Abstract for the talk at 07.05.2013 (16:45 h)

Oberseminar ANALYSIS - PROBABILITY

Yinshan Chang (Université Paris-Sud)
Loop covering probabilities

Given a sequence of undirected connected graphs G_n of n vertices with bounded degrees and weights and an additional sequence of killing parameters c_n, one can define a sequence μ_n of natural non-normalized measures on non-trivial loops of G_n. Denote by Q_n the transition matrix associated with G_n. Assuming that the empirical distributions ν_n of the eigenvalues of Q_n converge, we determine the limit of the μ_n-proportion of {loops which cover every vertex} as n →∞. Let _n be the Poissonian “loop soup” with intensity . A a corollary, we determine the limit law of the number of the loops covering the whole graph. As two concrete examples, we consider the closed balls in a regular graph and the discrete tori.