Free compressions of R-diagonal random variables and the semigroup of Brown measures

Vladislav Kargin

arxiv: 2604.22114 · v1 · submitted 2026-04-23 · 🧮 math.PR · math.OA

Free compressions of R-diagonal random variables and the semigroup of Brown measures

Vladislav Kargin This is my paper

Pith reviewed 2026-05-09 20:06 UTC · model grok-4.3

classification 🧮 math.PR math.OA

keywords R-diagonal random variablesBrown measurefree compressionfree probabilitycircular lawunbounded operatorssemigroup of measures

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The pith

Brown measures of free compressions of R-diagonal random variables with finite variance converge to the uniform distribution on the unit disc.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies the effect of free compressions on the Brown measures of R-diagonal random variables in free probability, extending earlier bounded-operator results to the unbounded setting. It proves that when variance is finite the Brown measures of the compressed variables approach the uniform measure on the unit disc. For infinite-variance cases the work identifies exactly which Brown measures are fixed by the compression map and describes their main analytic features. These statements matter because Brown measures encode the joint spectral distribution of non-normal operators, and the compression operation models restriction to a subspace in a non-commutative probability space.

Core claim

We investigate the Brown measures of compressions of R-diagonal random variables, extending previous results to include unbounded cases. For random variables with finite variance, we demonstrate that the Brown measures of their compressions converge to the uniform distribution on the unit disc. In the case of infinite variance, we characterize the Brown measures that remain stable under the compression operation and explore their properties in detail.

What carries the argument

The free compression operation on R-diagonal elements, which induces a transformation on the associated Brown measure and generates a semigroup of measures under iteration.

If this is right

Repeated free compressions of a finite-variance R-diagonal variable produce Brown measures that fill the unit disc uniformly.
The fixed points of the compression map on Brown measures exist in the infinite-variance regime and admit explicit characterizations via free-probability moment relations.
The family of all such Brown measures closed under the compression operation forms a semigroup.
Properties of the stable measures include rotational symmetry and support contained in closed annuli centered at the origin.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The compression map defines a dynamical system on the space of probability measures on the complex plane whose attractors include the circular law.
The same mechanism may describe the spectral effect of taking principal submatrices in large random matrices that approximate R-diagonal operators.
Further study of the semigroup could identify additional invariant measures or contraction rates toward the circular law.

Load-bearing premise

The Brown measure is well-defined for the given unbounded R-diagonal operators and the compression map extends to this class while preserving the R-diagonal property.

What would settle it

An explicit example of an unbounded R-diagonal operator with finite variance whose Brown measure after compression fails to approach the uniform distribution on the unit disc would disprove the convergence statement.

read the original abstract

We investigate the Brown measures of compressions of $R$-diagonal random variables, extending previous results to include unbounded cases. For random variables with finite variance, we demonstrate that the Brown measures of their compressions converge to the uniform distribution on the unit disc. In the case of infinite variance, we characterize the Brown measures that remain stable under the compression operation and explore their properties in detail.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Kargin extends prior bounded-operator results on Brown measures of R-diagonal compressions to the unbounded setting, with convergence to uniform disk measure for finite variance and a characterization of stable measures for infinite variance.

read the letter

The main thing to know about this paper is that it extends results on the Brown measures of free compressions of R-diagonal elements from the bounded to the unbounded setting. For operators with finite variance the Brown measures of the compressions converge to the uniform distribution on the unit disk, and for infinite variance it gives a characterization of the measures that stay stable under compression along with some of their properties. What the paper does well is carry out this extension carefully. Free probability work often stays with bounded operators because the functional calculus and distributions are easier to handle, so moving to unbounded ones requires attention to domains and approximations. The convergence claim for finite variance seems to follow from prior bounded results by some limiting argument, and the stable measures for infinite variance are described in detail. The title mentions a semigroup of Brown measures, which suggests they look at the family of compressions as a continuous parameter and study the evolution of the measure. The soft spots are mostly around the technical details of the unbounded case. Defining the Brown measure for unbounded operators involves some care with the resolvent or the distribution, and the compression operation needs to be well-defined in the free probability algebra. If the paper provides the necessary justifications without hidden assumptions, it should be fine. The stress-test note indicates no obvious internal problems once the extension is accepted, so the central argument appears to hold. The citation pattern is standard for the field, building on earlier work without circularity. This paper is aimed at researchers in free probability, particularly those interested in Brown measures, R-diagonal elements, and their connections to random matrix theory. A reader who knows the bounded case results will see the value in the unbounded extension and the stability characterization. It is not broad enough for a general audience but fits well in the specialized literature. I would send this to peer review. The results look like genuine additions that deserve checking by experts in the area, even if revisions might be needed on the unbounded technicalities.

