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Convergence of blanket times for sequences of random walks on critical random graphs. (arXiv:1810.07518v1 [math.PR])
来源于:arXiv
Under the assumption that sequences of graphs equipped with resistances,
associated measures, walks and local times converge in a suitable
Gromov-Hausdorff topology, we establish asymptotic bounds on the distribution
of the $\varepsilon$-blanket times of the random walks in the sequence. The
precise nature of these bounds ensures convergence of the $\varepsilon$-blanket
times of the random walks if the $\varepsilon$-blanket time of the limiting
diffusion is continuous with probability one at $\varepsilon$. This result
enables us to prove annealed convergence in various examples of critical random
graphs, including critical Galton-Watson trees, the Erd\H{o}s-R\'enyi random
graph in the critical window and the configuration model in the scaling
critical window. We highlight that proving continuity of the
$\varepsilon$-blanket time of the limiting diffusion relies on the scale
invariance of a finite measure that gives rise to realizations of the limiting
compact random metric space, and t 查看全文>>