Density of Brown measure of free circular Brownian motion
Erdös L, Ji HC. 2025. Density of Brown measure of free circular Brownian motion. Documenta Mathematica. 30(2), 417–453.
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Author
Erdös, LászlóISTA
;
Ji, Hong Chang

Corresponding author has ISTA affiliation
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Abstract
We consider the Brown measure of the free circular Brownian motion, a+t√x , with an arbitrary initial condition a , i.e. a is a general non-normal operator and x is a circular element ∗ -free from a . We prove that, under a mild assumption on a , the density of the Brown measure has one of the following two types of behavior around each point on the boundary of its support -- either (i) sharp cut, i.e. a jump discontinuity along the boundary, or (ii) quadratic decay at certain critical points on the boundary. Our result is in direct analogy with the previously known phenomenon for the spectral density of free semicircular Brownian motion, whose singularities are either a square-root edge or a cubic cusp. We also provide several examples and counterexamples, one of which shows that our assumption on a is necessary.
Publishing Year
Date Published
2025-03-20
Journal Title
Documenta Mathematica
Publisher
EMS Press
Acknowledgement
We thank Ping Zhong for pointing out references [15,19] and providing helpful comments. We also thank the anonymous referee for many valuable comments and proposals to streamline the presentation. This work was partially supported by ERC Advanced Grant “RMTBeyond” No. 10102033.
Volume
30
Issue
2
Page
417-453
ISSN
eISSN
IST-REx-ID
Cite this
Erdös L, Ji HC. Density of Brown measure of free circular Brownian motion. Documenta Mathematica. 2025;30(2):417-453. doi:10.4171/DM/999
Erdös, L., & Ji, H. C. (2025). Density of Brown measure of free circular Brownian motion. Documenta Mathematica. EMS Press. https://doi.org/10.4171/DM/999
Erdös, László, and Hong Chang Ji. “Density of Brown Measure of Free Circular Brownian Motion.” Documenta Mathematica. EMS Press, 2025. https://doi.org/10.4171/DM/999.
L. Erdös and H. C. Ji, “Density of Brown measure of free circular Brownian motion,” Documenta Mathematica, vol. 30, no. 2. EMS Press, pp. 417–453, 2025.
Erdös L, Ji HC. 2025. Density of Brown measure of free circular Brownian motion. Documenta Mathematica. 30(2), 417–453.
Erdös, László, and Hong Chang Ji. “Density of Brown Measure of Free Circular Brownian Motion.” Documenta Mathematica, vol. 30, no. 2, EMS Press, 2025, pp. 417–53, doi:10.4171/DM/999.
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arXiv 2307.08626