arXiv subjects

Yingxin Mu

Publications and source records attributed to Yingxin Mu.

9 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.

math.PR↗

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 $0$, $Q_x/\delta_{\|x\|}$ 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 $\delta_r$ is the visibility window introduced in arXiv:2304.10298, a length scale of correlations in the visible set at distance $r$ from $0$.

math.PR↗

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.

math.PR↗

arXiv2023

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.

math.PR↗

arXiv2023

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.

math.PR↗

arXiv2023

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

Scaling limit of DLA on a long line segment

In this paper, we prove that the bulk of DLA starting from a long line segment on the $x$-axis has a scaling limit to the stationary DLA process (SDLA). The main phenomenological difficulty is the multi-scale, non-monotone interaction of the DLA arms. We overcome this via a coupling scheme between the two processes and an intermediate DLA process with absorbing mesoscopic boundary segments.

math.PR↗

arXiv2018

On some threshold-one attractive interacting particle systems on homogeneous trees

In this paper, we consider the threshold-one contact process and the threshold-one voter model w/o spontaneous death on homogeneous trees $\mathbb{T}_d$, $d\ge 2$. Mainly inspired by the corresponding arguments for ordinary contact processes, we prove that the complete convergence theorem holds for these three systems under strong survival. When the systems survives weakly, complete convergence may also hold under certain transition and/or initial conditions.

math.PR↗