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Register a new userRecurrence versus Transience for Weight-Dependent Random Connection Models
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We investigate a large class of random graphs on the points of a Poisson process in $d$-dimensional space, which combine scale-free degree distributions and long-range effects. Every Poisson point carries an independent random weight and given weight and position of the points we form an edge between two points independently with a probability depending on the two weights and the distance of the points. In dimensions $din{1,2}$ we completely characterise recurrence vs transience of random walks on the infinite cluster. In $dgeq 3$ we prove transience in all cases except for a regime where we conjecture that scale-free and long-range effects play no role. Our results are particularly interesting for the special case of the age-dependent random connection model recently introduced in [P. Gracar et al., The age-dependent random connection model, Queueing Syst. {bf 93} (2019), no.~3-4, 309--331. MR4032928].
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We consider reversible random walks in random environment obtained from symmetric long--range jump rates on a random point process. We prove almost sure transience and recurrence results under suitable assumptions on the point process and the jump rate function. For recurrent models we obtain almost sure estimates on effective resistances in finite boxes. For transient models we construct explicit fluxes with finite energy on the associated electrical network.
Begin continuous time random walks from every vertex of a graph and have particles coalesce when they collide. We use a duality relation with the voter model to prove the process is site recurrent on bounded degree graphs, and for Galton-Watson trees whose offspring distribution has exponential tail. We prove bounds on the occupation probability of a site, as well as a general 0-1 law. Similar conclusions hold for a coalescing process on trees where particles do not backtrack.
We consider random walks in a random environment that is given by i.i.d. Dirichlet distributions at each vertex of Z^d or, equivalently, oriented edge reinforced random walks on Z^d. The parameters of the distribution are a 2d-uplet of positive real numbers indexed by the unit vectors of Z^d. We prove that, as soon as these weights are nonsymmetric, the random walk in this random environment is transient in a direction with positive probability. In dimension 2, this result can be strenghened to an almost sure directional transience thanks to the 0-1 law from [ZM01]. Our proof relies on the property of stability of Dirichlet environment by time reversal proved in [Sa09]. In a first part of this paper, we also give a probabilistic proof of this property as an alternative to the change of variable computation used in that article.
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 $lambda$. In the homogenous 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 $| cdot |$ is the Euclidean norm. In the inhomogenous version of the model, points of $mathcal{P}_{lambda}$ are endowed with weights that are non-negative independent random variables with distribution $P(W>w)= w^{-beta}1_{[1,infty)}(w)$, $beta>0$. Vertices located at $x,y$ with weights $W_x,W_y$ are connected with probability $1 - expleft( - frac{eta W_xW_y}{|x-y|^{alpha}} right)$, $eta, alpha > 0$, 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.
The paper deals with a random connection model, a random graph whose vertices are given by a homogeneous Poisson point process on $mathbb{R}^d$, and edges are independently drawn with probability depending on the locations of the two end points. We establish central limit theorems (CLT) for general functionals on this graph under minimal assumptions that are a combination of the weak stabilization for the-one cost and a $(2+delta)$-moment condition. As a consequence, CLTs for isomorphic subgraph counts, isomorphic component counts, the number of connected components are then derived. In addition, CLTs for Betti numbers and the size of biggest component are also proved for the first time.
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