Finite Free S-Transform Overview

• Finite free S-transform is a discrete analogue of Voiculescu's S-transform, linearizing finite free multiplicative convolution.
• It employs combinatorial methods using normalized elementary symmetric sums to recursively recover coefficients in finite settings.
• Its properties—multiplicativity, monotonicity, and duality—enable precise analytic control in random matrix and free probability models.

The finite free S-transform is a discrete analogue of Voiculescu's classical S-transform from free probability, adapted to settings where objects (typically random matrices, polynomials, or measures) are of finite size or degree. It provides an analytic and combinatorial tool for linearizing finite free multiplicative convolution, establishing a concrete bridge between classical (finite) polynomial or matrix models and infinite-dimensional limiting distributions.

1. Motivation and Definition

At its core, the finite free S-transform arises from the paper of monic real-rooted polynomials of fixed degree $d$ with roots in $[0,\infty)$ or strictly in $(0,\infty)$. Let

$$p(x) = \sum_{k=0}^d (-1)^k \binom{d}{k} \widetilde{e}_k(p) x^{d-k}$$

where $\widetilde{e}_k(p)$ is the normalized $k$-th elementary symmetric sum of the roots. The finite free S-transform $S_p^{(d)}$ is a discrete function on the mesh $-\frac{k}{d}$ for $k=1,2,\dots,d-r$ (with $r$ the multiplicity of the root $0$), given by

$$S_p^{(d)}\left(-\frac{k}{d}\right) = \frac{\frac{k-1}{d}\, \widetilde{e}_{k-1}(p)}{\widetilde{e}_k(p)}.$$

Alternatively, one introduces the piecewise-constant function

$$\mathcal{S}_p^{(d)}(z) = S_p^{(d)}\left(-\frac{\lceil dz\rceil}{d}\right),\qquad z\in(0,1-r/d),$$

and the associated finite T-transform, normalized by

$$T_p^{(d)}(z) = \begin{cases} 0, & 0<z<r/d, \ \dfrac{d-k+1}{k} \,\dfrac{e_{d-k+1}(p)}{e_{d-k}(p)}, & \frac{k-1}{d} \leq z < \frac{k}{d},\quad k=r+1,\dots,d. \end{cases}$$

These transforms recover all normalized coefficients recursively and structure the step from finite combinatorics to analytic free probability.

2. Properties and Convergence to the Classical S-Transform

The finite S-transform $S_p^{(d)}$ satisfies analogues of the fundamental properties of the Voiculescu S-transform:

• Multiplicativity: For $p,q \in_d([0,\infty))$,

$$S_{p \boxtimes_d q}^{(d)}\left(-\frac{k}{d}\right) = S_p^{(d)}\left(-\frac{k}{d}\right) S_q^{(d)}\left(-\frac{k}{d}\right).$$

• Monotonicity: If $p$ has at least two distinct positive roots,

$$S_p^{(d)}\left(-\frac{k+1}{d}\right) > S_p^{(d)}\left(-\frac{k}{d}\right),$$

i.e., the function is strictly decreasing in $k$.

• Reversal (Duality) Identity: For the reversed polynomial $p^\vee(x) = x^d p(1/x)$,

$$S_p^{(d)}\left(-\frac{k}{d}\right) \cdot S_{p^\vee}^{(d)}\left(-\frac{d+1-k}{d}\right) = 1.$$

As $d\to\infty$, with empirical root distributions $\mu_d$ converging weakly to a law $\mu\ne\delta_0$, for any $t\in(0,1-\mu(\{0\}))$ and $k(d)/d\to t$,

$$\lim_{d\to\infty} S_{p_d}^{(d)}\left(-\frac{k(d)}{d}\right) = S_\mu(-t),$$

where $S_\mu$ is Voiculescu's S-transform of $\mu$ on $(-1+\mu(\{0\}),0)$, i.e.,

$$S_\mu(z) = \frac{1+z}{z}\, \Psi_\mu^{-1}(z), \quad \Psi_\mu(w) = \int \frac{tw}{1-tw}\,\mu(dt).$$

3. S-Transform for Finite Random Matrices and Multiplicative Spherical Integrals

In the context of finite $N \times N$ Hermitian matrices $X_N$ with positive spectrum, the finite free S-transform $S_N(\theta)$ serves as a functional parameterization of spherical integrals that generalize the Harish-Chandra–Itzykson–Zuber (HCIZ) integral:

• Define the empirical law $\mu_N$ and its Stieltjes transform $G_{\mu_N}(z)$;
• The finite-$N$ T-transform is $T_{\mu_N}(z) = z G_{\mu_N}(z) - 1$;
• The finite-$N$ S-transform is constructed by inverting $T_{\mu_N}$ up to the maximal eigenvalue cutoff:

$$S_N(\theta) = \begin{cases} T_{\mu_N}^{-1}(\theta), & 0 < \theta < T_{\mu_N}(\lambda_N), \ \lambda_N, & \theta \geq T_{\mu_N}(\lambda_N), \end{cases}$$

where $\lambda_N$ is the largest eigenvalue.

The large-$N$ limit produces

$$\lim_{N\to\infty} S_N(\theta) = \min\left\{\widehat{S}_\mu(\theta), \lambda_*\right\}$$

with $\widehat{S}_\mu(\theta)=T_\mu^{-1}(\theta)$ and $\lambda_* = \lim \lambda_N(X_N)$. The principal tool in the proof is a rigorous large deviations analysis leveraging Varadhan's lemma and the method of successive conditioning.

