Uniqueness of the mobility edge for Lévy matrices

Abstract

We prove that the mobility edge equation introduced by Tarquini, Biroli, and Tarzia for Lévy matrices has a unique positive solution for every tail exponent $\alpha\in(0,1)$. Combined with a previous result by Aggarwal, Bordenave, and the author, this establishes for every $\alpha \in (0,1)$ the predicted transition between delocalized and localized spectral regimes. The proof reduces the mobility edge condition to the intersection of a one-dimensional curve with a fixed circle and establishes uniqueness by showing that this curve can cross the circle only in one direction. The proof relies on a new contour representation of the integrals in the fixed-point equation for the order parameter of the model. Combined with elementary moment inequalities, it yields control of the phase and effective bounds. The remaining quantitative estimate is reduced to an explicit one-dimensional inequality and verified using rigorous interval arithmetic.