The Brown measure of the sum of a self-adjoint element and an imaginary multiple of a semicircular element

Hall, Brian C.; Ho, Ching-Wei

doi:10.1007/s11005-022-01516-3

The Brown measure of the sum of a self-adjoint element and an imaginary multiple of a semicircular element

Jun 12, 2020

50 pages

Published in:

Lett.Math.Phys. 112 (2022) 2, 19

Published: Mar 7, 2022

e-Print:

2006.07168 [math.PR]

DOI:

10.1007/s11005-022-01516-3

View in:

pdf

reference search3 citations

Citations per year

Abstract: (Springer)

We compute the Brown measure of

x_{0}+i\sigma _{t}

, where

\sigma _{t}

is a free semicircular Brownian motion and

x_{0}

is a freely independent self-adjoint element that is not a multiple of the identity. The Brown measure is supported in the closure of a certain bounded region

\Omega _{t}

in the plane. In

\Omega _{t},

the Brown measure is absolutely continuous with respect to Lebesgue measure, with a density that is constant in the vertical direction. Our results refine and rigorize results of Janik, Nowak, Papp, Wambach, and Zahed and of Jarosz and Nowak in the physics literature. We also show that pushing forward the Brown measure of

x_{0}+i\sigma _{t}

by a certain map

Q_{t}:\Omega _{t} \rightarrow {\mathbb {R}}

gives the distribution of

x_{0}+\sigma _{t}.

We also establish a similar result relating the Brown measure of

x_{0}+i\sigma _{t}

to the Brown measure of

x_{0}+c_{t}

, where

c_{t}

is the free circular Brownian motion.

Note:

50 pages and 9 figures. Minor revisions in this version. To appear on Letters in Mathematical Physics

Brownian motion
density

References(40)

Figures(9)

[1]

Circular law

ZD Bai

- Annals Probab. 25 (1997) 494-529
•
DOI:
- 10.1214/aop/1024404298

[2]

On the free convolution with a semi-circular distribution

P. Biane

- Indiana Univ.Math.J. 46 (1997) 705-718
•
DOI:
- 10.1512/iumj.1997.46.1467

[3]

Processes with free increments

P. Biane

- Math.Z. 227 (1998) 143-174
•
DOI:
- 10.1007/PL00004363

[4]

Computation of some examples of Brown's spectral measure in free probability

P. Biane
,
F. Lehner

- Colloq.Math. 90 (2001) 181-211
•
DOI:
- 10.4064/cm90-2-3

[5]

Stochastic calculus with respect to free Brownian motion and analysis on Wigner space

P. Biane
,
R. Speicher

- Probab. Theory Related Fields 112 (1998) 373
•
DOI:
- 10.1007/s004400050194

[6]

Analytic subordination theory of operator-valued free additive convolution and the solution of a general random matrix problem

ST Belinschi
,
T. Mai
,
R. Speicher

- J.Reine Angew.Math. 732 (2017) 21-53
•
DOI:
- 10.1515/crelle-2014-0138

[7]

Eigenvalues of non-Hermitian random matrices and Brown measure of non-normal operators: Hermitian reduction and linearization method

- Linear Algebra Appl. 537 (2018) 48-83,
- Linear Algebra and its Applications Volume 537, 15 January 2018, Pages 48-83
•
e-Print:
- 1506.02017
•
DOI:
- 10.1016/j.laa.2017.09.024

[8]

: Lidskiĭ's theorem in the type II case. In: Geometric methods in operator algebras (Kyoto, 1983), 1-35

L.G. Brown

[9]

: Dysonian dynamics of the Ginibre ensemble, Phys. Rev. Letters 113, article 104102

Z. Burda
,
J. Grela
,
M.A. Nowak
,
W. Tarnowski
,
P. Warcho

[10]

Unveiling the significance of eigenvectors in diffusing non-hermitian matrices by identifying the underlying Burgers dynamics

Zdzislaw Burda
(
- AGH-UST, Cracow
)
,
Jacek Grela
(
- Jagiellonian U.
)
,
Maciej A. Nowak
(
- Jagiellonian U.
)
,
Wojciech Tarnowski
(
- Jagiellonian U.
)
,
Piotr Warchoł
(
- Jagiellonian U.
)

- Nucl.Phys.B 897 (2015) 421-447
•
e-Print:
- 1503.06846
•
DOI:
- 10.1016/j.nuclphysb.2015.06.002

[11]

: Support of the Brown measure of the product of a free unitary Brownian motion by a free self-adjoint projection, J. Funct

N. Demni
,
T. Hamdi

The Brown measure of the free multiplicative Brownian motion

Bruce K. Driver
(
- UC, San Diego
)
,
Brian C. Hall
(
- U. Notre Dame, Dept. Math.
)
,
Todd Kemp
(
- UC, San Diego
)

- Probab.Theor.Related Fields 184 (2022) 1-2, 209-273,
- Probab.Theor.Related Fields 184 (2022) 209-273
•
e-Print:
- 1903.11015
•
DOI:
- 10.1007/s00440-022-01142-z

[13]

: Partial differential equations. Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, xxii+749 pp

L.C. Evans

[14]

The circular law

VL Girko

- Theory Probab. Appl. 29 (1984) 694
•
DOI:
- 10.1137/1129095

[15]

Elliptic law

VL Girko

- Theory Probab. Appl. 30 (1985) 677
•
DOI:
- 10.1137/1130089

[16]

: Eikonal formulation of large dynamical random matrix models, Phys. Rev. E

J. Grela
,
M.A. Nowak
,
W. Tarnowski

[17]

The single ring theorem

A. Guionnet
,
M. Krishnapur
,
O. Zeitouni

- Annals Math. 2 (2011) 1189-1217
•
DOI:
- 10.4007/annals.2011.174.2.10

[18]

PDE methods in random matrix theory

MT Rassias
,
BC Hall
,
MT Rassias

- Anal.Appl. 77 (2021) 124
•
DOI:
- 10.1007/978-3-030-61887-2_5

The Brown measure of a family of free multiplicative Brownian motions

Brian C. Hall
(
- Notre Dame U.
)
,
Ching-Wei Ho
(
- Notre Dame U. and
- Academia Sinica, Taiwan
)

- Probab.Theor.Related Fields 186 (2023) 3-4, 1081-1166,
- Probab.Theor.Related Fields 186 (2023) 1081
•
e-Print:
- 2104.07859
•
DOI:
- 10.1007/s00440-022-01166-5

e-Print:
- 2007.06100

e-Print:
- 2011.14222

[22]

Ho, C.-W

P. Zhong

[23]

NonHermitian random matrix models: A Free random variable approach

Romuald A. Janik
(
- Jagiellonian U.
)
,
Maciej A. Nowak
(
- Darmstadt, GSI and
- Darmstadt, Tech. Hochsch. and
- Jagiellonian U.
)
,
Gabor Papp
(
- Darmstadt, GSI and
- Eotvos U.
)
,
Jochen Wambach
(
- Darmstadt, Tech. Hochsch.
)
,
Ismail Zahed
(
- SUNY, Stony Brook
)

- Phys.Rev.E 55 (1997) 4100-4106
•
e-Print:
- hep-ph/9609491
•
DOI:
- 10.1103/PhysRevE.55.4100

[25]

Random Hermitian versus random non-Hermitian operators-unexpected links

A. Jarosz
,
MA Nowak

- J.Phys.A 39 (2006) 10107-10122
•
DOI:
- 10.1088/0305-4470/39/32/S12

1-25 of 40
1
2
25 / page