Stable transports between stationary random measures

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 figure

Point-Map-Probabilities of a Point Process and Mecke's Invariant Measure Equation

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 figure

Doeblin Trees

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 figure

Unimodular Hausdorff and Minkowski Dimensions

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

Optimal Paths on the Space-Time SINR Random Graph

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

Routing on random geometric graphs

The two first chapters are focused on preliminaries.In Chapter 3 of this thesis, we analyze a class of Signal to Interference and Noise Ratio (SINR) random graphs. These random graphs arise in the modeling of packet transmissions in wireless networks. In contrast to previous studies on SINR graphs, we consider both a space and a time dimension.We study 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 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.In the second part, we develop a framework for studying point-map invariant measures. We focus on the case of a not necessarily bijective point-map f. We introduce the notion of Point-map Palm version of the point process , which satisfies the desired invariance property when it exists and we give sufficient conditions for it to exist.Chapter 5, explains the connection between Chapters 3 and 4. It generalizes the notion of point-map Palm measures for stochastic point-maps and time dependent point-maps. As we will see in the end of the Chapter 3, the optimal path in the time- space SINR graph is not computable locally in time. This fact leads us to considering suboptimal local algorithms.Cette thèse se concentre sur les propriétés de routages ou navigations sur les graphes aléatoires associés à des processus ponctuels et la théorie des fonctionnelles ponctuelles et des mesures de Palm. Les deux premiers chapitres se concentrent sur des définitions et des résultats préliminaires. Dans le chapitre 3, nous analysons des navigations sur une nouvelle classe de graphes aléatoires SINR. Nous considérons à la fois une dimension spatiale et une dimension temporelle. Nous étudions les chemins optimaux dans ces graphes. Le principal résultat négatif est que cette constante de temps est infinie sur le graphe aléatoire associé à un processus de Poisson sous des hypothèses naturelles sur les caractéristiques des canaux sans fil. Le principal résultat positif est que l'ajout d'une infrastructure de noeud périodique de densité arbitrairement petite rend la constante de temps positive et finie. Dans la deuxième partie, nous développons un cadre pour l'étude des mesures laissées invariantes par des fonctionnelles ponctuelles. Nous introduisons la notion mesure de Palm de fonctionnelle ponctuelle du processus ponctuel , qui satisfait, quand elle existe, la propriété d'invariance désirée. Le dernier chapitre généralise la notion de mesures Palm de fonctionnelle ponctuelle au cas de fonctionnelles stochastiques et de fonctionnelles dépendant du temps. Les chemins optimaux du graphe SINR spatio-temporel ne sont pas calculables. Les algorithmes de routage de la littérature sont donc fondés sur de algorithme locaux. Les mesures de Palm associées à ces fonctionnelles décrivent donc le paysage ponctuel "vu" par une navigation en temps long sur le processus ponctuel.PARIS-BIUSJ-Mathématiques rech (751052111) / SudocSudocFranceF

On the Dimension of Unimodular Discrete Spaces, Part II: Relations with Growth Rate

The notions of unimodular Minkowski and Hausdorff dimensions are defined in [5] 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 and Billingsley's lemma, which allow one to derive upper bounds on the unimodular Hausdorff dimension from the growth rate of suitable equivariant weight functions. Also, a unimodular version of Frostman's lemma is provided, which shows that the upper bound given by the unimodular Billingsley lemma is sharp. 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