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Artem Sapozhnikov

Publications and source records attributed to Artem Sapozhnikov.

At least 19 recordsLinked to original sources
arXiv2026

On the maximum visibility in a ball through the vacant set of Poissonian obstacles

We study the maximum visibility in a ball inside the vacant set of three obstacle models in $\mathbb R^d$ with slow decay of spatial correlations and disparate obstacle geometries: Poisson Boolean models with general i.i.d. radii distributions, Poisson cylinders and Brownian interlacements. Let $M_r$ be the maximum distance between points $x$ and $y$ in the ball $B(r)$ such that $x$ is visible from $y$. We prove that $M_r$ divided by $q_r$ converges in probability to an explicit model dependent constant, where $q_r=\log r$, except for the Brownian interlacements in dimension $d=3$, where $q_r = \log r\log\log r$.

math.PR↗
arXiv2026

Multi-hop visibility through the vacant set of Poissonian obstacles

We study multi-hop visibility inside the vacant set of three obstacle models in $\mathbb R^d$ with slow decay of spatial correlations and disparate obstacle geometries: Poisson-Boolean models with general i.i.d. radii distributions, Poisson cylinders and Brownian interlacements. For any $N\geq 0$, we obtain sharp bounds on the probability $P_{\mathrm{vis}}^N(r)$ of visibility to distance $r$ via $(N+1)$ hops in terms of the (explicit) probability of direct visibility to distance $r$ in a given direction, generalizing our earlier result from arXiv:2304.10298 for $N=0$. We observe a universal behavior of $P_{\mathrm{vis}}^N(r)$ in terms of two characteristic scales. In the three models of interest, these scales are generally the same, but with some anomalous exceptions in low dimensions.

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arXiv2026

Critical curve of loop percolation on the $d$-regular tree

We consider clusters formed by a Poisson ensemble of random walk loops on the $d$-regular tree with an intensity parameter $α>0$ and a killing parameter $κ>-1$; the latter penalizes ($κ> 0$) or favors ($κ<0$) the appearance of large loops. We obtain an implicit formula for the critical curve $κ\mapsto α_c(κ)$ for the percolation phase transition; the curve is positive if and only if $κ>κ_c = \frac{2\sqrt{d-1}}{d}-1$, differentiable away from $κ_c$, and has order $\sqrt{κ-κ_c}$ as $κ\downarrowκ_c$ and order $(1+κ)^2$ as $κ\to\infty$. We show that for each $κ>-1$, an infinite cluster exists exactly when $α>α_c(κ)$. Finally, we identify the near-critical behavior of the susceptibility and the percolation probability: for $κ>κ_c$, the critical exponents take the mean-field values, while for $κ=κ_c$, the phase transition is of a higher order with the percolation probability decaying quadratically in $α-α_c$.

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arXiv2025

On the visibility window for Brownian interlacements, Poisson cylinders and Boolean models

We study visibility inside the vacant set of three models in $\mathbb R^d$ with slow decay of spatial correlations: Brownian interlacements, Poisson cylinders and Poisson-Boolean models. Let $Q_x$ be the radius of the largest ball centered at $x$ every point of which is visible from $0$ through the vacant set of one of these models. We prove that conditioned on $x$ being visible from $0$, $Q_x/δ_{\|x\|}$ converges weakly, as $x\to\infty$, to the exponential distribution with an explicit intensity, which depends on the parameters of the respective model. The scaling function $δ_r$ is the visibility window introduced in arXiv:2304.10298, a length scale of correlations in the visible set at distance $r$ from $0$.

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arXiv2025

Indistinguishability of unbounded components in the occupied and vacant sets of Boolean models on symmetric spaces

We study Boolean models on Riemannian symmetric spaces driven by homogeneous insertion- or deletion-tolerant point processes. We prove that in both the set covered by the balls (the occupied set) and its complement (the vacant set), one cannot distinguish unbounded components from each other by any isometry invariant component property. This implies the uniqueness monotonicity for the occupied and vacant sets of Poisson-Boolean models and an equivalence of non-uniqueness to the decay of connectivity for both sets. These results are continuum analogues of those by Lyons and Schramm arXiv:math/9811170. However, unlike the proof of the indistinguishability in arXiv:math/9811170, our proof does not rely on transience of unbounded components. We also prove the existence of a percolation phase transition for independent Poisson-Boolean model on unbounded connected components of both occupied and vacant sets and show transience of a random walk on the occupied set. Apart from some technical differences, we treat the occupied and the vacant sets of Boolean models within a single framework.

