Degree-dependent threshold-based random sequential adsorption on random trees

Abstract

We consider a special version of random sequential adsorption (RSA) with nearest-neighbor interaction on infinite tree graphs. In classical RSA, starting with a graph with initially inactive nodes, each of the nodes of the graph is inspected in a random order and is irreversibly activated if none of its nearest neighbors are active yet. We generalize this nearest-neighbor blocking effect to a degree-dependent threshold-based blocking effect. That is, each node of the graph is assumed to have its own degree-dependent threshold and if, upon inspection of a node, the number of active direct neighbors is less than that node's threshold, the node will become irreversibly active. We analyze the activation probability of nodes on an infinite tree graph, given the degree distribution of the tree and the degree-dependent thresholds. We also show how to calculate the correlation between the activity of nodes as a function of their distance. Finally, we propose an algorithm which can be used to solve the inverse problem of determining how to set the degree-dependent thresholds in infinite tree graphs in order to reach some desired activation probabilities.

Modeling Rydberg gases using random sequential adsorption on random graphs

“…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 [24,[36][37][38][39][40][41][42]. The tools necessary from probability theory to analyze said processes include, e.g., random graph couplings, fluid and diffusion limits, stochastic differential equations, and martingale analyses, and the current work extends this framework in two ways.…”

Modeling Rydberg Gases using Random Sequential Adsorption on Random Graphs

“…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.…”