---
DOAJ_listed: '1'
OA_place: publisher
OA_type: gold
_id: '19500'
abstract:
- lang: eng
  text: 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.
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.
article_processing_charge: Yes
article_type: original
arxiv: 1
author:
- first_name: László
  full_name: Erdös, László
  id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
  last_name: Erdös
  orcid: 0000-0001-5366-9603
- first_name: Hong Chang
  full_name: Ji, Hong Chang
  last_name: Ji
citation:
  ama: Erdös L, Ji HC. Density of Brown measure of free circular Brownian motion.
    <i>Documenta Mathematica</i>. 2025;30(2):417-453. doi:<a href="https://doi.org/10.4171/DM/999">10.4171/DM/999</a>
  apa: Erdös, L., &#38; Ji, H. C. (2025). Density of Brown measure of free circular
    Brownian motion. <i>Documenta Mathematica</i>. EMS Press. <a href="https://doi.org/10.4171/DM/999">https://doi.org/10.4171/DM/999</a>
  chicago: Erdös, László, and Hong Chang Ji. “Density of Brown Measure of Free Circular
    Brownian Motion.” <i>Documenta Mathematica</i>. EMS Press, 2025. <a href="https://doi.org/10.4171/DM/999">https://doi.org/10.4171/DM/999</a>.
  ieee: L. Erdös and H. C. Ji, “Density of Brown measure of free circular Brownian
    motion,” <i>Documenta Mathematica</i>, vol. 30, no. 2. EMS Press, pp. 417–453,
    2025.
  ista: Erdös L, Ji HC. 2025. Density of Brown measure of free circular Brownian motion.
    Documenta Mathematica. 30(2), 417–453.
  mla: Erdös, László, and Hong Chang Ji. “Density of Brown Measure of Free Circular
    Brownian Motion.” <i>Documenta Mathematica</i>, vol. 30, no. 2, EMS Press, 2025,
    pp. 417–53, doi:<a href="https://doi.org/10.4171/DM/999">10.4171/DM/999</a>.
  short: L. Erdös, H.C. Ji, Documenta Mathematica 30 (2025) 417–453.
corr_author: '1'
date_created: 2025-04-06T22:01:32Z
date_published: 2025-03-20T00:00:00Z
date_updated: 2025-09-30T11:28:02Z
day: '20'
ddc:
- '510'
department:
- _id: LaEr
doi: 10.4171/DM/999
ec_funded: 1
external_id:
  arxiv:
  - '2307.08626'
  isi:
  - '001450119900005'
file:
- access_level: open_access
  checksum: 97a02d18c05f2b9f2048747b140e7d43
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  creator: dernst
  date_created: 2025-04-07T11:21:13Z
  date_updated: 2025-04-07T11:21:13Z
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file_date_updated: 2025-04-07T11:21:13Z
has_accepted_license: '1'
intvolume: '        30'
isi: 1
issue: '2'
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '03'
oa: 1
oa_version: Published Version
page: 417-453
project:
- _id: 62796744-2b32-11ec-9570-940b20777f1d
  call_identifier: H2020
  grant_number: '101020331'
  name: Random matrices beyond Wigner-Dyson-Mehta
publication: Documenta Mathematica
publication_identifier:
  eissn:
  - 1431-0643
  issn:
  - 1431-0635
publication_status: published
publisher: EMS Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: Density of Brown measure of free circular Brownian motion
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 30
year: '2025'
...