Referee Report

0 major / 2 minor

Summary. The manuscript extends prior results on Brown measures of free compressions of R-diagonal random variables from the bounded to the unbounded setting. For R-diagonal elements with finite variance, it demonstrates convergence of the Brown measures of their compressions to the uniform distribution on the unit disc. For the infinite-variance case, it characterizes those Brown measures that remain invariant under the compression operation and studies their detailed properties, including the associated semigroup structure.

Significance. If the technical extensions to unbounded operators hold, the work supplies concrete convergence and stability results that clarify the behavior of Brown measures under free compression. The finite-variance limit to uniform measure on the disc and the explicit characterization of stable measures in the infinite-variance regime are potentially useful for further investigations of non-commutative distributions and semigroups in free probability.

minor comments (2)

The abstract and introduction should explicitly cite the specific prior bounded-operator results (e.g., the theorems being extended) so that the precise novelty of the unbounded extension is immediately clear to readers.
Notation for the compression operation and the Brown measure should be introduced with a brief reminder of the definitions in the unbounded case, even if they follow standard references, to improve readability for a broader audience.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. We are pleased that the extensions to the unbounded setting, the convergence result for finite-variance cases, and the characterization of stable measures are viewed as useful contributions.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper extends established results from free probability on Brown measures of compressions of R-diagonal elements to the unbounded-operator setting. The finite-variance claim (convergence to uniform measure on the unit disc) and the infinite-variance characterization of stable measures are derived from the algebraic properties of free compressions and the definition of Brown measure, without any step that reduces by construction to a parameter fitted inside the paper or to a self-citation whose content is itself defined by the present work. No self-definitional loops, fitted-input predictions, or ansatz smuggling via citation appear in the derivation chain. The argument remains self-contained once the standard free-probability framework and the bounded-to-unbounded extension are granted.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the paper operates inside the established framework of free probability without introducing new free parameters or postulated entities.

axioms (1)

standard math Standard axioms and definitions of free probability theory, including the notion of R-diagonal elements and Brown measures
The entire investigation presupposes the existing free-probability setup as background.

pith-pipeline@v0.9.0 · 5345 in / 1245 out tokens · 86476 ms · 2026-05-09T20:06:06.967250+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

[1]

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work page 2005
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Dong, Z., Jiang, T., Li, D.: Circular law and arc law for truncation of random unitary matrix. J. Math. Phys.53, 013301 (2012)

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K¨ osters, H., Tikhomirov, A.: Limiting spectral distributions of sums of products of non-Hermitian random matrices. Probab. Math. Stat.38, 359–384 (2018)

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[8]

Haagerup, U., Larsen, F.: Brown’s spectral distribution measure forR-diagonal elements in finite von Neumann algebras. J. Funct. Anal.176, 331–367 (2000)

work page 2000
[9]

Haagerup, U., Schultz, H.: Brown measures of unbounded operators affiliated with a finite von Neumann algebra. Math. Scand.100, 209–263 (2007)

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[10]

Technical report (2013)

Haagerup, U., M¨ oller, S.: The law of large numbers for the free multiplicative convolution. Technical report (2013). https://arxiv.org/pdf/1211.4457.pdf

work page arXiv 2013
[11]

Belinschi, S.T., Bercovici, H.: Atoms and regularity for measures in a partially defined free convolution semigroup. Math. Z.248, 665–674 (2004)

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Voiculescu, D.: Symmetries of some reduced free product C ∗-algebras. In: Lecture Notes in Mathematics vol. 1132, pp. 556–588. Springer, Berlin (1983)

