We construct the uniform infinite planar map (UIPM), obtained as the n → ∞ local limit of planar maps with n edges, chosen uniformly at random. We then describe how the UIPM can be sampled using a “peeling” process, in a similar way as for uniform triangulations. This process allows us to prove that for bond and site percolation on the UIPM, the percolation thresholds are p c = 1/2 and p site c...

A double percolation transition is identified in a binary network composed of nanoparticles of MgB2 superconductor and CrO2 half-metallic ferromagnet. Anomalously high-resistance or insulating state, as compared to the conducting or superconducting states in single-component systems of either constituent, is observed between two distinct percolation thresholds. This double percolation effect, w...

The study of thermodynamic properties of classical spin models on infinite graphs naturally leads to consider the new combinatorial problems of random-walks and percolation on the average. Indeed, spin models with O n continuous symmetry present spontaneous magnetization only on transient on the average graphs, while models with discrete symmetry (Ising and Potts) are spontaneously magnetized o...

Abstract. This letter investigates the epidemic spreading in two-vectors propagation network (TPN). We propose detailed theoretical analysis that allows us to accurately calculate the epidemic threshold and outbreak size. It is found that the epidemics can spread across the TPN even if two sub-single-vector propagation networks (SPNs) of TPN are well below their respective epidemic thresholds. ...

Bounds for the diameter and expansion of the graphs created by long-range percolation on the cycle Z/NZ, are given.

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We prove the statement in the title of the paper.