15 research outputs found
The one-arm exponent for mean-field long-range percolation
Get PDFConsider a long-range percolation model on Zd where the probability that an edge {x; y} 2 Zd × Zd is open is proportional to ║x-y║2 -d-α for some α > 0 and where d > 3 min{2; α }. We prove that in this case the one-arm exponent equals ½ min{4; α}. We also prove that the maximal displacement for critical branching random walk scales with the same exponent. This establishes that both models undergo a phase transition in the parameter α when α = 4
High-dimensional incipient infinite clusters revisited
Get PDFThe incipient infinite cluster (IIC) measure is the percolation measure at criticality conditioned on the cluster of the origin to be infinite. Using the lace expansion, we construct the IIC measure for high-dimensional percolation models in three different ways, extending previous work by the second-named author and Járai. We show that each construction yields the same measure, indicating that the IIC is a robust object. Furthermore, our constructions apply to spread-out versions of both finite-range and long-range percolation models. We also get estimates on structural properties of the IIC, such as the volume of the intersection between the IIC and Euclidean balls. Keywords: Percolation; Incipient infinite cluster; Lace expansion; Critical behavio
Structures in supercritical scale-free percolation
Get PDFScale-free percolation is a percolation model on Zd which can be used to model real-world networks. We prove bounds for the graph distance in the regime where vertices have infinite degrees. We fully characterize transience versus recurrence for dimension 1 and 2 and give sufficient conditions for transience in dimension 3 and higher. Finally, we show the existence of a hierarchical structure for parameters where vertices have degrees with infinite variance and obtain bounds on the cluster density
{O}(k)-robust spanners in one dimension
Get PDFA geometric t-spanner on a set of points in Euclidean space is a graph containing for every pair of points a path of length at most t times the Euclidean distance between the points. Informally, a spanner is O(k)-robust if deleting k vertices only harms O(k) other vertices. We show that on any one-dimensional set of n points, for any ε>0, there exists an O(k)-robust 1-spanner with O(n1+ε) edges. Previously it was only known that O(k)-robust spanners with O(n2) edges exists and that there are point sets on which any O(k)-robust spanner has Ω(nlogn) edges
{O}(k)-robust spanners in one dimension
No full textA geometric t-spanner on a set of points in Euclidean space is a graph containing for every pair of points a path of length at most t times the Euclidean distance between the points. Informally, a spanner is O(k)-robust if deleting k vertices only harms O(k) other vertices. We show that on any one-dimensional set of n points, for any ε>0, there exists an O(k)-robust 1-spanner with O(n1+ε) edges. Previously it was only known that O(k)-robust spanners with O(n2) edges exists and that there are point sets on which any O(k)-robust spanner has Ω(nlogn) edges
Random walk on the high-dimensional IIC
Get PDFWe study the asymptotic behavior of the exit times of random walk from Euclidean balls around the origin of the incipient infinite cluster in a manner inspired by Kumagai and Misumi (J Theor Probab 21:910–935, 2008). We do this by getting bounds on the effective resistance between the origin and the boundary of these Euclidean balls. We show that the geometric properties of long-range percolation clusters are significantly different from those of finite-range clusters. We also study the behavior of random walk on the backbone of the IIC and we prove that the Alexander–Orbach conjecture holds for the incipient infinite cluster in high dimensions, both for long-range percolation and for finite-range percolation
Connectivity threshold for random subgraphs of the Hamming graph
Get PDFWe study the connectivity of random subgraphs of the -dimensional Hamming graph , which is the Cartesian product of complete graphs on vertices. We sample the random subgraph with an i.i.d.\ Bernoulli bond percolation on with parameter . We identify the window of the transition: when the probability that the graph is connected goes to , while when it converges to . We also investigate the connectivity probability inside the critical window, namely when . We find that the threshold does not depend on , unlike the phase transition of the giant connected component the Hamming graph (see [Bor et al, 2005]). Keywords: connectivity threshold, percolation, random graph, critical windo
Up and beyond: Building a mountain in the Netherlands
Get PDFWe discuss the idea of building a 2 km high mountain in the Netherlands. In this paper, we give suggestions on three important areas for the completion of this project. Issues like location, structure and sustainability are investigated and discussed in detail
Trajetórias de criação do mamulengo do professor Benedito em chão de estrelas e mais além
Get PDFDissertação (mestrado) - Universidade Federal de Santa Catarina, Centro de Filosofia e Ciências Humanas