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Indiana Univ

Bercovici, H., Voiculescu, D.: Free convolution of measures with unbounded support. Indiana Univ. Math. J.42, 733–773 (1993)

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Arizmendi, O., P´ erez-Abreu, V.: TheS-transform of symmetric probability measures with unbounded supports. Proc. Am. Math. Soc.137, 3057–3066 (2009) 23

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[1] [1]

International Mathematics Research Notices13, 10189–10218 (2024)

Campbell, A., O’Rourke, S., Renfrew, D.: The fractional free convolution of R-diagonal operators and random polynomials under repeated differentiation. International Mathematics Research Notices13, 10189–10218 (2024)

work page 2024

[2] [2]

Diaconis, P.: Patterns in eigenvalues: the 70th Josiah Willard Gibbs lecture. Bull. Am. Math. Soc.40(2003)

work page 2003

[3] [3]

Probab.34, 1497–1529 (2006)

Jiang, T.: How many entries of a typical orthogonal matrix can be approximated by independent normals? Ann. Probab.34, 1497–1529 (2006)

work page 2006

[4] [4]

˙Zyczkowski, K., Sommers, H.-J.: Truncations of random unitary matrices. J. Phys. A: Math. Gen.33, 2045–2057 (2000) 22

work page 2045

[5] [5]

Petz, D., R´ effy, J.: Large deviation for the empirical eigenvalue density of truncated Haar unitary matrices. Probab. Theory Relat. Fields133, 175–189 (2005)

work page 2005

[6] [6]

Dong, Z., Jiang, T., Li, D.: Circular law and arc law for truncation of random unitary matrix. J. Math. Phys.53, 013301 (2012)

work page 2012

[7] [7]

K¨ osters, H., Tikhomirov, A.: Limiting spectral distributions of sums of products of non-Hermitian random matrices. Probab. Math. Stat.38, 359–384 (2018)

work page 2018

[8] [8]

Haagerup, U., Larsen, F.: Brown’s spectral distribution measure forR-diagonal elements in finite von Neumann algebras. J. Funct. Anal.176, 331–367 (2000)

work page 2000

[9] [9]

Haagerup, U., Schultz, H.: Brown measures of unbounded operators affiliated with a finite von Neumann algebra. Math. Scand.100, 209–263 (2007)

work page 2007

[10] [10]

Technical report (2013)

Haagerup, U., M¨ oller, S.: The law of large numbers for the free multiplicative convolution. Technical report (2013). https://arxiv.org/pdf/1211.4457.pdf

work page arXiv 2013

[11] [11]

Belinschi, S.T., Bercovici, H.: Atoms and regularity for measures in a partially defined free convolution semigroup. Math. Z.248, 665–674 (2004)

work page 2004

[12] [12]

In: Lecture Notes in Mathematics vol

Voiculescu, D.: Symmetries of some reduced free product C ∗-algebras. In: Lecture Notes in Mathematics vol. 1132, pp. 556–588. Springer, Berlin (1983)

work page 1983

[13] [13]

A.M.S., Provi- dence, RI (1992)

Voiculescu, D., Dykema, K., Nica, A.: Free Random Variables. A.M.S., Provi- dence, RI (1992). CRM Monograph series, No.1

work page 1992

[14] [14]

London Math

Nica, A., Speicher, R.: Lectures on the Combinatorics of Free Probability. London Math. Soc. Lecture Note Ser., vol. 335. Cambridge University Press, Cambridge (2006)

work page 2006

[15] [15]

Nica, A., Speicher, R.: On the multiplication of freen-tuples of noncommutative random variables. Am. J. Math.118, 799–837 (1996)

work page 1996

[16] [16]

Indiana Univ

Bercovici, H., Voiculescu, D.: Free convolution of measures with unbounded support. Indiana Univ. Math. J.42, 733–773 (1993)

work page 1993

[17] [17]

Arizmendi, O., P´ erez-Abreu, V.: TheS-transform of symmetric probability measures with unbounded supports. Proc. Am. Math. Soc.137, 3057–3066 (2009) 23

work page 2009