Xiaofeng Xue
In this paper we are concerned with threshold-one contact processes on lattices. We show that the probability that the origin is infected converges to 0 at an exponential rate I in the subcritical case. Furthermore, we give a limit theorem for I as the degree of the lattice grows to infinity. Our results also hold for classic contact processes on lattices.
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The subcritical contact process seen from the rightmost infected site has no invariant measures. We prove that nevertheless it converges in distribution to a quasi-stationary measure supported on finite configurations.
In this paper we describe the subcritical contact process on $\Z^d$ for large times, starting with all sites infected. The configuration is described in terms of the macroscopic locations of infected regions in space and the relative positions of infected sites in each such region.
We show that the quasi-stationary distribution of the subcritical contact process on $\mathbb{Z}^d$ is unique. This is in contrast with other processes which also do not come down from infinity, like stable queues and Galton-Watson, and it seems to be the first such example.
We consider the subcritical contact branching random walk on Zd in continuous time with the arbitrary number of offspring and with immigration. We prove the existence of the steady state (statistical equilibrium).
We study a non-equilibrium dynamical model: a marked continuous contact model in $d$-dimensional space, $d \ge 1$. In contrast with the continuous contact model in a critical regime, see \cite{KKP}, \cite{KPZ}, the model under consideration is in the subcritical regime and it contains an additional spontaneous spatially homogeneous birth from an external source. We prove that this system has an invariant measure. We prove also that the process starting from any initial distribution converges to this invariant measure.
The Contact Process Can Survive on a Slightly Subcritical Dynamical Percolation Cluster
该研究证明在略微亚临界动态渗透簇上，接触过程可以生存，扩展了一维情况下的结果。
This paper gives a new, simple proof of the known fact that for contact processes on general lattices, in the subcritical regime the expected number of infected sites decays exponentially fast as time tends to infinity. The proof also yields an explicit bound on the survival probability below the critical recovery rate, which shows that the critical exponent associated with this function is bounded from below by its mean-field value. The main idea of the proof is that if the expected number of infected sites decays slower than exponentially, then this implies the existence of a harmonic functi…
Subcritical contact processes seen from a typical infected site
一个接触过程从一个感染点开始，按照格子上的计数测度分布，其规律的长期行为是什么？这个问题是有关的配置，从一个典型的感染网站，并产生了所谓的特征测度的定义，这可能是无限的措施，对一组非空的配置下保存的动态乘法常数。本文研究了亚临界区一般可数群上接触过程的特征测度。我们证明，在这个政权，该过程有一个独特的空间齐次本征测度。作为应用，我们证明了指数增长性是连续可微的，且是恢复率的严格递减函数，并给出了接触过程及其对偶的特征测度的导数公式.
The contact process on dynamic regular graphs: monotonicity and subcritical phase
本文研究动态随机 $d$-正则图上的接触过程及 herds 过程，证明临界感染率随边动态参数严格递减、亚临界灭绝时间呈指数尾，并推出全感染初始态下灭绝时间对顶点数为对数阶。
In this paper we are concerned with the contact process with random recovery rates and edge weights on complete graph with $n$ vertices. We show that the model has a critical value which is inversely proportional to the product of the mean of the edge weight and the mean of the inverse of the recovery rate. In the subcritical case, the process dies out before a moment with order $O(\log n)$ with high probability as $n\rightarrow+\infty$. In the supercritical case, the process survives at a moment with order $\exp\{O(n)\}$ with high probability as $n\rightarrow+\infty$. Our proof for the subcri…
