I would like to read some of the physics literature on two-dimensional percolation, however in attempting this I have run into two problems. (1) Physics papers on percolation are (relatively) hard for me to locate, and when I do locate them I don't know which are the important ones I should read. (2) Given my lack of substantial background in mathematical physics, the physics papers I find are hard for me to read.

I have become interested in two-dimensional percolation since reading some of the (relatively) recent been spectacular progress in the (mathematical) theory of two-dimensional percolation due to Smirnov, Schramm, Lawler, Werner, and others (which confirms many conjectures from the physics literature,though many open questions still remain).

The best I currently can do is to look at references to the physics literature in mathematics papers on percolation. I am looking for:

  1. Specific outstanding physics papers about two-dimensional percolation which I should read.
  2. Useful tips for mathematicians trying to read such papers.

Thanks very much!

  • $\begingroup$ You are in good company. Robert Langlands himself has tried (and failed) to understand Percolation theory publications.ias.edu/rpl/section/27 It can't be that difficult you're just randomly changing things white and black. $\endgroup$ – john mangual Mar 12 '16 at 20:22

(1) Look first at the references in Schramm and Smironv and Lowler. They refer to some important physics papers. Also look at the survey of Langlands, Pouliot, Saint-Aubin, Conformal invariance in two-dimensional percolation. Bull. Amer. Math. Soc. (N.S.) 30 (1994), no. 1, 1–61. This is almost a "physics paper", but Langlands is a mathematician, and it has a lot of references on other physics papers.

The large book Geoffrey Grimmett, Percolation. Springer-Verlag, New York, 1989 is written for mathematicians, but has a lot of references on physicists.

(2) This is a special case of a more general question, "how can a mathematician read physicist's papers?" It is really VERY difficult (my own experience). There is no general answer. There are some papers written specially for mathematicians, for example, Harry Kesten, Percolation theory for mathematicians, Birkhäuser, Boston, Mass., 1982. iv+423 pp.

On the last questions: look at the papers of John Cardy which have "percolation" in their title. He is a physicist, and some of his papers are among the most important in the subject.

Useful tips? It will take too long to explain. Just read and practice. Use patience. Do not try to understand every word or every sentence. Eventually you will discover that some physicists write in more intelligible style than others. Look at mathematicians papers on the same subject and compare. You will eventually learn the meaning of some words in this way. But this is indeed difficult, for a mathematician to read physicists papers, there is no doubt about it.

  • 6
    $\begingroup$ Since when is Kesten a physicist. $\endgroup$ – Ori Gurel-Gurevich Aug 22 '15 at 1:39
  • $\begingroup$ Reading Cardy is the best piece of advice here. He also has a book on renormalization (amazon.com/…) that touches briefly on percolation. $\endgroup$ – Steve Huntsman Aug 22 '15 at 2:44
  • $\begingroup$ @Ori Gurel-Gurewich: I thank you and Misha Sodin for this remark. I corrected my answer. $\endgroup$ – Alexandre Eremenko Aug 22 '15 at 23:56
  • $\begingroup$ @Steve Huntsman: Some mathematicians find Cardy's arguments incomprehensible. It all depends on your background. But he stated one of the main conjectures in the subject, and even guessed the correct answer. $\endgroup$ – Alexandre Eremenko Aug 22 '15 at 23:59
  • $\begingroup$ @AlexandreEremenko--I spent a lot of time with physics and physicists. So I am willing to deal with their way of reasoning. (I wish that was a two way street!) $\endgroup$ – Steve Huntsman Aug 23 '15 at 0:26

This is just one reference; by no means a comprehensive answer to your broad question. This paper,

Jacob J.H. Simmons. "Logarithmic operator intervals in the boundary theory of critical percolation." Journal of Physics A: Mathematical and Theoretical 46, no. 49 (2013): 494015. (Journal link.) (Earlier arXiv abstract.)

uses conformal field theory (CFT) "to probe the problems of critical two dimensional percolation." The author calculates "crossing probabilities and expectation values of crossing cluster numbers" in hexagons for continuum percolation at criticality.

In the figure caption, "$O(n)$" refers to the $O(n)$ loop model, referencing

Domany E, Mukamel D, Nienhuis B and Schwimmer A 1981 Nucl. Phys. B190 279–287.


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