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Serguei Popov

Publications and source records attributed to Serguei Popov.

At least 37 records · Page 2Linked to original sources
arXiv2019

Conditioned two-dimensional simple random walk: Green's function and harmonic measure

We study the Doob's $h$-transform of the two-dimensional simple random walk with respect to its potential kernel, which can be thought of as the two-dimensional simple random walk conditioned on never hitting the origin. We derive an explicit formula for the Green's function of this random walk, and also prove a quantitative result on the speed of convergence of the (conditional) entrance measure to the harmonic measure (for the conditioned walk) on a finite set.

math.PR↗
arXiv2019

An improved decoupling inequality for random interlacements

In this paper we obtain a decoupling feature of the random interlacements process $\mathcal{I}^u \subset \mathbb{Z}^d$, at level $u$, $d\geq 3$. More precisely, we show that the trace of the random interlacements process on two disjoint finite sets, $\textsf{F}$ and its translated $\textsf{F}+x$, can be coupled with high probability of success, when $\|x\|$ is large, with the trace of a process of independent excursions, which we call the noodle soup process. As a consequence, we obtain an upper bound on the covariance between two $[0,1]$-valued functions depending on the configuration of the random interlacements on $\textsf{F}$ and $\textsf{F}+x$, respectively. This improves a previous bound obtained by Sznitman in [12].

math.PR↗
arXiv2019

Equilibria in the Tangle

We analyse the Tangle --- a DAG-valued stochastic process where new vertices get attached to the graph at Poissonian times, and the attachment's locations are chosen by means of random walks on that graph. These new vertices, also thought of as "transactions", are issued by many players (which are the nodes of the network), independently. The main application of this model is that it is used as a base for the IOTA cryptocurrency system (www.iota.org). We prove existence of "almost symmetric" Nash equilibria for the system where a part of players tries to optimize their attachment strategies. Then, we also present simulations that show that the "selfish" players will nevertheless cooperate with the network by choosing attachment strategies that are similar to the "recommended" one.

math.PR↗
arXiv2019

On the range of a two-dimensional conditioned simple random walk

We consider the two-dimensional simple random walk conditioned on never hitting the origin. This process is a Markov chain, namely it is the Doob $h$-transform of the simple random walk with respect to the potential kernel. It is known to be transient and we show that it is "almost recurrent" in the sense that each infinite set is visited infinitely often, almost surely. We prove that, for a "large" set, the proportion of its sites visited by the conditioned walk is approximately a Uniform$[0,1]$ random variable. Also, given a set $G\subset\mathbb{R}^2$ that does not "surround" the origin, we prove that a.s.\ there is an infinite number of $k$'s such that $kG\cap \mathbb{Z}^2$ is unvisited. These results suggest that the range of the conditioned walk has "fractal" behavior.

math.PR↗
arXiv2019

Two-dimensional Brownian random interlacements

We introduce the model of two-dimensional continuous random interlacements, which is constructed using the Brownian trajectories conditioned on not hitting a fixed set (usually, a disk). This model yields the local picture of Wiener sausage on the torus around a late point. As such, it can be seen as a continuous analogue of discrete two-dimensional random interlacements [Comets, Popov, Vachkovskaia, 2016]. At the same time, one can view it as (restricted) Brownian loops through infinity. We establish a number of results analogous to these of [Comets, Popov, Vachkovskaia, 2016; Comets, Popov, 2016], as well as the results specific to the continuous case.

math.PR↗
arXiv2019

On uniform closeness of local times of Markov chains and i.i.d. sequences

In this paper we consider the field of local times of a discrete-time Markov chain on a general state space, and obtain uniform (in time) upper bounds on the total variation distance between this field and the one of a sequence of $n$ i.i.d. random variables with law given by the invariant measure of that Markov chain. The proof of this result uses a refinement of the soft local time method of [11].

math.PR↗
arXiv2018

Recurrence and Transience of Frogs with Drift on $\mathbb{Z}^d$

We study the frog model on $\mathbb{Z}^d$ with drift in dimension $d \geq 2$ and establish the existence of transient and recurrent regimes depending on the transition probabilities. We focus on a model in which the particles perform nearest neighbour random walks which are balanced in all but one direction. This gives a model with two parameters. We present conditions on the parameters for recurrence and transience, revealing interesting differences between dimension $d=2$ and dimension $d \geq 3$. Our proofs make use of (refined) couplings with branching random walks for the transience, and with percolation for the recurrence.

math.PR↗
arXiv2017

Biased random walk on the interlacement set

We study a biased random walk on the interlacement set of $\mathbb{Z}^d$ for $d\geq 3$. Although the walk is always transient, we can show, in the case $d=3$, that for any value of the bias the walk has a zero limiting speed and actually moves slower than any power.