The survival of large dimensional threshold contact processes
研究了带有感染参数$λ$的$\mathbb{Z}^d$上的阈值$θ$接触过程.我们发现，临界点$λ_{\mathrm{c}}$，定义为阈值的生存开始从每个网站占领，消失为$d\to\infty$。这意味着当$d$较大时，$\mathbb{Z}^d$上的阈值$θ$选举模型具有非退化极值不变测度。
Targeted Immunisation Thresholds for the Contact Process on Power-Law Trees
针对幂律树/无标度网络中均匀免疫无法阻断低感染率流行病传播的问题，研究靶向免疫最大度顶点的效果，发现最大允许度高于阈值时免疫效果微弱，低于同量级阈值时流行病存活概率骤降甚至清零。
We study the discrete-time threshold-$θ\geq 2$ contact process on random graphs of general degrees. For random graphs with a given degree distribution $μ$, we show that if $μ$ is lower bounded by $θ+2$ and has finite $k$th moments for all $k>0$, then the discrete-time threshold-$θ$ contact process on the random graph exhibits a discontinuous phase transition in the emergence of metastability, thus answering a question of Chatterjee and Durrett \cite{cd13}. To be specific, we establish that (i) for any large enough infection probability $p>p_1$, the process started from the all-infected state w…
Persistence of Activity in Threshold Contact Processes, an "Annealed Approximation" of Random Boolean Networks
我们考虑一个基因调控网络模型，它是Kauffmann（1969）随机布尔网络的一个改进。有三个参数：$n =$表示节点数，$r =$表示每个节点的输入数，$p =$表示每个节点处布尔函数中1的期望分数。遵循物理学文献中的标准实践，我们在$n$节点上的随机图上使用阈值接触过程，其中每个节点都具有$r$度，以近似其动态。我们证明，如果$r\ge 3$和$r \cdot 2p(1-p)>1$，那么阈值接触过程持续很长一段时间，这对应于布尔网络的混沌行为。不幸的是，我们只能证明当$r\cdot 2p(1-p)> 1$时，$b(p)>0$的持续时间是$\ge \exp(cn^{b(p)})$，当$(r-1)\cdot 2p(1-p)>1$时，$b(p)=1$的持续时间是$b(p)=1$。
Branching Processes in Random Environments with Thresholds
本文研究了具有阈值的随机环境分支过程，通过识别再生结构和极限分布，探讨了超临界和亚临界状态的持续时间和停留比例。
Asymptotic Behavior of Critical Infection Rates for Threshold-one Contact Processes on Lattices and Regular Trees
本文研究格和正则树上的阈值1接触过程。得到了临界感染率随图的度增长到无穷大的渐近性态。定义λ c为使过程在平衡点灭绝的传染率的上确界，证明了nλ c^{T^n} \rightarrow 1和2dλ c^{Z^d} \rightarrow 1为n，d \rightarrow + \infty .我们的结果是对\cite{Dur1991}中λ_c^{Z^d} \leq \frac{2.18}{d}的结论的发展。为了证明我们的主要结果，得到了一个关于格上简单随机游动返回零的概率的重要引理。详细地，引理是\lim_{d\rightarrow+\infty} 2dP \big（\exists n \geq 1，S_n^{（d）}=0 \big）=1，其中S_n^{（d）}是Z^d上的简单随机游动，其中S_0^{（d）}=0。
We study the stationary distribution of the (spread-out) $d$-dimensional contact process from the point of view of site percolation. In this process, vertices of $\mathbb{Z}^d$ can be healthy (state 0) or infected (state 1). With rate one infected sites recover, and with rate $λ$ they transmit the infection to some other vertex chosen uniformly within a ball of radius $R$. The classical phase transition result for this process states that there is a critical value $λ_c(R)$ such that the process has a non-trivial stationary distribution if and only if $λ> λ_c(R)$. In configurations sampled from…
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.
In this work we consider the two-dimensional percolation model arising from the majority dynamics process at a given time $t\in\mathbb{R}_+$. We show the emergence of a sharp threshold phenomenon for the box crossing event at the critical probability parameter $p_c(t)$ with polynomial size window. We then use this result in order to obtain stretched-exponential bounds on the one-arm event probability in the subcritical phase. Our results are based on differential inequalities derived from the OSSS inequality, inspired by the recent developments by Ahlberg, Broman, Griffiths, and Morris and by …
The stacked contact process is a three-state spin system that describes the co-evolution of a population of hosts together with their symbionts. In a nutshell, the hosts evolve according to a contact process while the symbionts evolve according to a contact process on the dynamic subset of the lattice occupied by the host population, indicating that the symbiont can only live within a host. This paper is concerned with a generalization of this system in which the symbionts may affect the fitness of the hosts by either decreasing (pathogen) or increasing (mutualist) their birth rate. Standard c…
