Section: New Results

Stochastic Geometry

40. On the Dimension of Unimodular Discrete Spaces, Part I: Definitions and Basic Properties [39] 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 lemmas are instrumental in deriving upper bounds on dimensions, whereas lower bounds are obtained from specific coverings. The notions of unimodular Hausdorff measure and unimodular dimension function are also introduced. This toolbox is used to connect the unimodular dimensions to various other notions such as growth rate, scaling limits, discrete dimension and amenability. 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.

41. On the Dimension of Unimodular Discrete Spaces, Part II: Relations with Growth Rate [40] The notions of unimodular Minkowski and Hausdorff dimensions are defined in [39] for unimodular random discrete metric spaces. This work 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.

42. Doeblin trees [4] This work 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.

43. The Stochastic Geometry of Unconstrained One-Bit Compression [5] A stationary stochastic geometric model is proposed for analyzing the data compression method used in one-bit compressed sensing. The data set is an unconstrained stationary set, for instance all of ${ℝ}^n$ or a stationary Poisson point process in ${ℝ}^n$. It is compressed using a stationary and isotropic Poisson hyperplane tessellation, assumed independent of the data. That is, each data point is compressed using one bit with respect to each hyperplane, which is the side of the hyperplane it lies on. This model allows one to determine how the intensity of the hyperplanes must scale with the dimension $n$ to ensure sufficient separation of different data by the hyperplanes as well as sufficient proximity of the data compressed together. The results have direct implications in compressed sensing and in source coding.

44. Limit theory for geometric statistics of point processes having fast decay of correlations [7] We develop a limit theory (Laws of Large Numbers and Central Limit Theorems) for functionals of spatially correlated point processes. The “strength” of data correlation is captured and controlled by the speed of decay of the additive error in the asymptotic factorization the correlation functions, when the separation distance increases. In this way, the classical theory of Poisson and Bernoulli processes is extended to a larger class of data inputs, such as determinantal point processes with fast decreasing kernels, including the $\alpha$-Ginibre ensembles, permanental point processes as well as the zero set of Gaussian entire functions. Both linear (U-statistics) and non-linear geometric statistics (such as clique counts, the number of Morse critical points, intrinsic volumes of the Boolean model, and total edge length of the $k$-nearest neighbor graph) are considered.