Lipschitz-Type Estimate for the Frog Model with Bernoulli Initial Configuration

IF 0.9 3区数学 Q3 MATHEMATICS, APPLIED

Mathematical Physics, Analysis and Geometry Pub Date : 2025-01-02 DOI:10.1007/s11040-024-09497-6

Van Hao Can, Naoki Kubota, Shuta Nakajima

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引用次数: 0

Abstract

We consider the frog model with Bernoulli initial configuration, which is an interacting particle system on the multidimensional lattice consisting of two states of particles: active and sleeping. Active particles perform independent simple random walks. On the other hand, although sleeping particles do not move at first, they become active and can move around when touched by active particles. Initially, only the origin has one active particle, and the other sites have sleeping particles according to a Bernoulli distribution. Then, starting from the original active particle, active ones are gradually generated and propagate across the lattice, with time. It is of interest to know how the propagation of active particles behaves as the parameter of the Bernoulli distribution varies. In this paper, we treat the so-called time constant describing the speed of propagation, and prove that the absolute difference between the time constants for parameters \(p,q \in (0,1]\) is bounded from above and below by multiples of \(|p-q|\).

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具有Bernoulli初始构型的Frog模型的lipschitz型估计

我们考虑具有伯努利初始构型的青蛙模型，它是一个多维晶格上的相互作用粒子系统，由两种状态的粒子组成：活动状态和睡眠状态。活动粒子进行独立的简单随机游动。另一方面，虽然睡眠粒子一开始不动，但它们变得活跃起来，当被活跃粒子触摸时，它们可以四处移动。最初，根据伯努利分布，只有原点有一个活动粒子，其他位置有睡眠粒子。然后，从原始的活跃粒子开始，随着时间的推移，逐渐产生活跃粒子并在晶格中传播。当伯努利分布的参数变化时，活性粒子的传播是如何变化的，这是很有意义的。本文讨论了描述传播速度的所谓时间常数，并证明了参数\(p,q \in (0,1]\)的时间常数之间的绝对差以\(|p-q|\)的倍数从上到下有界。

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

Mathematical Physics, Analysis and Geometry 数学-物理：数学物理

CiteScore

2.10

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： MPAG is a peer-reviewed journal organized in sections. Each section is editorially independent and provides a high forum for research articles in the respective areas. The entire editorial board commits itself to combine the requirements of an accurate and fast refereeing process. The section on Probability and Statistical Physics focuses on probabilistic models and spatial stochastic processes arising in statistical physics. Examples include: interacting particle systems, non-equilibrium statistical mechanics, integrable probability, random graphs and percolation, critical phenomena and conformal theories. Applications of probability theory and statistical physics to other areas of mathematics, such as analysis (stochastic pde''s), random geometry, combinatorial aspects are also addressed. The section on Quantum Theory publishes research papers on developments in geometry, probability and analysis that are relevant to quantum theory. Topics that are covered in this section include: classical and algebraic quantum field theories, deformation and geometric quantisation, index theory, Lie algebras and Hopf algebras, non-commutative geometry, spectral theory for quantum systems, disordered quantum systems (Anderson localization, quantum diffusion), many-body quantum physics with applications to condensed matter theory, partial differential equations emerging from quantum theory, quantum lattice systems, topological phases of matter, equilibrium and non-equilibrium quantum statistical mechanics, multiscale analysis, rigorous renormalisation group.