In 1980, Kesten showed the exponential decay of percolation probability in the subcritical phase for the two-dimensional percolation model. This result implies his celebrated computation that pc = 0.5 for bond percolation in the square lattice, and site percolation in the triangular lattice, respectively. In this paper, we present a simpler proof for Kesten’s theorem.

A percolation model is an infinite random graph model for phase transitions and critical phenomena. The percolation threshold corresponds to a phase transition point, such as a melting or freezing temperature. The exact value of the percolation threshold is not known for any three-dimensional percolation models, which are important for physical applications. Furthermore, rigorous bounds for the...

In this paper we apply scaling laws from percolation theory to the problem of estimating the time for a fluid injected into an oil field to breakthrough into a production well. The main contribution is to show that, when these previously published results are used on realistic data, they are in good agreement with results calculated in a more conventional way but they can be obtained significan...

Sea ice exhibits a marked transition in its fluid transport properties at a critical brine volume fraction pc of about 5 percent, or temperature Tc of about -5 degreesC for salinity of 5 parts per thousand. For temperatures warmer than Tc, brine carrying heat and nutrients can move through the ice, whereas for colder temperatures the ice is impermeable. This transition plays a key role in the g...

In recent years, many variants of percolation have been used to study network structure and the behavior of processes spreading on networks. These include bond percolation, site percolation, k-core percolation, bootstrap percolation, the generalized epidemic process, and the Watts threshold model (WTM). We show that-except for bond percolation-each of these processes arises as a special case of...

Consider the following process, known as r-bootstrap percolation, on a graph G. Designate some initial infected set A and infect any vertex with at least r infected neighbors, continuing until no new vertices can be infected. We say A percolates if it eventually infects the entire graph. We say A is a minimal percolating set if A percolates, but no proper subset percolates. We compute the size ...

Five methods to assess percolation rate from alternative earthen final covers (AEFCs) are described in the context of the precision with which the percolation rate can be estimated: trend analysis, tracer methods, water balance method, Darcy’s Law calculations, and lysimetry. Trend evaluation of water content data is the least precise method because it cannot be used alone to assess the percola...

We consider a random field (Xn)n∈Zd and investigate when the set Ah = {k ∈ Zd; ‖Xk‖ ≥ h} has infinite clusters. The main problem is to decide whether the critical level hc = sup{h ∈ R; P (Ah has an infinite cluster) > 0} is neither 0 nor +∞. Thus, we say that a percolation transition occurs. In a first time, we show that weakly dependent Gaussian fields satisfy to a wellknown criterion implying...

We consider a type of dependent percolation introduced in [2], where it is shown that certain “enhancements” of independent (Bernoulli) percolation, called essential, make the percolation critical probability strictly smaller. In this paper we first prove that, for two-dimensional enhancements with a natural monotonicity property, being essential is also a necessary condition to shift the criti...

Percolation theory concerns the emergence of connected clusters that percolate through a networked system. Previous studies ignored the effect that a node outside the percolating cluster may actively induce its inside neighbours to exit the percolating cluster. Here we study this inducing effect on the classical site percolation and K-core percolation, showing that the inducing effect always ca...