While rarely mentioned in elementary introductions to random matrix theory, statistical field theory techniques used for evaluating partition functions provide a powerful tool for analysing random matrices and random graphs. I will review the applications of these techniques to eigenvalue spectra.
Starting from the resolvent representation, one first converts it to an integral over vectors with both bosonic and fermionic components, and ultimately arrives at an explicit functional saddle point that controls the eigenvalue distribution of large matrices.
With this functional saddle point equation, for the ordinary Wigner random matrix, one obtains a two-line derivation of the Wigner semicircle law.
The same equation gives analytic control over the much more complicated problem of eigenvalue distributions of sparse graph Laplacians.
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