We investigate geometric percolation and scaling relations in suspensions of nanorods, covering the entire range of aspect ratios from spheres to extremely slender needles. A new version of connectedness percolation theory is introduced and tested against specialised Monte Carlo simulations. The theory accurately predicts percolation thresholds for aspect ratios of rod length to width as low as...

Article history: Magmas often contain mu Received 5 March 2008 Accepted 21 July 2008 Available online 27 July 2008

The concept of city or urban resilience has emerged as one of the key challenges for the next decades. As a consequence, institutions like the United Nations or Rockefeller Foundation have embraced initiatives that increase or improve it. These efforts translate into funded programs both for action“on the ground”and to develop quantification of resilience, under the for of an index. Ironically,...

Given a locally finite connected infinite graph G, let the interval [pmin(G), pmax(G)] be the smallest interval such that if p > pmax(G) then every 1-independent bond percolation model on G with bond probability p percolates, and for p < pmin(G) none does. We determine this interval for trees in terms of the branching number of the tree. We also give some general bounds for other graphs G, in p...

Oded Schramm (1961–2008) influenced greatly the development of percolation theory beyond the usual Z setting, in particular the case of nonamenable lattices. Here we review some of his work in this field.

We describe the percolation model and some of the principal results and open problems in percolation theory. We also discuss briefly the spectacular recent progress by Lawler, Schramm, Smirnov and Werner towards understanding the phase transition of percolation (on the triangular lattice). 2000 Mathematics Subject Classification: 60K35, 82B43.