4. Analytic Toolkit and Functional Relations

The finite free S-transform fits into a larger framework of analytic transforms in free and finite free probability:

• The Cauchy (Stieltjes) transform: $G_\mu(z) = \int \frac{1}{z-x}\,\mu(dx)$;
• The F-transform: $F_\mu(z) = 1/G_\mu(z)$;
• The $\psi$- and $\eta$-transforms: $\psi_\mu(z) = \frac{1}{z}G_\mu(1/z)-1$ and $\eta_\mu(z) = 1 - \frac{z}{G_\mu(1/z)}$;
• The Voiculescu R-transform: $C_\mu(z)=zG_\mu^{-1}(z)-1$;
• The finite S- and T-transforms as detailed above.

Subordination functions $\omega_1, \omega_2$ satisfying

$$\eta_{\mu \boxtimes \nu}(z) = \eta_\mu(\omega_1(z)) = \eta_\nu(\omega_2(z)) = \frac{\omega_1(z)\omega_2(z)}{z}$$

serve as the analytic key to extending many infinite-dimensional convolution identities and regularity results to the finite setting.

The fundamental inversion formula for the classical S-transform,

$\psi_\mu\left(\frac{u}{1+u} S_\mu(u)\right) = u$

finds a discrete counterpart in the finite setting, linking ratios of elementary symmetric sums to Stieltjes/Cauchy values.

5. Applications to Free Stable Laws and Random Matrix Models

Finite free S-transform techniques enable concrete computations and limit transitions for several classes of laws and models:

• Free and Boolean stable laws: Their S-transforms can be computed and manipulated in closed form:

$$S_{b_{\alpha,\rho}}(u) = -e^{i\rho \pi} \left(\frac{-u}{1+u}\right)^{\frac{1-\alpha}{\alpha}},\quad S_{f_{\alpha,\rho}}(u) = -e^{i\rho\pi} (-u)^{\frac{1-\alpha}{\alpha}}$$

for $-1 < u < 0$.

• Reproducing identities for convolution powers, e.g.,

$$b_{\alpha,\rho}(b_{\beta,1})^{1/\alpha} = b_{\alpha\beta,\rho}, \qquad f_{\alpha,\rho}(f_{\beta,1})^{1/\alpha} = f_{\alpha\beta,\rho}$$

follow directly from multiplicativity.

• Jacobi processes: By analyzing averaged characteristic polynomials and the evolution under Jacobi diffusions, the finite free S-transform provides a discrete-to-continuous route for understanding crystallization and bulk limit phenomena in random matrix ensembles. The finite difference convergence result

$$\nabla^{(d)}[\mathcal{S}_{p_d}^{(d)}](v) \to \partial_v S_\mu(v)$$

with $\nabla^{(d)}f(v)=d[f(v+1/d)-f(v)]$ establishes rigorous links between finite and infinite objects.

6. Algebraic and Combinatorial Framework

The S-transform in finite and arbitrary dimensions admits an algebraic-geometric description:

• Moments and free cumulants are encoded in non-commutative power series rings.
• The boxed convolution $\boxtimes$ structures the group law underpinning multiplicative free convolution.
• For $s$-tuples, the S-transform is defined as the image under a minimal faithful representation $\rho_s$ of the underlying affine group scheme, embedding them in Borel subgroups.
• In one dimension, this machinery reduces to

$S(z) = \frac{1+z}{z} M^{\langle -1 \rangle}(z)$

with $M^{\langle -1 \rangle}$ the functional inverse of the moment series.

This framework ensures that the only image-groups seen under the S-transform are solvable (pro-)algebraic groups or their semi-direct products with tori, giving a precise algebraic taxonomy of the "symmetries" generated by finite free convolution.

7. Summary Table: Key Definitions and Relations

Object	Notation / Formula	Context / Domain
Finite free S-transform	$$S_p^{(d)}(-k/d) = \frac{\frac{k-1}{d} \widetilde{e}_{k-1}(p)}{ \widetilde{e}_k(p) }$$	Polynomials of degree $d$, $k=1,\dots,d-r$
Finite free T-transform	$T_p^{(d)}(t) = 1/S_p^{(d)}(-\lfloor td\rfloor/d)$	Step function on $(0,1)$
Classical S-transform	$S_\mu(z) = \frac{1+z}{z}\, \Psi_\mu^{-1}(z)$, $\Psi_\mu(w)=\int \frac{tw}{1-tw} \,\mu(dt)$	Measures on $[0,\infty)$
Multiplicativity	$$S_{p \boxtimes_d q}^{(d)}(-k/d) = S_p^{(d)}(-k/d)\,S_q^{(d)}(-k/d)$$	All $k$ in admissible range
Convergence (finite to free)	$S_{p_d}^{(d)}(-k(d)/d) \to S_\mu(-t)$ for $k(d)/d\to t$, $\mu_{p_d}\to\mu$	$d\to\infty$, weak convergence

The finite free S-transform thus constitutes a unifying analytic and combinatorial apparatus for understanding finite models and their asymptotics within free probability, preserving the salient algebraic and analytic features of the infinite-dimensional S-transform and enabling precise control over convergence, regularity, and multiplicative identities.