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M. Mezard, G. Parisi and coworkers have written about replica symmetry and its breaking in spin glasses, structural glasses, and hard computational problems.

I am just getting acquainted with this literature. Where are the best places to start understanding replica symmetry?

Can anyone here explain the concept?

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Try having a look at "replica trick". This is the mathematical technique and has been used in also in neural networks, error-correcting codes or eigenvalue distribution of random matrices. Also the book "Statistical physics of spin glasses and information processing : an introduction." by Nishimori, Hidetoshi can be a good starting point. –  DaniH Apr 5 '12 at 21:07
    
Thanks DaniH. I am reading Nishimori and will update with any progress I make. –  jwco Apr 7 '12 at 16:17
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A good starting point could be the excellent article "Spin Glass Theory for Pedestrian" by Tommaso Castellani and Andrea Cavagna, that provides a beautiful and clear introduction to the world of spin-glasses.

http://arxiv.org/abs/cond-mat/0505032

If you want a more complete and historically complete reference, you could also try "Spin Glass Theory and Beyond", written by Parisi, Mezard and Virasoro.

http://www.worldscientific.com/worldscibooks/10.1142/0271

In both these references you can find a clear and complete explanation of the concept of replica symmetry, and its breaking in spin-glass-like systems.

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