Supersymmetric methods in random matrix theory

Abstract

Randomness and chaos are key ingredients in the description of nature and are therefore central elements in mathematics and physics. A conducting metal becomes an insulator if there are enough random defects in its structure. This phase transition generated by randomness (also called Anderson transition) is a central point of study. It is an unproven conjecture that in dimension 3 there is such a phase transition between diffusive and isolated states while in dimension 1 there are proofs that only localization occurs. <br /> This doctoral thesis provides insights into supersymmetric methods relevant for the study of two prominent random matrix models describing disordered materials: random Schrödinger operators and random band matrices. <br /> The main idea in the following is - using the supersymmetric approach - to establish dual representations for the quantity of interest, which in turn can be studied via analytic tools, inspired by statistical mechanics.