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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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AncientSwordRage
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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). – Did Jan 11 '12 at 21:25
  • Yes I do. Is there a way to derive some 'figures' from that (the cross thing you mentioned)? Criticality, percolation threshold etc? – AncientSwordRage Jan 11 '12 at 21:54
  • 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 on Models and Methods for Random Networks. – Did Jan 11 '12 at 22:14
  • 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. – AncientSwordRage Jan 11 '12 at 22:23
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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). – Did Jan 11 '12 at 22:26
  • @DidierPiau I know, but right now I'm writing a report that will reference those papers! That comes first ;) – AncientSwordRage Jan 11 '12 at 22:30
  • 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? – Dan Brumleve Jan 12 '12 at 03:10
  • @DanBrumleve, D2 to C3 is possible. – AncientSwordRage Jan 12 '12 at 07:53
  • If there is an update to this question, please do not hesitate to post it here. – Did Apr 02 '14 at 17:06
  • @did... I'll see if I remember when I get home – AncientSwordRage Apr 02 '14 at 17:10

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Yes it's called variable range hoping. You can find papers by Faggionato, Mathieu on arxiv.

percojazz
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    Hi Antoine. As your answer stands, it is very minimal. It might be helpful to future viewers if you were to expand your answer with more context and details--perhaps linking to specific papers which address the OP's question – eepperly16 Jan 17 '18 at 05:50