensional self-organized critical forest-fire model". Comm. Math. Phys. 253 (2005)] and [Volkov, Stanislav. "Forest fires on $\mathbb{Z}_+$ with ignition only at 0". ALEA 6 (2009)] by allowing the rates at which the tree grow to depend on their location, introducing long-range burning, as well as continuous-space generalization of the model. We establish that in all the models in consideration the time required to reach site at distance $x$ from the origin is of order at most $(\log x)^{(\log 2)^{-1}+δ}$ for any $δ>0$.

math.PR↗

arXiv2021

Rigorous upper bound for the discrete Bak-Sneppen model

Fix some $p\in[0,1]$ and a positive integer $n$. The discrete Bak-Sneppen model is a Markov chain on the space of zero-one sequences of length $n$ with periodic boundary conditions. At each moment of time a minimum element (typically, zero) is chosen with equal probability, and is then replaced together with both its neighbours by independent Bernoulli(p) random variables. Let $ν^{(n)}(p)$ be the probability that an element of this sequence equals one under the stationary distribution of this Markov chain. It was shown in [Barbay, Kenyon (2001)] that $ν^{(n)}(p)\to 1$ as $n\to\infty$ when $p>0.54\dots$; the proof there is, alas, not rigorous. The complimentary fact that $\limsup ν^{(n)}(p)< 1$ for $p\in(0,p')$ for some $p'>0$ is much harder; this was eventually shown in [Meester, Znamenski (2002)]. The purpose of this note is to provide a rigorous proof of the result from Barbay et al, as well as to improve it, by showing that $ν^{(n)}(p)\to 1$ when $p>0.45$. In fact, our method with some finer tuning allows to show this fact even for all $p>0.419533$.

math.PR↗

arXiv2021

Vertex-reinforced jump process on the integers with nonlinear reinforcement

We consider a non-linear vertex-reinforced jump process (VRJP($w$)) on $\mathbb{Z}$ with an increasing measurable weight function $w:[1,\infty)\to [1,\infty)$ and initial weights equal to one. Our main goal is to study the asymptotic behaviour of VRJP($w$) depending on the integrability of the reciprocal of $w$. In particular, we prove that if $\int_1^{\infty} \frac{\text{d}u}{w(u)} =\infty$ then the process is recurrent, i.e. it visits each vertex infinitely often and all local times are unbounded. On the other hand, if $\int_1^{\infty} \frac{\text{d} u}{w(u)} <\infty$ and there exists a $ρ>0$ such that $t \mapsto w(t)^ρ\int_t^{\infty}\frac{\text{d}u}{w(u)}$ is non-increasing then the process will eventually get stuck on exactly three vertices, and there is only one vertex with unbounded local time. We also show that if the initial weights are all the same, VRJP on $\mathbb{Z}$ cannot be transient, i.e. there exists at least one vertex that is visited infinitely often. Our results extend the ones previously obtained by Davis and Volkov [Probab. Theory Relat. Fields (2002)] who showed that VRJP with linear reinforcement on $\mathbb{Z}$ is recurrent.

math.PR↗

arXiv2021

Linear competition processes and generalized Polya urns with removals

A competition process is a continuous time Markov chain that can be interpreted as a system of interacting birth-and-death processes, the components of which evolve subject to a competitive interaction. This paper is devoted to the study of the long-term behaviour of such a competition process, where a component of the process increases with a linear birth rate and decreases with a rate given by a linear function of other components. A zero is an absorbing state for each component, that is, when a component becomes zero, it stays zero forever (and we say that this component becomes extinct). We show that, with probability one, eventually only a random subset of non-interacting components of the process survives. A similar result also holds for the relevant generalized Polya urn model with removals.

math.PR↗

arXiv2020

A local barycentric version of the Bak-Sneppen model

We study the behaviour of the interacting particle system, arising from the Bak-Sneppen model and Jante's law process. Let $N$ vertices be placed on a circle, such that each vertex has exactly two neighbours. To each vertex assign a real number, called {\em fitness}. Now find the vertex which fitness deviates most from the average of the fitnesses of its two immediate neighbours (in case of a tie, draw uniformly among such vertices), and replace it by a random value drawn independently according to some distribution $ζ$. We show that in case where $ζ$ is a uniform or a discrete uniform distribution, all the fitnesses except one converge to the same value.

math.PR↗

arXiv2019

On a class of random walks in simplexes

We study the limit behaviour of a class of random walk models taking values in the $d$-dimensional unit standard simplex, $d\ge 1$, defined as follows. From an interior point $z$, the process chooses one of the $d+1$ vertices of the simplex, with probabilities depending on $z$, and then the particle randomly jumps to a new location $z'$ on the segment connecting $z$ to the chosen vertex. In some specific cases, using properties of the Beta distribution, we prove that the limiting distributions of the Markov chain are, in fact, Dirichlet. We also consider a related history-dependent random walk model in $[0,1]$ based on an urn-type scheme. We show that this random walk converges in distribution to the arcsine law.

math.PR↗

arXiv2019

Convergence in the $p$-contest

We study asymptotic properties of the following Markov system of $N \geq 3$ points in~$[0,1]$. At each time step, the point farthest from the current centre of mass, multiplied by a constant $p>0$, is removed and replaced by an independent $ζ$-distributed point; the problem, inspired by variants of the Bak--Sneppen model of evolution and called a $p$-contest, was posed in [Grinfeld, M, Knight, P.A., and Wade, A.R. Rank-driven Markov processes, J. Stat. Phys. 146 (2012)]. We obtain various criteria for the convergences of the system, both for $p<1$ and $p>1$. In particular, when $p<1$ and $ζ\sim U[0,1]$, we show that the limiting configuration converges to zero. When $p>1$, we show that the configuration must converge to either zero or one, and we present an example where both outcomes are possible. Finally, when $p>1$, $N=3$ and $ζ$ satisfies certain conditions (e.g.~$ζ\sim U[0,1]$), we prove that the configuration can only converge to one a.s. Our paper substantially extends the results of [Grinfeld, M., Volkov, S., and Wade, A.R. Convergence in a multidimensional randomized Keynesian beauty contest. Adv. in Appl. Probab. 47 (2015)] and [Kennerberg, P., and Volkov, S. Jante's law process. Adv. in Appl. Probab. 50 (2018)] where it was assumed that $p=1$. Unlike the previous models, one can no longer use the Lyapunov function based just on the radius of gyration; when $0<p<1$ one has to find a much finer tuned function which turns out to be a supermartingale; the proof of this fact constitutes an unwieldy, albeit necessary, part of the paper.

math.PR↗

arXiv2019

The coin-turning walk and its scaling limit

Let $S$ be the random walk obtained from "coin turning" with some sequence $\{p_n\}_{n\ge 1}$, as introduced in [6]. In this paper we investigate the scaling limits of $S$ in the spirit of the classical Donsker invariance principle, both for the heating and for the cooling dynamics. We prove that an invariance principle, albeit with a non-classical scaling, holds for "not too small" sequences, the order const$\cdot n^{-1}$ (critical cooling regime) being the threshold. At and below this critical order, the scaling behavior is dramatically different from the one above it. The same order is also the critical one for the Weak Law of Large Numbers to hold. In the critical cooling regime, an interesting process emerges: it is a continuous, piecewise linear, recurrent process, for which the one-dimensional marginals are Beta-distributed. We also investigate the recurrence of the walk and its scaling limit, as well as the ergodicity and mixing of the $n$th step of the walk.

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