Monotone max-convolution and subordination functions for free max-convolution

Abstract: We show that the distribution of the spectral maximum of monotonically independent self-adjoint operators coincides with the classical max-convolution of their distributions. In free probability, it was proven that for any probability measures $σ,μ$ on $\mathbb{R}$ there is a unique probability measure $\mathbb{A}σ(μ)$ satisfying $σ\boxplus μ= σ\triangleright \mathbb{A}σ(μ)$, where $\boxplus$ and $\triangleright$ are free and monotone additive convolutions, respectively. We recall that the reciprocal Cauchy transform of $\mathbb{A}_σ(μ)$ is the subordination function for free additive convolution. Motivated by this analogy, we introduce subordination functions for free max-convolution and prove their existence and structural properties.

• The paper introduces a formal framework for monotone max-convolution, proving its coincidence with classical max-convolution and highlighting its commutative properties.
• It constructs unique subordination functions for free max-convolution analogous to the additive case, demonstrating composition, distributivity, and power stability.
• The work bridges extreme value theory with non-commutative probability, offering novel analytical tools and setting the stage for advances in operator-valued distributions.

Monotone Max-Convolution and Subordination Functions for Free Max-Convolution

Introduction and Context

This work rigorously investigates max-convolutions within the framework of monotone independence for non-commutative random variables, simultaneously establishing new connections to free probability and extreme value theory. The core contribution lies in identifying and classifying the structure of the spectral maximum for monotonically independent self-adjoint operators, then extending the apparatus of subordination functions—traditionally instrumental in additive free convolution—to the theory of free max-convolution. The analysis elucidates the relationships between monotone, free, and Boolean max-convolutions, with consequential implications for the classification and combinatorics of extreme value distributions in non-commutative probability.

Monotone Max-Convolution: Spectral Maxima and Classical Correspondence

Monotone independence, as introduced by Muraki, is characterized by an asymmetric, universal combinatorics of independence distinct from tensor, free, or Boolean independence. For self-adjoint operators $X$ and $Y$ that are monotonically independent with respect to a vector state $\varphi_\xi$, this work shows that the spectral distribution of their spectral maximum (according to Ando-Olson's operator order) is given by the classical max-convolution of their individual distributions:

$$\varphi_\xi(E_{X\lor Y}((-\infty,x])) = F_\mu(x)F_\nu(x)$$

where $F_\mu, F_\nu$ are the distribution functions of $X$ and $Y$. In other words, the monotone max-convolution

$F_{\mu\triangledown \nu}(x) = F_\mu(x)F_\nu(x)$

coincides exactly with its classical (commutative) counterpart, a property that is absent for monotone additive convolution. The result is commutativity for the monotone max-convolution, unlike the generic noncommutativity of monotone additive convolution.

This structural alignment with classical max-convolution sharply contrasts with the functional forms for free and Boolean max-convolutions, highlighting a unique commutative regime within monotone max-operations on the operator level.

Subordination for Free Max-Convolution

The analytic study of free additive convolution $\boxplus$ relies heavily on the concept of subordination functions, mapping the complexities of free sums onto analytically tractable domains. In additive theory, for $\sigma,\mu$ probability measures, there exists a unique measure $\mathbb{A}_{\sigma}(\mu)$ (Belinschi-Bercovici subordination) such that

$$\sigma \boxplus \mu = \sigma \triangleright \mathbb{A}_\sigma(\mu)$$

with subordination functions acting via reciprocal Cauchy transforms.

The main technical achievement here is the construction of subordination functions for free max-convolution. For probability measures $\sigma, \mu$, the free max-convolution is defined by

$$F_{\sigma \Box \lor \mu}(x) = \max\{ F_\sigma(x) + F_\mu(x) - 1, 0 \}$$

and the author shows that, analogous to the additive case, for each pair $(\sigma,\mu)$ there exists a unique probability measure $\mathbb{A}_\sigma^\lor(\mu)$ such that:

$$\sigma \Box \lor \mu = \sigma \lor \mathbb{A}_\sigma^\lor(\mu)$$

where the subordination function is

$$F_{\mathbb{A}_\sigma^\lor(\mu)}(x) = \max\left\{1 - \frac{\overline{F}_\mu(x)}{F_\sigma(x)},\,0\right\}$$

with $\overline{F}_\mu=1-F_\mu$. This definition is justified by showing the recursive and structural properties required of subordination functions, including composition, distributivity over free max-convolution powers, and compatibility with Boolean max-convolution under certain restrictions.

Structural and Algebraic Properties

The subordination function $\mathbb{A}_\sigma^\lor$ exhibits several key properties, directly paralleling those of additive subordination:

1. Composition: $$\mathbb{A}_{\sigma_1}^\lor \circ \mathbb{A}_{\sigma_2}^\lor = \mathbb{A}_{\sigma_1\lor\sigma_2}^\lor$$
2. Distribution under Convolution: $$\mathbb{A}_\sigma^\lor(\mu_1 \Box \lor \mu_2) = \mathbb{A}_\sigma^\lor(\mu_1)\Box \lor \mathbb{A}_\sigma^\lor(\mu_2)$$
3. Power Stability: $$\mathbb{A}_\sigma^\lor(\mu^{\Box \lor t}) = \mathbb{A}_\sigma^\lor(\mu)^{\Box \lor t}$$ for convolution powers $t\geq 1$
4. Boolean Max-Convolution Correspondence: For non-negative support, $$\sigma\Box\lor\mu = \mathbb{A}_\sigma^\lor(\mu) \cup\lor \mathbb{A}_\mu^\lor(\sigma)$$

These properties provide a robust algebraic framework for extreme value behavior in the non-commutative setting, and crucially, facilitate hierarchies of operations paralleling those in free, Boolean, and monotone probability.

Implications and Future Directions

The complete identification of monotone max-convolution with the classical case is a strong structural claim, positioning monotone independence as a unique bridge between commutative and noncommutative extreme value phenomena. The subordination framework for free max-convolution opens analytic avenues for limit theorems, universality classes, and operator inequalities, paralleling the foundational role played by their additive analogues in free probability.

Open problems posed include:

• Extending the relationships between additive and max-convolutions to the monotone setting, generalizing results known for free, Boolean, and classical convolutions.
• Investigating whether the commutativity of monotone max-convolution has a conceptual explanation via graph product structures or broader universal products (e.g., via the recently introduced BMT independence [Arizmendi et al., 2025]).

These directions suggest that a unified extreme value theory for all universal independences may be attainable, with deep combinatorial and analytical consequences.

Conclusion

This work integrates the interplay between monotone, free, and Boolean max-convolutions in non-commutative probability, characterizing the spectral distribution of maximums via monotone independence and advancing the analytic apparatus for free max-convolution using subordination functions. The clarification of algebraic and functional properties paves the way for novel developments in extreme value theory, with potential impact on the general understanding of non-commutative probability, random matrices, and the combinatorics of operator-valued distributions.