In this paper, we prove that the fluctuations of the graph distance and the effective resistance on the trace of a random walk in four and five dimensions converge in distribution to a stable law. In previous work, the first and second authors proved that the corresponding fluctuations converge to a Gaussian distribution in dimensions six and higher. Taken together, these results reveal a phase transition between dimensions five and six. Our proof develops a novel coupling with long range percolation, and we expect this technique to find applications in a broad class of related models.
In this paper, we investigate models of percolation of words on $\mathbb{Z}_+$ with long-range connections. The underlying graph is oriented and constructed according to a sequence of non-negative ranges. In the first model, the sequence of ranges is considered random and, in the second model, deterministic and non-decreasing. In both cases, we establish a phase transition for the occurrence of percolation of all words simultaneously from the origin.
We prove that on the metric graph of $\mathbb{Z}^d$ for $d\in \{3,4,5\}$, clusters of the critical loop soup (or equivalently, clusters of the critical Gaussian free field level-set) admit a scaling limit. To the best of our knowledge, this presents the first scaling limit result for a percolation model on a three-dimensional lattice.
We prove locality of the critical probability for Bernoulli site percolation on infinite, connected, locally finite, vertex-transitive graphs, under the usual assumption that the critical probabilities stay uniformly below one. The proof develops site-percolation forms of the two-ghost inequality, sharp-threshold and snowballing estimates, and the nonunimodular height method. We also establish the plentiful-tube geometry needed for the multiscale argument from quantitative structure and random-walk estimates.
This work demonstrates how equilibrium measures within the framework of Schr¨odinger random walks on networks can be leveraged to compute key network parameters such as the Mean First Passage Time (MFPT) and Kemeny's constant by expressing these parameters in terms of generalized inverses of the associated M-matrix.
Á. Carmona, A. Encinas, M. J. Jiménez et al.· The Electronic Journal of Li...· 0 citations
The stochastic block model is a widely studied model of community structure in networks. Here we study the component structure and percolation properties of networks generated from this model and its variants, using exact methods based on probability generating functions. In particular, we derive expressions for the size of the giant component and the distribution of small components in such networks and for the size of the percolating cluster and position of the percolation threshold for both node and edge percolation, for the original stochastic block model and for its degree-corrected versions. In passing, we also develop a mapping between generating functions for microcanonical and canonical block models that allows us to generalize results for the former to the latter with minimal effort.
We study percolation on lattices with directed bonds, focusing on the behavior of strongly-connected percolation clusters -- clusters in which every site is reachable from every other along a directed path. We consider the two-dimensional square lattice and various globally isotropic arrangements of the directions of the bonds. Performing simulations using a range of algorithmic approaches, we calculate high-precision values for critical exponents, fractal dimensions, crossing probabilities, and percolation thresholds for bond percolation with each bond arrangement. We find that the critical behavior is in a distinctly different universality class from that of traditional undirected percolation, but that all bond arrangements appear to fall in the same universality class.
M. Newman, P. Grassberger, R. Ziff· 0 citations
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