Anomalous singularity of the solution of the vector Dyson equation in the critical case

TL;DR

This paper analyzes the behavior of solutions to the vector Dyson equation with a block staircase structure, revealing fractional power laws near zero and characterizing the density of states' singularity.

Contribution

It provides a detailed analysis of the vector Dyson equation's solutions in critical block-structured cases, including fractional power behavior and density of states near zero.

Findings

Components of m behave as fractional powers of z near zero

Density of states scales as |E|^{-(n-1)/(n+1)} near zero

Uniform estimates for m components in non-constant block cases

Abstract

We consider the solution of the vector Dyson equation $-1/m=z+Sm$ in the case when $S$ has a block staircase structure with $(n-1)$ different critical zero blocks below the strictly positive anti-diagonal and all elements right above the anti-diagonal are strictly positive. We prove that the components of $m$ behave as fractional powers of $z$ in the neighbourhood of zero and show that the self-consistent density of states $\rho(E)$ behaves as $\vert E\vert^{-\frac{n-1}{n+1}}$ as $E$ tends to zero, where $n^2$ is a number of blocks. Both constant block and non-constant block cases are considered. In the non-constant case uniform estimates for the components of $m$ are obtained.