Scaling Limits and Generic Bounds for Exploration Processes

Abstract

We consider exploration algorithms of the random sequential adsorption type both for homogeneous random graphs and random geometric graphs based on spatial Poisson processes. At each step, a vertex of the graph becomes active and its neighboring nodes become explored. Given an initial number of vertices $N$ growing to infinity, we study statistical properties of the proportion of explored nodes in time using scaling limits. We obtain exact limits for homogeneous graphs and prove an explicit central limit theorem for the final proportion of active nodes, known as the \emph{jamming constant}, through a diffusion approximation for the exploration process. We then focus on bounding the trajectories of such exploration processes on random geometric graphs, i.e. random sequential adsorption. As opposed to homogeneous random graphs, these do not allow for a reduction in dimensionality. Instead we build on a fundamental relationship between the number of explored nodes and the discovered volume in the spatial process, and obtain generic bounds: bounds that are independent of the dimension of space and the detailed shape of the volume associated to the discovered node. Lastly, we give two trajectorial interpretations of our bounds by constructing two coupled processes that have the same fluid limits.Comment: 25 pages, 3 figure

Large Deviation Principle for the Greedy Exploration Algorithm over Erdös-Rényi Graphs

“…Thanks to the great amount of independence and symmetry of the collection of edges in a sparse ER graph G(N, c/N ), the greedy exploration algorithm is characterized by Z N k k a simple one-dimensional Markov process. As a consequence, a functional law of large numbers described by a differential equation can be employed to get the macroscopic size of the constructed independent set when the number of nodes goes to infinity (see Bermolen et al (2017a) and the references in McDiarmid (1990)). Diffusion approximations for the process and central limit theorem derived from it for the size T * N of the associated independent set are also known, see Bermolen et al (2017a).…”

“…As a consequence, a functional law of large numbers described by a differential equation can be employed to get the macroscopic size of the constructed independent set when the number of nodes goes to infinity (see Bermolen et al (2017a) and the references in McDiarmid (1990)). Diffusion approximations for the process and central limit theorem derived from it for the size T * N of the associated independent set are also known, see Bermolen et al (2017a). However, there is to the best of our knowledge no characterization of a large deviation principle (LDP) for both the discrete time Markov process Z N k k and the random variable T * N , which can give various types of useful information both on the greedy exploration and on the independent sets landscape.…”

Large Deviations in One-Dimensional Random Sequential Adsorption

“…graphs with few loops, can be also tractable. For instance, the RSA model on the sparse Erdős-Rényi random graphs with b = 1 has been studied, see [39][40][41][42]. It would be interesting to determine the full counting statistics of the occupation number for this RSA model.…”

Modeling Rydberg Gases using Random Sequential Adsorption on Random Graphs

“…, n} at which all vertices are either labeled excited or unaffected. Borrowing terminology from [20,33,34], the resulting state is called a jamming limit, and T n is a so-called hitting time. Compared to the QMMRG, we note that a vertex that is unaffected within the RSA process corresponds to an atom that is still in a ground state because it has not excited yet, and moreover, this atom has not yet been blocked by another atom.…”

“…Theorems 1-3 also contribute to the mathematical perspective. The study of exploration processes of the RSA type on random graphs has seen interest in recent years [20,[32][33][34][35][36][37][38]. The tools necessary from probability theory to analyze said processes include e.g.…”