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This ultimately relates to a physics question, but It wasn't getting any answers, on physics.stackexchange. As it crosses the boundaries of each subject I was advised to post it here:

Has a system where conducting sites can percolate by hopping over/tunnelling through a non-conducting site been described? If so what are the characteristics, and where can I find more details (such as a paper on the subject)?

In the image below, if the edge of a black square touches the edge of another black it 'conducts' across. That could be described as singularly percolated.

I'm trying to describe a system whereby the 'conduction' can hop over a white square.

Does this have a name? Is it formally described in a paper anywhere?

Percolation grid

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    Do you mean that, in your figure, black squares (2,2) and (2,4), for example, are in the same black component because only white square (2,3) separates them? But that black (11,1) and black (11,4) are not in the same black component, because two white squares separate them? Then you could consider the equivalent Boolean percolation based on crosses (a cross being one square plus its four neighboring half-squares).2012-01-11
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    Yes I do. Is there a way to derive some 'figures' from that (the cross thing you mentioned)? Criticality, [percolation threshold](http://en.wikipedia.org/wiki/Percolation_threshold) etc?2012-01-11
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    Exact values, certainly not, unless some unlikely miracle occurs. Nontriviality of the phase transition and so on, yes, thanks to general results. You might want to have a look at [some recent lecture notes](http://icawww1.epfl.ch/class-nooc/) on *Models and Methods for Random Networks*.2012-01-11
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    Thanks! In actually finding the link on the percolation threshold I found some papers on the subject so I should be able to self-answer in a day or two.2012-01-11
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    Great. Do not hesitate to post here the result of your investigations as an answer to your own question (this is accepted, and even recommended on this site).2012-01-11
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    @DidierPiau I know, but right now I'm writing a report that will reference those papers! That comes first ;)2012-01-11
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    Is it possible for the "electron" to "turn" while it is hopping? i.e., if two black squares are touching at a corner and are completely surrounded by white squares, is there a conductive path between them? With reference to your figure, if we start at D2 (near the lower left), can we tunnel to C3, and then to either of C5 or D4? Or is it the case that from D2 we are permitted only to move to D4?2012-01-12
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    @DanBrumleve, D2 to C3 is possible.2012-01-12
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    If there is an update to this question, please do not hesitate to post it here.2014-04-02
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    @did... I'll see if I remember when I get home2014-04-02

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