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

Free Compressions of R-Diagonal Random Variables and the Semigroup of Brown Measures

Introduction and Motivation

This paper investigates the Brown measures of compressions of R-diagonal random variables, significantly extending existing results to encompass unbounded operators. The study is motivated by developments in free probability theory and random matrix theory, particularly the behavior of eigenvalues under truncations of large unitarily invariant ensembles (e.g., truncated Haar unitary matrices). The analysis not only generalizes classical results to a broader class of operators (allowing infinite variance) but also provides an explicit characterization of stable Brown measures under the action of free compressions.

Brown Measures, R-Diagonal Operators, and Free Compression

Central to the paper is the interplay between free probability objects—R-diagonal operators, compressions, and additive convolution semigroups—and their spectral distributions described by Brown measures. R-diagonal operators serve as an abstraction of unitarily invariant non-normal random matrices and include, as special cases, Haar unitaries and circular elements.

Given a tracial $W^*$-probability space $(\mathcal{A}, \tau)$ and an R-diagonal (potentially unbounded) operator $x$ with $\tau(|x|^2) = 1$, the compression operation with respect to a projection $P_s$, free from $x$ and of normalized trace $1/s$, yields the compressed operator $s[P_s x P_s]$. In the limit $s \rightarrow \infty$, the Brown measure of such compressions converges weakly to the circular law—i.e., the uniform measure on the unit disk. This result generalizes the random matrix analogy where small truncations of large Haar unitary matrices yield limiting eigenvalue distributions that are circular.

The Semigroup Structure and Stability of Brown Measures

A critical innovation in the paper is the construction of a semigroup of Brown measures $\{\mu^{\boxplus s}\}_{s \geq 1}$ generated by an initial Brown measure $(\mathcal{A}, \tau)$0 of an R-diagonal operator $(\mathcal{A}, \tau)$1. The semigroup is constructed via analytic methods rather than free cumulants, allowing unbounded operators with merely finite (or even infinite) variance. The core properties of this semigroup are:

• Identity: $(\mathcal{A}, \tau)$2
• Associativity: $(\mathcal{A}, \tau)$3 coincides with the Brown measure of the sum of two free R-diagonal elements with Brown measures $(\mathcal{A}, \tau)$4 and $(\mathcal{A}, \tau)$5
• Continuity: The semigroup parameter $(\mathcal{A}, \tau)$6 is weakly continuous

The class of stable Brown measures is precisely characterized: $(\mathcal{A}, \tau)$7 is stable under this semigroup if and only if it is a dilation $(\mathcal{A}, \tau)$8 of a “canonical” stable measure $(\mathcal{A}, \tau)$9 (parameterized by $x$0), with explicit radial distribution function:

$x$1

This characterization strengthens previous works and shows that stability under free additive convolution coincides with stability under compression. In the finite variance case, the stable measure reduces to the uniform measure on the unit disk (the circular law).

Analytic Techniques and Explicit Results

The proof strategy avoids free cumulants (which are unavailable for unbounded operators) and instead utilizes analytic techniques, leveraging the S-transform and R-transform machinery developed by Haagerup and collaborators. The Brown measure of an $x$2-diagonal operator $x$3 (with $x$4 Haar unitary free from $x$5) is determined by the S-transform of $x$6. The compression operation on $x$7 is related to a semigroup acting on the spectral measure of $x$8, realized concretely through the compression of $x$9 and subsequent multiplication by a free Haar unitary.

Key explicit formulas developed in the paper include:

• For the S-transform under free compression:

$\tau(|x|^2) = 1$0

• For stable Brown measures, the S-transform is of the form:

$\tau(|x|^2) = 1$1

and the corresponding density exhibits a power-law tail with parameter determined by $\tau(|x|^2) = 1$2.

Quantitative results on the moments and tail behavior of these stable measures reveal a transition: for $\tau(|x|^2) = 1$3, the first moment is finite; for $\tau(|x|^2) = 1$4, it diverges. The density of the stable Brown measure $\tau(|x|^2) = 1$5 satisfies $\tau(|x|^2) = 1$6 for large $\tau(|x|^2) = 1$7.

Contrasts and Extensions

The study notably extends Campbell et al. (2024), which treated only bounded R-diagonal variables, by removing boundedness assumptions completely and providing new structural insights. In addition to strengthening the connection to random matrix theory (e.g., by relating compression to truncation of Haar unitaries), it offers new explicit formulas for the Brown measures of non-Hermitian polynomials and the corresponding spectral radii.

The explicit characterization of stable measures in terms of analytic functionals of the S-transform sets the groundwork for systematic analysis of limiting spectral distributions in random matrix products and sums, relevant for the investigation of non-Hermitian random systems beyond classical cases.

Implications and Future Directions

The work provides a comprehensive analytic framework for understanding the spectral (Brown measure) behavior of compressions of R-diagonal operators, including those with unbounded moments. Given the ubiquity of R-diagonal elements in the asymptotic theory of random non-Hermitian matrices, the techniques developed here are likely to impact several research threads:

• Analysis of sums and products of random matrices with heavy-tailed entries (infinite variance)
• Study of limiting eigenvalue distributions in truncated unitary and orthogonal ensembles
• Further development of free probability, especially infinite divisibility and stability in the noncommutative context
• Applications to zeros of random polynomials and operator-valued free probability

Potential extensions include the treatment of more general non-normal operators, multidimensional free convolution semigroups, and finer properties of outlier eigenvalues and singular values (particularly for heavy-tailed regimes).

Conclusion

This paper establishes a detailed analytic theory for the Brown measures of compressions of (possibly unbounded) R-diagonal random variables, fully characterizing the semigroup structure generated by compression and identifying the corresponding class of stable measures. The results bridge and generalize aspects of free probability and random matrix theory, with explicit operator-theoretic and probabilistic consequences. The analytic machinery developed herein paves the way for further rigorous studies of spectral properties in large random systems and noncommutative probability frameworks.