arXiv subjects
Publications and source records attributed to Serguei Popov.
Let $\mathcal{T}_n$ be the cover time of two-dimensional discrete torus $\mathbb{Z}^2_n=\mathbb{Z}^2/n\mathbb{Z}^2$. We prove that $\mathbb{P}[\mathcal{T}_n\leq \frac{4}πγn^2\ln^2 n]=\exp(-n^{2(1-\sqrtγ)+o(1)})$ for $γ\in (0,1)$. One of the main methods used in the proofs is the decoupling of the walker's trace into independent excursions by means of soft local times.
We consider a continuous time random walk $X$ in random environment on $\Z^+$ such that its potential can be approximated by the function $V: \R^+\to \R$ given by $V(x)=\sig W(x) -\frac{b}{1-\alf}x^{1-\alf}$ where $\sig W$ a Brownian motion with diffusion coefficient $\sig>0$ and parameters $b$, $\alf$ are such that $b>0$ and $0<\alf<1/2$. We show that $¶$-a.s.\ (where $¶$ is the averaged law) $\lim_{t\to \infty} \frac{X_t}{(C^*(\ln\ln t)^{-1}\ln t)^{\frac{1}{\alf}}}=1$ with $C^*=\frac{2\alf b}{\sig^2(1-2\alf)}$. In fact, we prove that by showing that there is a trap located around $(C^*(\ln\ln t)^{-1}\ln t)^{\frac{1}{\alf}}$ (with corrections of smaller order) where the particle typically stays up to time $t$. This is in sharp contrast to what happens in the "pure" Sinai's regime, where the location of this trap is random on the scale $\ln^2 t$.
We consider a branching random walk on $\Z$, where the particles behave differently in visited and unvisited sites. Informally, each site on the positive half-line contains initially a cookie. On the first visit of a site its cookie is removed and particles at positions with a cookie reproduce and move differently from particles on sites without cookies. Therefore, the movement and the reproduction of the particles depend on the previous behaviour of the population of particles. We study the question if the process is recurrent or transient, i.e., whether infinitely many particles visit the origin or not.
We study a one-dimensional random walk among random conductances, with unbounded jumps. Assuming the ergodicity of the collection of conductances and a few other technical conditions (uniform ellipticity and polynomial bounds on the tails of the jumps) we prove a quenched conditional invariance principle for the random walk, under the condition that it remains positive until time $n$. As a corollary of this result, we study the effect of conditioning the random walk to exceed level $n$ before returning to 0 as $n\to \infty$.
Consider a random walk among random conductances on $\mathbb{Z}^d$ with $d\geq 2$. We study the quenched limit law under the usual diffusive scaling of the random walk conditioned to have its first coordinate positive. We show that the conditional limit law is the product of a Brownian meander and a $(d-1)$-dimensional Brownian motion.
In this paper we study a substantial generalization of the model of excited random walk introduced in [Electron. Commun. Probab. 8 (2003) 86-92] by Benjamini and Wilson. We consider a discrete-time stochastic process $(X_n,n=0,1,2,...)$ taking values on ${\mathbb{Z}}^d$, $d\geq2$, described as follows: when the particle visits a site for the first time, it has a uniformly-positive drift in a given direction $\ell$; when the particle is at a site which was already visited before, it has zero drift. Assuming uniform ellipticity and that the jumps of the process are uniformly bounded, we prove that the process is ballistic in the direction $\ell$ so that $\liminf_{n\to\infty}\frac{X_n\cdot \ell}{n}>0$. A key ingredient in the proof of this result is an estimate on the probability that the process visits less than $n^{{1/2}+α}$ distinct sites by time n, where $α$ is some positive number depending on the parameters of the model. This approach completely avoids the use of tan points and coupling methods specific to the excited random walk. Furthermore, we apply this technique to prove that the excited random walk in an i.i.d. random environment satisfies a ballistic law of large numbers and a central limit theorem.
