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The Doeblin Graph of a countable state space Markov chain describes the joint pathwise evolutions of the Markov dynamics starting from all possible initial conditions, with two paths coalescing when they reach the same point of the state space at the same time. Its Bridge Doeblin subgraph only contains the paths starting from a tagged point of the state space at all possible times. In the irreducible, aperiodic, and positive recurrent case, the following results are known: the Bridge Doeblin Graph is an infinite tree that is unimodularizable. Moreover, it contains a single bi-infinite path, which allows one to build a perfect sample of the stationary state of the Markov chain. The present paper is focused on the null recurrent case. It is shown that when assuming irreducibility and aperiodicity again, the Bridge Doeblin Graph is either an infinite tree or a forest made of a countable collection of infinite trees. In the first case, the infinite tree in question has a single end, is not unimodularizable in general, but is always locally unimodular. These key properties are used to study the stationary regime of several measure-valued random dynamics on this Bridge Doeblin Tree. The most important ones are the taboo random dynamics, which admits as steady state a random measure with mean measure equal to the invariant measure of the Markov chain, and the potential random dynamics which is a random extension of the classical potential measure, with a mean measure equal to infinity at every point of the state space., Comment: 31 Pages, 8 figures

This paper is centered on the random graph generated by a Doeblin-type coupling of discrete time processes on a countable state space whereby when two paths meet, they merge. This random graph is studied through a novel subgraph, called a bridge graph, generated by paths started in a fixed state at any time. The bridge graph is made into a unimodular network by marking it and selecting a root in a specified fashion. The unimodularity of this network is leveraged to discern global properties of the larger Doeblin graph. Bi-recurrence, i.e., recurrence both forwards and backwards in time, is introduced and shown to be a key property in uniquely distinguishing paths in the Doeblin graph, and also a decisive property for Markov chains indexed by $\mathbb{Z}$. Properties related to simulating the bridge graph are also studied., Comment: 44 pages, 4 figures

The notions of unimodular Minkowski and Hausdorff dimensions are defined in [arXiv:1807.02980] for unimodular random discrete metric spaces. The present paper is focused on the connections between these notions and the polynomial growth rate of the underlying space. It is shown that bounding the dimension is closely related to finding suitable equivariant weight functions (i.e., measures) on the underlying discrete space. The main results are unimodular versions of the mass distribution principle, Billingsley's lemma and Frostman's lemma, which allow one to derive upper bounds on the unimodular Hausdorff dimension from the growth rate of suitable equivariant weight functions. These results allow one to compute or bound both types of unimodular dimensions in a large set of examples in the theory of point processes, unimodular random graphs, and self-similarity. Further results of independent interest are also presented, like a version of the max-flow min-cut theorem for unimodular one-ended trees and a weak form of pointwise ergodic theorems for all unimodular discrete spaces., Comment: 39 pages. This paper is now merged with the latest version of arXiv:1807.02980. A previous version of this submission was titled `On the Dimension of Unimodular Discrete Spaces, Part II: Relations with Growth Rate'

This work introduces two new notions of dimension, namely the unimodular Minkowski and Hausdorff dimensions, which are inspired from the classical analogous notions. These dimensions are defined for unimodular discrete spaces, introduced in this work, which provide a common generalization to stationary point processes under their Palm version and unimodular random rooted graphs. The use of unimodularity in the definitions of dimension is novel. Also, a toolbox of results is presented for the analysis of these dimensions. In particular, analogues of Billingsley's lemma and Frostman's lemma are presented. These last lemmas are instrumental in deriving upper bounds on dimensions, whereas lower bounds are obtained from specific coverings. The notions of unimodular Hausdorff size, which is a discrete analogue of the Hausdorff measure, and unimodular dimension function are also introduced. This toolbox allows one to connect the unimodular dimensions to other notions such as volume growth rate, discrete dimension and scaling limits. It is also used to analyze the dimensions of a set of examples pertaining to point processes, branching processes, random graphs, random walks, and self-similar discrete random spaces. Further results of independent interest are also presented, like a version of the max-flow min-cut theorem for unimodular one-ended trees and a weak form of pointwise ergodic theorems for all unimodular discrete spaces., Comment: 89 pages, 1 figure. This version of the paper is a merging of the previous version with arXiv:1808.02551. Earlier versions of this paper were titled `On the Dimension of Unimodular Discrete Spaces, Part I: Definitions and Basic Properties'