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arXiv2024

Visibility in Brownain interlacements, Poisson cylinders and Boolean models

We study visibility inside the vacant set of three models in $\mathbb R^d$ with slow decay of spatial correlations: Brownian interlacements, Poisson cylinders and Boolean model. For each of them, we obtain sharp asymptotic bounds on the probability of visibility to distance $r$ in some direction in terms of the probability of visibility to distance $r$ in a given direction. In dimensions $d\geq 4$, the ratio of the two probabilities has the same scaling $r^{2(d-1)}$ for all three models, but in lower dimensions the scalings are different. In particular, we improve some main results from arXiv:0905.4874 and arXiv:1709.09052.

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arXiv2024

On questions of uniqueness for the vacant set of Wiener sausages and Brownian interlacements

We consider connectivity properties of the vacant set of (random) ensembles of Wiener sausages in $\mathbb R^d$ in the transient dimensions $d \geq 3$. We prove that the vacant set of Brownian interlacements contains at most one infinite connected component almost surely. For finite ensembles of Wiener sausages, we provide sharp polynomial bounds on the probability that their vacant set contains at least $2$ connected components in microscopic balls. The main proof ingredient is a sharp polynomial bound on the probability that several Brownian motions visit jointly all hemiballs of the unit ball while avoiding a slightly smaller ball.

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arXiv2024

Uniqueness of the infinite connected component for the vacant set of random interlacements on amenable transient graphs

We prove the uniqueness of the infinite connected component for the vacant set of random interlacements on general vertex-transitive amenable transient graphs. Our approach is based on connectedness of random interlacements and differs from the one used by Teixera arXiv:0805.4106 to prove the uniqueness of the infinite connected component for the vacant set of random interlacements on $\mathbb Z^d$.

math.PR↗
arXiv2019

Decoupling inequalities and supercritical percolation for the vacant set of random walk loop soup

It has been recently understood (arXiv:1212.2885, arXiv:1310.4764, arXiv:1410.0605) that for a general class of percolation models on $\mathbb{Z}^d$ satisfying suitable decoupling inequalities, which includes i.a.\ Bernoulli percolation, random interlacements and level sets of the Gaussian free field, large scale geometry of the unique infinite cluster in strongly percolative regime is qualitatively the same; in particular, the random walk on the infinite cluster satisfies the quenched invariance principle, Gaussian heat-kernel bounds and local CLT. In this paper we consider the random walk loop soup on $\mathbb{Z}^d$ in dimensions $d\geq 3$. An interesting aspect of this model is that despite its similarity and connections to random interlacements and the Gaussian free field, it does not fall into the above mentioned general class of percolation models, since the required decoupling inequalities are not valid. We identify weaker (and more natural) decoupling inequalities and prove that (a) they do hold for the random walk loop soup and (b) all the results about the large scale geometry of the infinite percolation cluster proved for the above mentioned class of models hold also for models that satisfy the weaker decoupling inequalities. Particularly, all these results are new for the vacant set of the random walk loop soup. (The range of the random walk loop soup has been addressed by Chang arXiv:1504.07906 by a model specific approximation method, which does not apply to the vacant set.) Finally, we prove that the strongly supercritical regime for the vacant set of the random walk loop soup is non-trivial. It is expected, but open at the moment, that the strongly supercritical regime coincides with the whole supercritical regime.

math.PR↗
arXiv2016

Kesten's incipient infinite cluster and quasi-multiplicativity of crossing probabilities

In this paper we consider Bernoulli percolation on an infinite connected bounded degrees graph $G$. Assuming the uniqueness of the infinite open cluster and a quasi-multiplicativity of crossing probabilities, we prove the existence of Kesten's incipient infinite cluster. We show that our assumptions are satisfied if $G$ is a slab $\mathbb Z^2\times\{0,\ldots,k\}^{d-2}$ ($d\geq 2$, $k\geq 0$). We also argue that the quasi-multiplicativity assumption is fulfilled for $G=\mathbb Z^d$ if and only if $d<6$.

math.PR↗
arXiv2015

On Brownian motion, simple paths, and loops

We provide a decomposition of the trace of the Brownian motion into a simple path and an independent Brownian soup of loops that intersect the simple path. More precisely, we prove that any subsequential scaling limit of the loop erased random walk is a simple path (a new result in three dimensions), which can be taken as the simple path of the decomposition. In three dimensions, we also prove that the Hausdorff dimension of any such subsequential scaling limit lies in $(1,\frac53]$. We conjecture that our decomposition characterizes uniquely the law of the simple path. If so, our results would give a new strategy to the existence of the scaling limit of the loop erased random walk and its rotational invariance.

math.PR↗
arXiv2015

Mixing time for the random walk on the range of the random walk on tori

Consider the subgraph of the discrete $d$-dimensional torus of size length $N$, $d\ge3$, induced by the range of the simple random walk on the torus run until the time $uN^d$. We prove that for all $d\ge 3$ and $u>0$, the mixing time for the random walk on this subgraph is of order $N^2$ with probability at least $1 - Ce^{-(\log N)^2}$.

