Finite Free Information Inequalities

Abstract: We develop finite free information theory for real-rooted polynomials, establishing finite free analogues of entropy and Fisher information monotonicity, as well as the Stam and entropy power inequalities. These results resolve conjectures by Shlyakhtenko and Gribinski and recover inequalities in free probability in the large-degree limit. Equivalently, our results may be interpreted as potential-theoretic inequalities for the zeros of real-rooted polynomials under differential operators which preserve real-rootedness. Our proofs leverage a new connection between score vectors and Jacobians of root maps, combined with convexity results for hyperbolic polynomials.

• The paper introduces a finite analogue of free probability inequalities by translating information-theoretic quantities into root distributions of real-rooted polynomials.
• It proves the monotonicity of finite free Fisher information and establishes a finite free Stam’s inequality, with Hermite polynomials serving as extremizers.
• It provides a rigorous bridge between finite and infinite-dimensional settings, opening new avenues for applications in random matrix theory and combinatorial optimization.

Finite Free Information Inequalities: A Technical Overview

Introduction and Motivation

The paper "Finite Free Information Inequalities" (2602.15822) addresses information-theoretic inequalities in the finite setting of free probability, articulated through the behavior of real-rooted polynomials and their zeros under certain differential operators. The impetus is twofold: first, a classical problem in the Pólya–Schur program regarding the quantitative effects of real-rootedness-preserving linear operators on zero spacing; second, the development of a finite analogue of free probability and its associated information-theoretic constructs (entropy, Fisher information, Stam's and entropy power inequalities).

Through a judicious translation of information-theoretic quantities to the context of root distributions of polynomials, the paper proves finite analogues of key free probability inequalities, settling conjectures by Shlyakhtenko and Gribinski, and rigorously recovering their infinite-dimensional free probability counterparts as degree ($n$) tends to infinity.

Main Definitions and Constructs

Letting $p(x)=\prod_{i=1}^n (x-\alpha_i)$ denote a monic real-rooted polynomial, the following finite free analogues are defined:

• Finite Free Score Vector: $$J(\alpha) = \left(\sum_{j\ne i}\frac{1}{\alpha_i-\alpha_j}: 1\le i\le n\right)$$.
• Finite Free Fisher Information: $$\Phi_n(p) = \frac{1}{n}\sum_{i=1}^{n}\left(\frac{2}{n-1}\sum_{j\ne i}\frac{1}{\alpha_i-\alpha_j}\right)^2$$.
• Finite Free Entropy: $\rchi_n[p] = \frac{2}{n(n-1)}\sum_{i < j} \log|\alpha_i-\alpha_j| = \frac{1}{2\binom{n}{2}}\log\Disc(p)$, where $\Disc(p)$ is the discriminant of $p$.
• Finite Free Convolution: For polynomials $p(x)$ and $q(x)$, $p\boxplus_n q$ is defined through differential operators and also via symmetrized root matching: $$p\boxplus_n q(x) = \frac{1}{n!}\sum_{\pi\in S_n}\prod_{i=1}^n(x-\alpha_i-\beta_{\pi(i)})$$.

These quantities are direct analogues of Voiculescu's free entropy and Fisher information, where the spectral distribution of an operator is replaced by the empirical root distribution of a polynomial.

Main Results

Fisher Information Monotonicity

Considering the differential operator $T=\partial_x$, the paper proves:

Finite Free Fisher Information Monotonicity:

If $p(x)$ is real-rooted of degree $n$, and $\widetilde{p'}(x)$ is the appropriately rescaled derivative so that $Var(\widetilde{p'})=Var(p)$, then

$\Phi_{n-1}(\widetilde{p'}) \leq \Phi_n(p)$

with equality for Hermite polynomials. The proof involves both an explicit matrix-analytic bound using the Gauss–Lucas matrix and an alternative geometric proof via convexity of root maps under hyperbolic polynomials.

