Physics – Condensed Matter – Disordered Systems and Neural Networks
Scientific paper
2004-06-12
Nucl. Phys. B 701 (2004) 409-480
Physics
Condensed Matter
Disordered Systems and Neural Networks
62 pages, 97 figures
Scientific paper
10.1016/j.nuclphysb.2004.08.022
In an earlier publication, we have introduced a method to obtain, at large N, the effective action for d-dimensional manifolds in a N-dimensional disordered environment. This allowed to obtain the Functional Renormalization Group (FRG) equation for N=infinity and was shown to reproduce, with no need for ultrametric replica symmetry breaking, the predictions of the Mezard-Parisi solution. Here we compute the corrections at order 1/N. We introduce two novel complementary methods, a diagrammatic and an algebraic one, to perform the complicated resummation of an infinite number of loops, and derive the beta-function of the theory to order 1/N. We present both the effective action and the corresponding functional renormalization group equations. The aim is to explain the conceptual basis and give a detailed account of the novel aspects of such calculations. The analysis of the FRG flow, comparison with other studies, and applications, e.g. to the strong-coupling phase of the Kardar-Parisi-Zhang equation are examined in a subsequent publication.
Doussal Pierre Le
Wiese Kay Joerg
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