We study a one-dimensional random walk among random conductances, with unbounded jumps. Assuming the ergodicity of the collection of conductances and a few other technical conditions (uniform ellipticity and polynomial bounds on the tails of the jumps) we prove a quenched \textit{conditional} invariance principle for the random walk, under the condition that it remains positive until time $n$. As a corollary of this result, we study the effect of conditioning the random walk to exceed level $n$ before returning to 0 as $n\to \infty$. One of the main tools for proving these conditional limit laws is the \textit{uniform} quenched functional Central Limit Theorem, that states that the convergence is uniform with respect to the starting point, provided that the starting point is chosen in a certain interval around the origin.
We study a one-dimensional random walk among random conductances, with unbounded jumps. Assuming the ergodicity of the collection of conductances and a few other technical conditions (uniform ellipticity and polynomial bounds on the tails of the jumps) we prove a quenched \textit{uniform} invariance principle for the random walk. This means that the rescaled trajectory of length $n$ is (in a certain sense) close enough to the Brownian motion, uniformly with respect to the choice of the starting location in an interval of length $O(\sqrt{n})$ around the origin.
We consider a discrete-time process adapted to some filtration which lives on a (typically countable) subset of $\mathbb{R}^d$, $d\geq 2$. For this process, we assume that it has uniformly bounded jumps, is uniformly elliptic (can advance by at least some fixed amount with respect to any direction, with uniformly positive probability). Also, we assume that the projection of this process on some fixed vector is a submartingale, and that a stronger additional condition on the direction of the drift holds (this condition does not exclude that the drift could be equal to 0 or be arbitrarily small). The main result is that with very high probability the number of visits to any fixed site by time $n$ is less than $n^{1/2-δ}$ for some $δ>0$. This in its turn implies that the number of different sites visited by the process by time $n$ should be at least $n^{1/2+δ}$.
We prove a shape theorem for the internal (graph) distance on the interlacement set $\mathcal{I}^u$ of the random interlacement model on $\mathbb Z^d$, $d\ge 3$. We provide large deviation estimates for the internal distance of distant points in this set, and use these estimates to study the internal distance on the range of a simple random walk on a discrete torus.
Let $μ_1,... μ_k$ be $d$-dimensional probability measures in $\R^d$ with mean 0. At each step we choose one of the measures based on the history of the process and take a step according to that measure. We give conditions for transience of such processes and also construct examples of recurrent processes of this type. In particular, in dimension 3 we give the complete picture: every walk generated by two measures is transient and there exists a recurrent walk generated by three measures.
We consider the random conductance model, where the underlying graph is an infinite supercritical Galton--Watson tree, the conductances are independent but their distribution may depend on the degree of the incident vertices. We prove that, if the mean conductance is finite, there is a deterministic, strictly positive speed $v$ such that $\lim_{n\to\infty} \frac{|X_n|}{n}= v$ a.s.\ (here, $|\cdot|$ stands for the distance from the root). We give a formula for $v$ in terms of the laws of certain effective conductances and show that, if the conductances share the same expected value, the speed is not larger than the speed of simple random walk on Galton--Watson trees. The proof relies on finding a reversible measure for the environment observed by the particle.
We consider a random walk in a stationary ergodic environment in $\mathbb Z$, with unbounded jumps. In addition to uniform ellipticity and a bound on the tails of the possible jumps, we assume a condition of strong transience to the right which implies that there are no "traps". We prove the law of large numbers with positive speed, as well as the ergodicity of the environment seen from the particle. Then, we consider Knudsen stochastic billiard with a drift in a random tube in ${\mathbb R}^d$, $d\geq 3$, which serves as environment. The tube is infinite in the first direction, and is a stationary and ergodic process indexed by the first coordinate. A particle is moving in straight line inside the tube, and has random bounces upon hitting the boundary, according to the following modification of the cosine reflection law: the jumps in the positive direction are always accepted while the jumps in the negative direction may be rejected. Using the results for the random walk in random environment together with an appropriate coupling, we deduce the law of large numbers for the stochastic billiard with a drift.
