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Assume we are simulating percolation on a 2d lattice. While the system is of finite size, we say that the critical state appears when a cluster connects two opposing ends of the lattice. The bigger the lattice the better our approximation.

My question is: assume we are doing the same thing but instead of a lattice we have a random graph with a known degree distribution. As we are increasing the occupation probability, who do we know that we reached percolation?

In other words, how does the 'connecting opposing ends' assumption translate to random graph topologies?

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  • $\begingroup$ This question is actually about mathematics. Anyway here and here and here you could find something useful. $\endgroup$ – valerio Nov 10 '16 at 8:18
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    $\begingroup$ As a far younger member and a non-physicist accept your remark, nonetheless I am surprised. Percolation in random networks is an active field of researcher with numerous publications in Physics journals. The works of Callaway, Newman, Gleeson, Watts and others focus on similar stuff. That is why I posted here. $\endgroup$ – Dionysios Georgiadis Nov 10 '16 at 9:10

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