Determine whether the Brown measure has full-disk support

Establish that the Brown measure \(\mu_d\) of the adjacency operator of the directed Poisson–Galton–Watson tree assigns positive mass to every annulus \(\{z:r_1<|z|<r_2\}\) with \(0\le r_1<r_2<\sqrt d\), thereby proving that every interior radius occurs in its support.

Background

The paper proves that the limiting Brown measure μd\mu_d is supported inside the closed disk of radius d\sqrt d, has positive mass outside every disk of radius r<dr<\sqrt d, and contains both the origin and the boundary circle {z:∣z∣=d}\{z:|z|=\sqrt d\} in its support. These results determine the outer radius but do not determine whether the support fills the disk radially.

The unresolved strengthening is positivity of μd\mu_d on every nonempty interior annulus, namely μd({z:r1<∣z∣<r2})>0\mu_d(\{z:r_1<|z|<r_2\})>0 for all 0≤r1<r2<d0\le r_1<r_2<\sqrt d. The paper notes that rotational invariance and positive mass outside every smaller disk do not imply this property, since a rotationally invariant measure could consist only of an atom at zero and mass on the outer circle.

References

These inclusions leave open whether every interior radius occurs in the support. Full-disk support would require

\mu_d\bigl({z:r_1<|z|<r_2\}\bigr)\>0 \qquad(0\le r_1<r_2<\sqrt d).

This is stronger than eq:main-positive-tail. For example, a mixture of an atom at zero and the uniform probability measure on the circle of radius \sqrt d, with both weights positive, is rotationally invariant and has positive mass outside every smaller disk, but no mass in an interior annulus. Our proof determines the outer radius and does not establish eq:no-annular-gaps.

eq:main-positive-tail:

$\mu_d\bigl(\{z\inC:|z|>r\}\bigr)>0. $

eq:no-annular-gaps:

μd({z:r1<∣z∣<r2})>0(0≤r1<r2<d).\mu_d\bigl(\{z:r_1<|z|<r_2\}\bigr)>0 \qquad(0\le r_1<r_2<\sqrt d).

— The spectral edge of sparse directed Erdös-Rényi graphs  (2609.18369 - Coste et al., 16 Sep 2026) in Section 1, paragraph “What the theorem says about the support”