Stam's Inequality (Finite Free Setting)

For the finite free convolution,

$$\frac{1}{\Phi_n(p)} + \frac{1}{\Phi_n(q)} \leq \frac{1}{\Phi_n(p\boxplus_n q)}$$

where $p$ and $q$ are any real-rooted polynomials of degree $n$. Once again, the inequality is tight for Hermite polynomials. The proof exploits a novel connection between score vectors and the Jacobian of the root maps, drawing from the geometry of hyperbolic polynomials and relying on a convexity result of Bauschke et al.

Finite Free Entropy Monotonicity

Integrating the monotonicity of Fisher information along the heat flow yields:

If $\widetilde{p'}$ is as above, then

$\rchi_n[p] < \rchi_{n-1}[\widetilde{p'}]$

Moreover, the deviation from equality is quantified in terms of normalized Hermite polynomials. This result matches the differential entropy and free entropy monotonicity in the classical and free settings in the $n \to \infty$ limit.

Finite Free Entropy Power Inequality

Defining the entropy power as $N_n(p) = \exp(2\rchi_n[p])$, it is shown that

$N_n(p\boxplus_n q) \geq N_n(p) + N_n(q)$

for all monic real-rooted degree $n$ polynomials $p$ and $q$, with sharpness at Hermites. This is the finite free analogue of Szarek and Voiculescu's free entropy power inequality, derived by adapting the classical arguments of Dembo, Cover, and Thomas, and invoking the Jacobian convexity derived earlier.

Technical Approach and Key Innovations

• Root Map Jacobians and Score Vectors: The paper establishes a method for precisely relating the score vectors before and after application of a root-transforming operator, via tight singular value bounds on explicit Jacobians derived from the Gauss–Lucas theorem and the structure of finite free convolution.
• Hyperbolic Polynomial Convexity: Building on multivariate hyperbolic polynomial theory, the spectral convexity results are systematically adapted to control discriminants and log-Coulomb energy under finite free evolution.
• Analogue to Conditional Expectation: The Jacobians in the root maps play a role analogous to conditional expectation in probabilistic interpolations, which is key in information-theoretic inequalities despite the lack of genuine probability couplings in this deterministic, algebraic context.
• Sharpness via Hermite Polynomials: In all main inequalities, Hermite polynomials serve as extremizers, and the exact deviation from equality can be quantified via these cases, paralleling the role of Gaussians (respectively, semicircles) in the classical and free settings.

Implications and Connections

The results rigorously bridge finite free probability and its classical/free analogues, providing new proof techniques for infinite-dimensional inequalities by taking the large-$n$ limit of the finite results. Practically, this framework gives an explicit, computable regime of information inequalities at finite $n$, which is directly relevant for:

• Finite-dimensional random matrix theory: expected characteristic polynomials and their root dynamics.
• Quantitative analysis of randomized algorithms and spectral graph theory, where root spacing is an explicit quantity of interest.
• Further development of finite free probability, which underpins important combinatorial and operator-algebraic techniques.

On the theoretical side, these inequalities may illuminate the structure of finite-dimensional approximations of free entropy and Fisher information, with potential for new invariants in operator algebras and further connections to the geometry of polynomials, hyperbolic programming, and optimization.

Directions for Future Research

• Extending operator types: Beyond differential operators, understanding root dynamics under difference and $q$-difference operators; see connections to $q$-multiplicative convolutions.
• Multivariate and non-commutative analogues: Extending these inequalities to the setting of multivariate or matrix-valued polynomials, possibly relating to advances in non-commutative probability and free analysis.
• Applications to randomness extractors, spectral graph methods, and combinatorial optimization where finite free analogues may prove algorithmically meaningful.
• Deeper study of extremizers and stability: Extending sharpness results, stability analysis, and characterizing near-extremal polynomials in these frameworks.

Conclusion

"Finite Free Information Inequalities" (2602.15822) rigorously establishes finite analogues of canonical free probability entropy and Fisher information inequalities, utilizing an intersection of root map analysis, hyperbolic polynomials, and operator-theoretic insights. The resolution of conjectures, explicit sharp inequalities, and technical connections to both probabilistic and algebraic combinatorics illustrate the richness and depth now accessible in finite free probability. The methods and results in this work open avenues for both new theoretical understanding and practical analysis at the interface of probability, linear algebra, and algebraic combinatorics.