Resolvent of Large Random Graphs

We analyze the convergence of the spectrum of large random graphs to the spectrum of a limit infinite graph. We apply these results to graphs converging locally to trees and derive a new formula for the Stieljes transform of the spectral measure of such graphs. We illustrate our results on the uniform regular graphs, Erdos-Renyi graphs and preferential attachment graphs. We sketch examples of application for weighted graphs, bipartite graphs and the uniform spanning tree of n vertices.Comment: 21 pages, 1 figur

Invariance principles for pruning processes of Galton-Watson trees

Pruning processes $(\mathcal{F}(\theta),\theta\geq 0)$ have been studied separately for Galton-Watson trees and for L\'evy trees/forests. We establish here a limit theory that strongly connects the two studies. This solves an open problem by Abraham and Delmas, also formulated as a conjecture by L\"ohr, Voisin and Winter. Specifically, we show that for any sequence of Galton-Watson forests $\mathcal{F}_n$, $n\geq 1$, in the domain of attraction of a L\'evy forest $\mathcal{F}$, suitably scaled pruning processes $(\mathcal{F}_n(\theta),\theta\geq 0)$ converge in the Skorohod topology on cadlag functions with values in the space of (isometry classes of) locally compact real trees to limiting pruning processes. We separately treat pruning at branch points and pruning at edges. We apply our results to study ascension times and Kesten trees and forests.Comment: 33 page

A cut-invariant law of large numbers for random heaps

Heap monoids equipped with Bernoulli measures are a model of probabilistic asynchronous systems. We introduce in this framework the notion of asynchronous stopping time, which is analogous to the notion of stopping time for classical probabilistic processes. A Strong Bernoulli property is proved. A notion of cut-invariance is formulated for convergent ergodic means. Then a version of the Strong law of large numbers is proved for heap monoids with Bernoulli measures. Finally, we study a sub-additive version of the Law of large numbers in this framework based on Kingman sub-additive Ergodic Theorem.Comment: 29 pages, 3 figures, 21 reference