math.PR↗
arXiv2017

The vacant set of two-dimensional critical random interlacement is infinite

For the model of two-dimensional random interlacements in the critical regime (i.e., $α=1$), we prove that the vacant set is a.s.\ infinite, thus solving an open problem from arXiv:1502.03470. Also, we prove that the entrance measure of simple random walk on annular domains has certain regularity properties; this result is useful when dealing with soft local times for excursion processes.

math.PR↗
arXiv2016

One-dimensional random interlacements

We base ourselves on the construction of the two-dimensional random interlacements [12] to define the one-dimensional version of the process. For this constructions we consider simple random walks conditioned on never hitting the origin, which makes them transient. We also compare this process to the conditional random walk on the ring graph. Our results are the convergence of the vacant set on the ring graph to the vacant set of one-dimensional random interlacements, a central limit theorem for the interlacements' local time for sites far from the origin and the convergence in law of the local times of the conditional walk on the ring graph to the interlacements' local times.

math.PR↗
arXiv2016

Conditional decoupling of random interlacements

We prove a conditional decoupling inequality for the model of random interlacements in dimension $d\geq 3$: the conditional law of random interlacements on a box (or a ball) $A_1$ given the (not very "bad") configuration on a "distant" set $A_2$ does not differ a lot from the unconditional law.

math.PR↗
arXiv2016

Total flooding time and rumor propagation on graphs

We study a model of rumor propagation in discrete time where each site in the graph has initially a distinct information; we are interested in the number of "conversations" before the entire graph knows all informations. This problem can be described as a flooding time problem with multiple liquids. For the complete graph we compare the ratio between the expected propagation time for all informations and the corresponding time for a single information, obtaining the asymptotic ratio $3/2$ between them.

math.PR↗
arXiv2016

Constrained information transmission on Erdös-Rényi graphs

We model the transmission of information of a message on the Erdös-Rény random graph with parameters $(n,p)$ and limited resources. The vertices of the graph represent servers that may broadcast a message at random. Each server has a random emission capital that decreases by one at each emission. We examine two natural dynamics: in the first dynamics, an informed server performs its attempts, then checks at each of them if the corresponding edge is open or not; in the second dynamics the informed server knows a priori who are its neighbors, and it performs all its attempts on its actual neighbors in the graph. In each case, we obtain first and second order asymptotics (law of large numbers and central limit theorem), when $n\to \infty$ and $p$ is fixed, for the final proportion of informed servers.

math.PR↗
arXiv2015

Reconstruction of a multidimensional scenery with a branching random walk

In this paper we consider a d-dimensional scenery seen along a simple symmetric branching random walk, where at each time each particle gives the color record it is seeing. We show that we can a.s. reconstruct the scenery up to equivalence from the color record of all the particles. For this we assume that the scenery has at least 2d + 1 colors which are i.i.d. with uniform probability. This is an improvement in comparison to [22] where the particles needed to see at each time a window around their current position. In [11] the reconstruction is done for d = 2 with only one particle instead of a branching random walk, but millions of colors are necessary.

math.PR↗
arXiv2015

Two-dimensional random interlacements and late points for random walks

We define the model of two-dimensional random interlacements using simple random walk trajectories conditioned on never hitting the origin, and then obtain some properties of this model. Also, for random walk on a large torus conditioned on not hitting the origin up to some time proportional to the mean cover time, we show that the law of the vacant set around the origin is close to that of random interlacements at the corresponding level. Thus, this new model provides a way to understand the structure of the set of late points of the covering process from a microscopic point of view.

math.PR↗
arXiv2015

Soft local times and decoupling of random interlacements

In this paper we establish a decoupling feature of the random interlacement process I^u in Z^d, at level u, for d \geq 3. Roughly speaking, we show that observations of I^u restricted to two disjoint subsets A_1 and A_2 of Z^d are approximately independent, once we add a sprinkling to the process I^u by slightly increasing the parameter u. Our results differ from previous ones in that we allow the mutual distance between the sets A_1 and A_2 to be much smaller than their diameters. We then provide an important application of this decoupling for which such flexibility is crucial. More precisely, we prove that, above a certain critical threshold u**, the probability of having long paths that avoid I^u is exponentially small, with logarithmic corrections for d=3. To obtain the above decoupling, we first develop a general method for comparing the trace left by two Markov chains on the same state space. This method is based in what we call the soft local time of a chain. In another crucial step towards our main result, we also prove that any discrete set can be "smoothened" into a slightly enlarged discrete set, for which its equilibrium measure behaves in a regular way. Both these auxiliary results are interesting in themselves and are presented independently from the rest of the paper.

math.PR↗
arXiv2015

Russo's formula for random interlacements

In this paper we obtain a couple of explicit expressions for the derivative of the probability of an increasing event in the random interlacements model. The event is supported in a finite subset of the lattice, and the derivative is with respect to the intensity parameter of the model.

math.PR↗