This paper is centered on covariant dynamics on unimodular random graphs and random networks, namely maps from the set of vertices to itself which are preserved by graph or network isomorphisms. Such dynamics are referred to as vertex-shifts here. The first result of the paper is a classification of vertex-shifts on unimodular random networks. Each such vertex-shift partitions the vertices into a collection of connected components and foils. The latter are discrete analogues the stable manifold of the dynamics. The classification is based on the cardinality of the connected components and foils. Up to an event of zero probability, there are three classes of foliations in a connected component: F/F (with finitely many finite foils), I/F (infinitely many finite foils), and I/I (infinitely many infinite foils). An infinite connected component of the graph of a vertex-shift on a random network forms an infinite tree with one selected end which is referred to as an Eternal Family Tree. Such trees can be seen as stochastic extensions of branching processes. Unimodular Eternal Family Trees can be seen as extensions of critical branching processes. The class of offspring-invariant Eternal Family Trees, which is introduced in the paper, allows one to analyze dynamics on networks which are not necessarily unimodular. These can be seen as extensions of non-necessarily critical branching processes. Several construction techniques of Eternal Family Trees are proposed, like the joining of trees or moving the root to a far descendant. Eternal Galton-Watson Trees and Eternal Multi-type Galton-Watson Trees are also introduced as special cases of Eternal Family Trees satisfying additional independence properties. These examples allow one to show that the results on Eternal Family Trees unify and extend to the dependent case several well known theorems of the literature on branching processes., Comment: 43 pages, 1 figure

A point-shift $F$ maps each point of a point process $\Phi$ to some point of $\Phi$. For all translation invariant point-shifts $F$, the $F$-foliation of $\Phi$ is a partition of the support of $\Phi$ which is the discrete analogue of the stable manifold of $F$ on $\Phi$. It is first shown that foliations lead to a classification of the behavior of point-shifts on point processes. Both qualitative and quantitative properties of foliations are then established. It is shown that for all point-shifts $F$, there exists a point-shift $F_\bot$, the orbits of which are the $F$-foils of $\Phi$, and which are measure-preserving. The foils are not always stationary point processes. Nevertheless, they admit relative intensities with respect to one another., Comment: 36 pages, 1 figure

We give an algorithm to construct a translation-invariant transport kernel between ergodic stationary random measures $\Phi$ and $\Psi$ on $\mathbb R^d$, given that they have equal intensities. As a result, this yields a construction of a shift-coupling of an ergodic stationary random measure and its Palm version. This algorithm constructs the transport kernel in a deterministic manner given realizations $\varphi$ and $\psi$ of the measures. The (non-constructive) existence of such a transport kernel was proved in [8]. Our algorithm is a generalization of the work of [3], in which a construction is provided for the Lebesgue measure and an ergodic simple point process. In the general case, we limit ourselves to what we call constrained densities and transport kernels. We give a definition of stability of constrained densities and introduce our construction algorithm inspired by the Gale-Shapley stable marriage algorithm. For stable constrained densities, we study existence, uniqueness, monotonicity w.r.t. the measures and boundedness., Comment: In the second version, we change the way of presentation of the main results in Section 4. The main results and their proofs are not changed significantly. We add Section 3 and Subsection 4.6. 25 pages and 2 figures

A compatible point-shift $F$ maps, in a translation invariant way, each point of a stationary point process $\Phi$ to some point of $\Phi$. It is fully determined by its associated point-map, $f$, which gives the image of the origin by $F$. It was proved by J. Mecke that if $F$ is bijective, then the Palm probability of $\Phi$ is left invariant by the translation of $-f$. The initial question motivating this paper is the following generalization of this invariance result: in the non-bijective case, what probability measures on the set of counting measures are left invariant by the translation of $-f$? The point-map probabilities of $\Phi$ are defined from the action of the semigroup of point-map translations on the space of Palm probabilities, and more precisely from the compactification of the orbits of this semigroup action. If the point-map probability exists, is uniquely defined, and if it satisfies certain continuity properties, it then provides a solution to this invariant measure problem. Point-map probabilities are objects of independent interest. They are shown to be a strict generalization of Palm probabilities: when $F$ is bijective, the point-map probability of $\Phi$ boils down to the Palm probability of $\Phi$. When it is not bijective, there exist cases where the point-map probability of $\Phi$ is singular with respect to its Palm probability. A tightness based criterion for the existence of the point-map probabilities of a stationary point process is given. An interpretation of the point-map probability as the conditional law of the point process given that the origin has $F$-pre-images of all orders is also provided. The results are illustrated by a few examples., Comment: 35 pages, 2 figures

We analyze a class of Signal-to-Interference-and-Noise-Ratio (SINR) random graphs. These random graphs arise in the modeling packet transmissions in wireless networks. In contrast to previous studies on the SINR graphs, we consider both a space and a time dimension. The spatial aspect originates from the random locations of the network nodes in the Euclidean plane. The time aspect stems from the random transmission policy followed by each network node and from the time variations of the wireless channel characteristics. The combination of these random space and time aspects leads to fluctuations of the SINR experienced by the wireless channels, which in turn determine the progression of packets in space and time in such a network. This paper studies optimal paths in such wireless networks in terms of first passage percolation on this random graph. We establish both "positive" and "negative" results on the associated time constant. The latter determines the asymptotics of the minimum delay required by a packet to progress from a source node to a destination node when the Euclidean distance between the two tends to infinity. The main negative result states that this time constant is infinite on the random graph associated with a Poisson point process under natural assumptions on the wireless channels. The main positive result states that when adding a periodic node infrastructure of arbitrarily small intensity to the Poisson point process, the time constant is positive and finite.