{"title":"增强随机连接模型中的相变和临界渗流","authors":"Srikanth K. Iyer, Sanjoy Kr. Jhawar","doi":"10.1007/s11040-021-09409-y","DOIUrl":null,"url":null,"abstract":"<div><p>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 <span>\\(\\mathcal {P}_{\\lambda }\\)</span> in <span>\\(\\mathbb {R}^{2}\\)</span> of intensity <i>λ</i>. In the homogeneous RCM, the vertices at <i>x</i>,<i>y</i> are connected with probability <i>g</i>(|<i>x</i> − <i>y</i>|), independent of everything else, where <span>\\(g:[0,\\infty ) \\to [0,1]\\)</span> and |⋅| is the Euclidean norm. In the inhomogeneous version of the model, points of <span>\\(\\mathcal {P}_{\\lambda }\\)</span> are endowed with weights that are non-negative independent random variables with distribution <span>\\(P(W>w)= w^{-\\beta }1_{[1,\\infty )}(w)\\)</span>, <i>β</i> > 0. Vertices located at <i>x</i>,<i>y</i> with weights <i>W</i><sub><i>x</i></sub>,<i>W</i><sub><i>y</i></sub> are connected with probability <span>\\(1 - \\exp \\left (- \\frac {\\eta W_{x}W_{y}}{|x-y|^{\\alpha }} \\right )\\)</span>, <i>η</i>,<i>α</i> > 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 <span>\\(\\mathcal {P}_{\\lambda }\\)</span>. 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 <span>\\(\\mathcal {P}_{\\lambda }\\)</span>. 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.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2022-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-021-09409-y.pdf","citationCount":"0","resultStr":"{\"title\":\"Phase Transitions and Percolation at Criticality in Enhanced Random Connection Models\",\"authors\":\"Srikanth K. Iyer, Sanjoy Kr. Jhawar\",\"doi\":\"10.1007/s11040-021-09409-y\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>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 <span>\\\\(\\\\mathcal {P}_{\\\\lambda }\\\\)</span> in <span>\\\\(\\\\mathbb {R}^{2}\\\\)</span> of intensity <i>λ</i>. In the homogeneous RCM, the vertices at <i>x</i>,<i>y</i> are connected with probability <i>g</i>(|<i>x</i> − <i>y</i>|), independent of everything else, where <span>\\\\(g:[0,\\\\infty ) \\\\to [0,1]\\\\)</span> and |⋅| is the Euclidean norm. In the inhomogeneous version of the model, points of <span>\\\\(\\\\mathcal {P}_{\\\\lambda }\\\\)</span> are endowed with weights that are non-negative independent random variables with distribution <span>\\\\(P(W>w)= w^{-\\\\beta }1_{[1,\\\\infty )}(w)\\\\)</span>, <i>β</i> > 0. Vertices located at <i>x</i>,<i>y</i> with weights <i>W</i><sub><i>x</i></sub>,<i>W</i><sub><i>y</i></sub> are connected with probability <span>\\\\(1 - \\\\exp \\\\left (- \\\\frac {\\\\eta W_{x}W_{y}}{|x-y|^{\\\\alpha }} \\\\right )\\\\)</span>, <i>η</i>,<i>α</i> > 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 <span>\\\\(\\\\mathcal {P}_{\\\\lambda }\\\\)</span>. 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 <span>\\\\(\\\\mathcal {P}_{\\\\lambda }\\\\)</span>. 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.</p></div>\",\"PeriodicalId\":694,\"journal\":{\"name\":\"Mathematical Physics, Analysis and Geometry\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2022-01-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://link.springer.com/content/pdf/10.1007/s11040-021-09409-y.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematical Physics, Analysis and Geometry\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s11040-021-09409-y\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Physics, Analysis and Geometry","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s11040-021-09409-y","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Phase Transitions and Percolation at Criticality in Enhanced Random Connection Models
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(|x − y|), 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.
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.