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arXiv2015

Random walks on infinite percolation clusters in models with long-range correlations

For a general class of percolation models with long-range correlations on $\mathbb Z^d$, $d\geq 2$, introduced in arXiv:1212.2885, we establish regularity conditions of Barlow arXiv:math/0302004 that mesoscopic subballs of all large enough balls in the unique infinite percolation cluster have regular volume growth and satisfy a weak Poincaré inequality. As immediate corollaries, we deduce quenched heat kernel bounds, parabolic Harnack inequality, and finiteness of the dimension of harmonic functions with at most polynomial growth. Heat kernel bounds and the quenched invariance principle of arXiv:1310.4764 allow to extend various other known results about Bernoulli percolation by mimicking their proofs, for instance, the local central limit theorem of arXiv:0810.2467 or the result of arXiv:1111.4853 that the dimension of at most linear harmonic functions on the infinite cluster is $d+1$. In terms of specific models, all these results are new for random interlacements at every level in any dimension $d\geq 3$, as well as for the vacant set of random interlacements arXiv:0704.2560, arXiv:0808.3344 and the level sets of the Gaussian free field arXiv:1202.5172 in the regime of the so-called local uniqueness (which is believed to coincide with the whole supercritical regime for these models).

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arXiv2015

Quenched invariance principle for simple random walk on clusters in correlated percolation models

We prove quenched invariance principle for simple random walk on the unique infinite percolation cluster for a general class of percolation models on Z^d, d>=2, with long-range correlations introduced in arXiv:1212.2885, solving one of the open problems from there. This gives new results for random interlacements in dimension d>=3 at every level, as well as for the vacant set of random interlacements and the level sets of the Gaussian free field in the regime of the so-called local uniqueness (which is believed to coincide with the whole supercritical regime). An essential ingredient of our proof is a new isoperimetric inequality for correlated percolation models.

math.PR↗
arXiv2014

On chemical distances and shape theorems in percolation models with long-range correlations

In this paper we provide general conditions on a one parameter family of random infinite subsets of Z^d to contain a unique infinite connected component for which the chemical distances are comparable to the Euclidean distances, focusing primarily on models with long-range correlations. Our results are in the spirit of those by Antal and Pisztora proved for Bernoulli percolation. We also prove a shape theorem for balls in the chemical distance under such conditions. Our general statements give novel results about the structure of the infinite connected component of the vacant set of random interlacements and the level sets of the Gaussian free field. We also obtain alternative proofs to the main results in arXiv:1111.3979. Finally, as a corollary, we obtain new results about the (chemical) diameter of the largest connected component in the complement of the trace of the random walk on the torus.

math.PR↗
arXiv2013

Local percolative properties of the vacant set of random interlacements with small intensity

Random interlacements at level u is a one parameter family of connected random subsets of Z^d, d>=3 introduced in arXiv:0704.2560. Its complement, the vacant set at level u, exhibits a non-trivial percolation phase transition in u, as shown in arXiv:0704.2560 and arXiv:0808.3344, and the infinite connected component, when it exists, is almost surely unique, see arXiv:0805.4106. In this paper we study local percolative properties of the vacant set of random interlacements at level u for all dimensions d>=3 and small intensity parameter u>0. We give a stretched exponential bound on the probability that a large (hyper)cube contains two distinct macroscopic components of the vacant set at level u. Our results imply that finite connected components of the vacant set at level u are unlikely to be large. These results were proved in arXiv:1002.4995 for d>=5. Our approach is different from that of arXiv:1002.4995 and works for all d>=3. One of the main ingredients in the proof is a certain conditional independence property of the random interlacements, which is interesting in its own right.

math.PR↗
arXiv2013

Cycle structure of percolation on high-dimensional tori

In the past years, many properties of the largest connected components of critical percolation on the high-dimensional torus, such as their sizes and diameter, have been established. The order of magnitude of these quantities equals the one for percolation on the complete graph or Erdos-Renyi random graph, raising the question whether the scaling limits of the largest connected components, as identified by Aldous (1997), are also equal. In this paper, we investigate the cycle structure of the largest critical components for high-dimensional percolation on the torus (Z/rZ)^d. While percolation clusters naturally have many short cycles, we show that the long cycles, i.e., cycles that pass through the boundary of the cube of width r/4 centered around each of their vertices, have length of order r^{d/3}, as on the critical Erdos-Renyi random graph. On the Erdos-Renyi random graph, cycles play an essential role in the scaling limit of the large critical clusters, as identified by Addario-Berry, Broutin and Goldschmidt (arXiv:0908.3629). Our proofs crucially rely on various new estimates of probabilities of the existence of open paths in critical Bernoulli percolation on Z^d with constraints on their lengths. We believe these estimates are interesting in their own right.

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