Phase Transitions and Percolation at Criticality in Enhanced Random Connection Models

IF 0.9 3区 数学 Q3 MATHEMATICS, APPLIED
Srikanth K. Iyer, Sanjoy Kr. Jhawar
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引用次数: 0

Abstract

We study phase transition and percolation at criticality for three random graph models on the plane, viz., the homogeneous and inhomogeneous enhanced random connection models (RCM) and the Poisson stick model. These models are built on a homogeneous Poisson point process \(\mathcal {P}_{\lambda }\) in \(\mathbb {R}^{2}\) of intensity λ. In the homogeneous RCM, the vertices at x,y are connected with probability g(|xy|), independent of everything else, where \(g:[0,\infty ) \to [0,1]\) and |⋅| is the Euclidean norm. In the inhomogeneous version of the model, points of \(\mathcal {P}_{\lambda }\) are endowed with weights that are non-negative independent random variables with distribution \(P(W>w)= w^{-\beta }1_{[1,\infty )}(w)\), β > 0. Vertices located at x,y with weights Wx,Wy are connected with probability \(1 - \exp \left (- \frac {\eta W_{x}W_{y}}{|x-y|^{\alpha }} \right )\), η,α > 0, independent of all else. The graphs are enhanced by considering the edges of the graph as straight line segments starting and ending at points of \(\mathcal {P}_{\lambda }\). A path in the graph is a continuous curve that is a subset of the union of all these line segments. The Poisson stick model consists of line segments of independent random lengths and orientation with the mid point of each segment located at a distinct point of \(\mathcal {P}_{\lambda }\). Intersecting lines form a path in the graph. A graph is said to percolate if there is an infinite connected component or path. We derive conditions for the existence of a phase transition and show that there is no percolation at criticality.

增强随机连接模型中的相变和临界渗流
本文研究了平面上三种随机图模型,即齐次和非齐次增强随机连接模型(RCM)和泊松棒模型的相变和临界渗流。这些模型建立在强度λ的\(\mathbb {R}^{2}\)中的齐次泊松点过程\(\mathcal {P}_{\lambda }\)上。在齐次RCM中,x,y处的顶点以概率g(|x−y|)连接,独立于其他一切,其中\(g:[0,\infty ) \to [0,1]\)和|⋅|是欧几里得范数。在模型的非齐次版本中,\(\mathcal {P}_{\lambda }\)的点被赋予权值为分布为\(P(W>w)= w^{-\beta }1_{[1,\infty )}(w)\), β &gt; 0的非负独立随机变量。位于x,y的权重为Wx,Wy的顶点以概率\(1 - \exp \left (- \frac {\eta W_{x}W_{y}}{|x-y|^{\alpha }} \right )\), η,α &gt; 0连接,独立于其他所有点。通过考虑图的边缘作为直线段开始和结束于\(\mathcal {P}_{\lambda }\)点来增强图。图中的路径是一条连续曲线,它是所有线段并集的子集。泊松棒模型由独立的随机长度和方向的线段组成,每个线段的中点位于不同的\(\mathcal {P}_{\lambda }\)点。相交的线在图中形成一条路径。如果一个图有无限个连通的分量或路径,我们就说它是渗透的。我们推导了相变存在的条件,并证明了临界时不存在渗流。
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来源期刊
Mathematical Physics, Analysis and Geometry
Mathematical Physics, Analysis and Geometry 数学-物理:数学物理
CiteScore
2.10
自引率
0.00%
发文量
26
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.
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