Phase transitions for contact processes on one-dimensional networks
研究一维网络上接触过程的相变，给出亚临界相存在的充分条件，证明在局部有限和长程渗流参数大于2时相变非平凡，并推广到高维情形。
Recent progress in the study of the contact process [2] has verified that the extinction-survival threshold $λ_1$ on a Galton-Watson tree is strictly positive if and only if the offspring distribution $ξ$ has an exponential tail. In this paper, we derive the first-order asymptotics of $λ_1$ for the contact process on Galton-Watson trees and its corresponding analog for random graphs. In particular, if $ξ$ is appropriately concentrated around its mean, we demonstrate that $λ_1(ξ) \sim 1/\mathbb{E} ξ$ as $\mathbb{E}ξ\rightarrow \infty$, which matches with the known asymptotics on the $d$-regular…
Threshold $theta geq 2$ contact processes on homogeneous trees
我们研究了$theta geq 2$接触过程的均匀树$T_b$度$kappa = b + 1$，感染参数$lambda geq 0$和开始从一个产品的测量密度$p$。相应的平均场模型在临界点$lambda_c^{MF}(kappa,theta)$处显示不连续的转变，并且对于$lambda geq lambda_c^{MF}(kappa,theta)$，它存活当且仅当$p geq p_c^{MF}(kappa,theta,lambda)$，其中该临界密度满足$0 < p_c^{MF}(kappa,theta,lambda) < 1$，$lim_{lambda to infty} p_c^{MF}(kappa,theta,lambda) = 0$。对于大的$b$，我们表明，$T_b$的过程具有定性相似的行为时，$lambda$是小的，包括在和接近的临界点$lambda_c(T_b,theta)$的行为。相比之下，对于大的$p_c(T_b,theta,infty) := lim_{lambda to infty} p_c(T_b,theta,lambda) > 0$，$T_b$上的过程的行为在性质上不同于平均场模型，因为临界密度具有$p_c(T_b,theta,infty) := lim_{lambda to infty} p_c(T_b,theta,lambda) > 0$。我们还表明…
Phase transitions for contact processes on sparse random graphs via metastability and local limits
通过亚稳态密度和局部收敛技术，研究稀疏有限图上接触过程的快慢灭绝渐近区域，给出临界阈值与局部极限中存活/灭绝阈值一致的条件，并证明稀疏图上快灭绝可能发生在拉伸指数时间尺度。
Motion helps the contact process: survival below the percolation threshold on Poisson Brownian motions
该研究针对泊松布朗运动上的SIS接触过程，证明当恢复速率足够小时，在低于静态布尔模型临界强度的条件下，感染可存活并以正速度传播。
The evolution of subcritical Achlioptas processes
在Achlioptas过程中，从一个空图开始，在每一步中随机均匀地选择两个潜在的边，并使用某种规则选择其中一个并添加到演化图中。虽然这种“本地”修改的Erd{ \H o}s-雷尼随机图过程的演变在过去十年中得到了相当大的关注，到目前为止，只有相当简单的规则很好地理解。事实上，主要的重点是“有界大小”的规则，其中所有组件的大小大于一些常数$B$被视为相同的方式，和更复杂的规则很少严格的结果是已知的。在本文中，我们研究Achlioptas过程（无界）的大小规则，如总和和产品的规则。使用一个变体的邻域探索过程和分支过程参数，我们表明，某些关键的统计数据是紧密集中的，至少直到易感性（预期大小的组件包含一个随机选择的顶点）发散。我们的收敛结果很可能是最好的可能为某些规则：在后来的演变中，小组件的顶点数可能不会集中。此外，我们认为，对于一个大类的规则的敏感性“爆炸”的临界时间与渗流阈值相一致。
A first order phase transition in the threshold-$θ\ge 2$ contact process on random $r$-regular graphs and $r$-trees
本文考虑n阶随机r-正则图上的离散时间阈值$θ\ge 2$接触过程。在这个过程中，在时间t至少有θ个邻居被占用的顶点将在时间t+1以概率p被占用，否则为空。我们表明，如果$θ\ge 2$和$r \ge θ+2$，$ε_1$是小的，p至少是$p_1(ε_1)$，然后从所有的顶点占用的分数的顶点停留在$1-2ε_1$以上的时间$\exp(γ_1(r)n)$与概率至少$1 - \exp(-γ_1(r)n)$。在另一个方向上，我们表明，$p_2 < 1$有一个$ε_2(p_2)>0$，使如果$p \le p_2$和初始配置中的占用顶点的数量是最多$ε_2(p_2)n$，然后以很高的概率，所有的顶点都是空的时间$C_2(p_2) \log(n)$。这两个结论意味着在随机r-正则图上不可能有一个占据顶点密度在0和$ε_2(p_1)$之间的准平稳分布，并允许我们得出r-树上的过程具有一阶相变的结论。
We derive macroscopic equations for a generalized contact process that is inspired by a neuronal integrate and fire model on the lattice $\mathbb{Z}^d$. The states at each lattice site can take values in $0,\ldots,k$. These can be interpreted as neuronal membrane potential, with the state $k$ corresponding to a firing threshold. In the terminology of the contact processes, which we shall use in this paper, the state $k$ corresponds to the individual being infectious (all other states are noninfectious). In order to reach the firing threshold, or to become infectious, the site must progress seq…
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