A spider consists of several, say $N$, particles. Particles can jump independently according to a random walk if the movement does not violate some given restriction rules. If the movement violates a rule it is not carried out. We consider random walk in random environment (RWRE) on $\Z$ as underlying random walk. We suppose the environment $ω=(ω_x)_{x \in \Z}$ to be elliptic, with positive drift and nestling, so that there exists a unique positive constant $κ$ such that $\E[((1-ω_0)/ω_0)^κ]=1$. The restriction rules are kept very general; we only assume transitivity and irreducibility of the spider. The main result is that the speed of a spider is positive if $κ/N>1$ and null if $κ/N<1$. In particular, if $κ/N <1$ a spider has null speed but the speed of a (single) RWRE is positive.
We consider a stochastic billiard in a random tube which stretches to infinity in the direction of the first coordinate. This random tube is stationary and ergodic, and also it is supposed to be in some sense well behaved. The stochastic billiard can be described as follows: when strictly inside the tube, the particle moves straight with constant speed. Upon hitting the boundary, it is reflected randomly, according to the cosine law: the density of the outgoing direction is proportional to the cosine of the angle between this direction and the normal vector. We also consider the discrete-time random walk formed by the particle's positions at the moments of hitting the boundary. Under the condition of existence of the second moment of the projected jump length with respect to the stationary measure for the environment seen from the particle, we prove the quenched invariance principles for the projected trajectories of the random walk and the stochastic billiard.
We consider transport diffusion in a stochastic billiard in a random tube which is elongated in the direction of the first coordinate (the tube axis). Inside the random tube, which is stationary and ergodic, non-interacting particles move straight with constant speed. Upon hitting the tube walls, they are reflected randomly, according to the cosine law: the density of the outgoing direction is proportional to the cosine of the angle between this direction and the normal vector. Steady state transport is studied by introducing an open tube segment as follows: We cut out a large finite segment of the tube with segment boundaries perpendicular to the tube axis. Particles which leave this piece through the segment boundaries disappear from the system. Through stationary injection of particles at one boundary of the segment a steady state with non-vanishing stationary particle current is maintained. We prove (i) that in the thermodynamic limit of an infinite open piece the coarse-grained density profile inside the segment is linear, and (ii) that the transport diffusion coefficient obtained from the ratio of stationary current and effective boundary density gradient equals the diffusion coefficient of a tagged particle in an infinite tube. Thus we prove Fick's law and equality of transport diffusion and self-diffusion coefficients for quite generic rough (random) tubes.
We consider transport diffusion in a stochastic billiard in a random tube which is elongated in the direction of the first coordinate (the tube axis). Inside the random tube, which is stationary and ergodic, non-interacting particles move straight with constant speed. Upon hitting the tube walls, they are reflected randomly, according to the cosine law: the density of the outgoing direction is proportional to the cosine of the angle between this direction and the normal vector. Steady state transport is studied by introducing an open tube segment as follows: We cut out a large finite segment of the tube with segment boundaries perpendicular to the tube axis. Particles which leave this piece through the segment boundaries disappear from the system. Through stationary injection of particles at one boundary of the segment a steady state with non-vanishing stationary particle current is maintained. We prove (i) that in the thermodynamic limit of an infinite open piece the coarse-grained density profile inside the segment is linear, and (ii) that the transport diffusion coefficient obtained from the ratio of stationary current and effective boundary density gradient equals the diffusion coefficient of a tagged particle in an infinite tube. Thus we prove Fick's law and equality of transport diffusion and self-diffusion coefficients for quite generic rough (random) tubes. We also study some properties of the crossing time and compute the Milne extrapolation length in dependence on the shape of the random tube.
Spider walks are systems of interacting particles. The particles move independently as long as their movement do not violate some given rules describing the relative position of the particles; moves that violate the rules are not realized. The goal of this paper is to study qualitative properties, as recurrence, transience, ergodicity, and positive rate of escape of these Markov processes.