Abstract
Diagrammatic techniques to compute perturbatively the spectral properties of Euclideanrandom matrices (ERM) in the high-density regime are introduced and discussed in detail.Such techniques are developed in two alternative and very different formulations of themathematical problem and are shown to give identical results up to second order in theperturbative expansion. One method, based on writing the so-called resolventfunction as a Taylor series, allows us to group the diagrams into a small number oftopological classes, providing a simple way to determine the infrared (small momenta)behaviour of the theory up to third order, which is of interest for the comparisonwith experiments. The other method, which reformulates the problem as a fieldtheory, can instead be used to study the infrared behaviour at any perturbativeorder.
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