Contact process in an evolving random environment

IF 1.3 3区 数学 Q2 STATISTICS & PROBABILITY Electronic Journal of Probability Pub Date : 2023-01-01 DOI:10.1214/23-ejp1002
Marco Seiler, Anja Sturm
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引用次数: 1

Abstract

In this paper we introduce a contact process in an evolving random environment (CPERE) on a connected and transitive graph with bounded degree, where we assume that this environment is described through an ergodic spin systems with finite range. We show that under a certain growth condition the phase transition of survival is independent of the initial configuration of the process. We study the invariant laws of the CPERE and show that under aforementioned growth condition the phase transition for survival coincides with the phase transition of non-triviality of the upper invariant law. Furthermore, we prove continuity properties for the survival probability and derive equivalent conditions for complete convergence, in an analogous way as for the classical contact process. We then focus on the special case, where the evolving random environment is described through a dynamical percolation. We show that the contact process on a dynamical percolation on the d-dimensional integers dies out almost surely at criticality and complete convergence holds for all parameter choices. In the end we derive some comparison results between a dynamical percolation and ergodic spin systems with finite range such that we get bounds on the survival probability of a contact process in an evolving random environment and we determine in this case that complete convergence holds in a certain parameter regime.
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随机环境下的接触过程
本文引入了有界度连通传递图上演化随机环境(CPERE)中的接触过程,并假设该环境可以用有限范围的遍历自旋系统来描述。我们证明了在一定的生长条件下,生存相变与过程的初始结构无关。我们研究了CPERE的不变定律,并证明了在上述生长条件下,生存相变与上不变定律的非平凡相变是一致的。进一步,我们证明了生存概率的连续性,并导出了与经典接触过程类似的完全收敛的等价条件。然后,我们将重点放在特殊情况下,其中通过动态渗透来描述进化的随机环境。我们证明了d维整数上动态渗流的接触过程在临界时几乎肯定会消失,并且对所有参数选择都保持完全收敛。最后,我们得到了有限范围的动态渗流系统和遍历自旋系统的比较结果,从而得到了在演化的随机环境中接触过程的生存概率的界限,并确定了在这种情况下,在一定的参数范围内是完全收敛的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Electronic Journal of Probability
Electronic Journal of Probability 数学-统计学与概率论
CiteScore
1.80
自引率
7.10%
发文量
119
审稿时长
4-8 weeks
期刊介绍: The Electronic Journal of Probability publishes full-size research articles in probability theory. The Electronic Communications in Probability (ECP), a sister journal of EJP, publishes short notes and research announcements in probability theory. Both ECP and EJP are official journals of the Institute of Mathematical Statistics and the Bernoulli society.
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