Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

$S$-transform in Finite Free Probability

Octavio Arizmendi, Katsunori Fujie, Daniel Perales, Yuki Ueda

TL;DR

This work builds a rigorous bridge between finite free probability and Voiculescu’s free probability by introducing a finite S-transform that encodes the coefficient structure of polynomials with nonnegative roots. It shows that, in the large-degree limit along diagonal index sequences, the finite S-transform converges to the classical S-transform, enabling a precise correspondence between the limiting root distribution and the multiplicative free convolution. The authors develop a comprehensive framework including finite S- and T-transforms, symmetry/unitarity extensions, derivative behavior via finite free cumulants, and a suite of finite analogues for key free-probability objects (Poisson laws, stable laws, max-convolution, etc.), with applications to approximation of free convolutions and to limit theorems of polynomial coefficients. These results deepen the conceptual link between polynomial root geometry, random matrices, and free probability, and supply concrete tools for finite-size approximations in spectral problems.

Abstract

We present a simplified explanation of why free fractional convolution corresponds to the differentiation of polynomials, by finding how the finite free cumulants of a polynomial behave under differentiation. This approach allows us to understand the limiting behaviour of the coefficients $\widetilde{\mathsf{e}}_k(p_d)$ of $p_d$ when the degree $d$ tends to infinity and the empirical root distribution of $p_d$ has a limiting distribution $μ$ on $[0,\infty)$. Specifically, we relate the asymptotic behaviour of the ratio of consecutive coefficients to Voiculescu's $S$-transform of $μ$. This prompts us to define a new notion of finite $S$-transform, which converges to Voiculescu's $S$-transform in the large $d$ limit. It also satisfies several analogous properties to those of the $S$-transform in free probability, including multiplicativity and monotonicity. This new insight has several applications that strengthen the connection between free and finite free probability. Most notably, we generalize the approximation of $\boxtimes_d$ to $\boxtimes$ and prove a finite approximation of the Tucci--Haagerup--Möller limit theorem in free probability, conjectured by two of the authors. We also provide finite analogues of the free multiplicative Poisson law, the free max-convolution powers and some free stable laws.

$S$-transform in Finite Free Probability

TL;DR

This work builds a rigorous bridge between finite free probability and Voiculescu’s free probability by introducing a finite S-transform that encodes the coefficient structure of polynomials with nonnegative roots. It shows that, in the large-degree limit along diagonal index sequences, the finite S-transform converges to the classical S-transform, enabling a precise correspondence between the limiting root distribution and the multiplicative free convolution. The authors develop a comprehensive framework including finite S- and T-transforms, symmetry/unitarity extensions, derivative behavior via finite free cumulants, and a suite of finite analogues for key free-probability objects (Poisson laws, stable laws, max-convolution, etc.), with applications to approximation of free convolutions and to limit theorems of polynomial coefficients. These results deepen the conceptual link between polynomial root geometry, random matrices, and free probability, and supply concrete tools for finite-size approximations in spectral problems.

Abstract

We present a simplified explanation of why free fractional convolution corresponds to the differentiation of polynomials, by finding how the finite free cumulants of a polynomial behave under differentiation. This approach allows us to understand the limiting behaviour of the coefficients of when the degree tends to infinity and the empirical root distribution of has a limiting distribution on . Specifically, we relate the asymptotic behaviour of the ratio of consecutive coefficients to Voiculescu's -transform of . This prompts us to define a new notion of finite -transform, which converges to Voiculescu's -transform in the large limit. It also satisfies several analogous properties to those of the -transform in free probability, including multiplicativity and monotonicity. This new insight has several applications that strengthen the connection between free and finite free probability. Most notably, we generalize the approximation of to and prove a finite approximation of the Tucci--Haagerup--Möller limit theorem in free probability, conjectured by two of the authors. We also provide finite analogues of the free multiplicative Poisson law, the free max-convolution powers and some free stable laws.
Paper Structure (41 sections, 51 theorems, 251 equations)

This paper contains 41 sections, 51 theorems, 251 equations.

Table of Contents

  1. Introduction
  2. -transform
  3. Quick overview of the proof
  4. Relation to Multiplicative Law of Large Numbers
  5. Organization of the paper
  6. Preliminaries
  7. Measures
  8. Polynomials
  9. Weak convergence
  10. Free Probability
  11. Free additive convolution and $R$-transform
  12. Free cumulants
  13. Free multiplicative convolution and $S$-transform
  14. Finite free probability
  15. Finite free convolutions
  16. ...and 26 more sections

Key Result

Theorem 1.1

Let $(p_d)_{d\in \mathbb{N}}$ be a sequence of polynomials with $p_d\in \mathcal{P}_d(\mathbb{R}_{\geq 0})$ and $\mu\in \mathcal{M}(\mathbb{R}_{\geq 0})\backslash \{\delta_0\}$. The following are equivalent:

Theorems & Definitions (121)

  • Theorem 1.1
  • Theorem 1.2: hoskins2020dynamics, AGP
  • Theorem 1.3
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Lemma 2.5: Ref. durrett2019probability
  • Definition 2.6: Free additive convolution
  • Definition 2.7: Fractional free convolution powers
  • Remark 2.8
  • ...and 111 more