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Qingsan Zhu

Publications and source records attributed to Qingsan Zhu.

6 recordsLinked to original sources
arXiv2021

Supercritical Spatial SIR Epidemics: Spreading Speed and Herd Immunity

We study supercritical spatial SIR epidemics on $\mathbb{Z}^2\times \{1,2,\ldots, N\}$, where each site in $\mathbb{Z}^2$ represents a village and $N$ stands for the village size. We establish several key asymptotic results as $N\to\infty$. In particular, we derive the probability that the epidemic will last forever if the epidemic is started by one infected individual. Moreover, conditional on that the epidemic lasts forever, we show that the epidemic spreads out linearly in all directions and derive an explicit formula for the spreading speed. Furthermore, we prove that the ultimate proportion of infection converges to a number that is constant over space and find its explicit value. An important message is that if there is no vaccination, then the ultimate proportion of population who will be infected can be \emph{much higher} than the vaccination proportion that is needed in order to prevent sustained spread of the infection.

math.PR↗
arXiv2019

Branching interlacements and tree-indexed random walks in tori

We introduce a model of branching interlacements for general critical offspring distributions. It consists of a countable collection of infinite tree-indexed random walk trajectories on $Z^d,d\geq5$. We show that this model turns out to be the local limit of the tree-indexed random walk in a discrete torus, conditioned on the size proportional to the volume of the torus. This generalizes the previous results of Angel, Ráth and the author, for the critical geometric offspring distribution. Our model also includes the model of random interlacements introduced by Sznitman as a degenerate case. To obtain the local convergence, we establish results on decomposing large random trees into small trees, local limits of random trees around a prefixed vertex, and asymptotics of the visiting probability of a set by a tree-indexed random walk with a given size in a torus. These auxiliary results are interesting in themselves. As another application, we show that when $d\geq5$ the cover time of a $d$-dimensional torus of side-length $N$ by tree-indexed random walks is concentrated at $N^d log N^d/\mathrm{BCap}(\{0\})$.

math.PR↗
arXiv2017

On the critical branching random walk I: Branching capacity and visiting probability

We extend the theory of discrete capacity to critical branching random walk. We introduce branching capacity for any finite subset of $\Z^d, d\geq5$. Analogous to the regular discrete capacity, branching capacity is closely related to the asymptotics of the probability of visiting a fixed finite set by a critical branching random walk starting from a distant point and the conditional distribution of the hitting point.

math.PR↗
arXiv2017

On the critical branching random walk III: the critical dimension

In this paper, we study the critical branching random walk in the critical dimension, $Z^4$. We provide the asymptotics of the probability of visiting a fixed finite subset and the range of the critical branching random walk conditioned on the total number of offsprings. We also prove that conditioned on visiting, the first visiting point converges in distribution.

math.PR↗