3,677 research outputs found
On the scaling limits of planar percolation
We prove Tsirelson's conjecture that any scaling limit of the critical planar
percolation is a black noise. Our theorems apply to a number of percolation
models, including site percolation on the triangular grid and any subsequential
scaling limit of bond percolation on the square grid. We also suggest a natural
construction for the scaling limit of planar percolation, and more generally of
any discrete planar model describing connectivity properties.Comment: With an Appendix by Christophe Garban. Published in at
http://dx.doi.org/10.1214/11-AOP659 the Annals of Probability
(http://www.imstat.org/aop/) by the Institute of Mathematical Statistics
(http://www.imstat.org
Geographical Embedding of Scale-Free Networks
A method for embedding graphs in Euclidean space is suggested. The method
connects nodes to their geographically closest neighbors and economizes on the
total physical length of links. The topological and geometrical properties of
scale-free networks embedded by the suggested algorithm are studied both
analytically and through simulations. Our findings indicate dramatic changes in
the embedded networks, in comparison to their off-lattice counterparts, and
call into question the applicability of off-lattice scale-free models to
realistic, everyday-life networks
Conformal Invariance of Spin Correlations in the Planar Ising Model
We rigorously prove the existence and the conformal invariance of scaling
limits of the magnetization and multi-point spin correlations in the critical
Ising model on arbitrary simply connected planar domains. This solves a number
of conjectures coming from the physical and the mathematical literature. The
proof relies on convergence results for discrete holomorphic spinor observables
and probabilistic techniques.Comment: Changes in this version: the explicit formula for n-point spin
correlations is proved in full generality. The appendix is rewritten
completely and contains this new proof, the introduction is changed
accordingly, the presentation in Sections 2.5 and 2.7(2.8) is rearranged
slightly. 42 pages, 2 figure
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