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I recently came across the "replica trick\method" in the paper "Optimal storage properties of neural network models" by Gardner and Derrida. I would really like to understand it, but the problem is that the intuition used there is just from physics. I tried to look for introductory texts about this method, but everything that I found so far was written by physicists for physicists, so not only they use intuition from problems in physics, they sometimes use it instead of giving proper definitions and notations (not to mention the fact that physicists like to invent new names for all the known mathematical objects).

My question is, if there is a text somewhere where a mathematician who is not a physicist can read about it. This method seems to have applications in other fields (e.g. learning theory, error correcting codes etc), so I would like to believe that there is some text out there for non physicists. While I don't have a problem learning some physics for that, most texts that I saw so far assume more or less that you finished at least first degree in physics, which I don't intend to do any time soon.

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    It is a bit unclear (to me) what this trick is about. Do you suggest us to go and find that article and read about it? Could you then, at least, give us a more precise link (and page reference), or explain what the method is about as far as you see? – 2017-02-20
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    @mickep I don't really know what this trick is about because I couldn't find even one source that explains it to non physicists. I can only say that I saw it appear in relation to problems in neural networks, error correcting codes etc which are fields which interest me. I was hoping that someone who is familiar with this trick can point me to a good introductory source. Anyway, I added the link to the paper and a link to the wikipedia article. – 2017-02-20

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A good book available on the internet which also covers uses of replica trick in information theory and error-correcting codes as well as the classic results in physics is "Statistical Physics of Spin Glasses and Information Processing: An Introduction" by Hidetoshi Nishimori.

A rigorous mathematical treatment of the replica trick (by an actual mathematician) can be found in Michel Talagrand's book Mean Field Models for Spin Glasses. Note that he treats individual models using the replica trick separately as the general trick is not always valid.