Variational formula for the logarithmic potential of free additive convolutions

David Belius; Francesco Concetti; Giuseppe Genovese

Variational formula for the logarithmic potential of free additive convolutions

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Pith's one-line read For any compactly supported $\mu,\nu$ with $\mu$ non-degenerate, the logarithmic potential of $\mu\boxplus\nu$ below the left edge of its support is the infimum of an explicit one-variable functional, attained uniquely at…

desk verdict A clean, useful variational formula for the logarithmic potential of free additive convolutions, with a long but coherent proof; the main soft spot is the analytic-continuation bridge, not the core argument. read the letter →

arxiv 2506.19064 v1 pith:RLT4PFRW submitted 2025-06-23 math.PR

Francesco Concetti , David Belius , Giuseppe Genovese This is my paper · ORCID

classification math.PR MSC 46L5431A1060B20

keywords freeadditiveconvolutionlogarithmicpotentialR-transformvariationalformulaStieltjestransformrandommatrixdeterminantsemicirclelawMarchenko-Pastur

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes a variational formula for the logarithmic potential $U_{\mu\boxplus\nu}(z)=\int\log|z-\lambda|\,(\mu\boxplus\nu)(d\lambda)$ of the free additive convolution of two compactly supported probability measures on $\mathbb{R}$. For every real $z$ below the left edge $z^*_{\mu\boxplus\nu}$ of the support of $\mu\boxplus\nu$, the potential is shown to equal the infimum, over $g\in(0,g^*_{\mu\boxplus\nu})$, of an explicit functional $E_{\mu,\nu,z}(g)$ built from the $R$-transform of $\mu$ and the logarithmic potential of $\nu$; the unique minimizer is the Stieltjes transform $G_{\mu\boxplus\nu}(z)$. When $\mu$ is the semicircle law or the Marchenko-Pastur law, the formula reduces to explicit one-line expressions. Logarithmic potentials of additive convolutions control the exponential asymptotics of determinants of sums of independent random matrices, which appear in random-landscape problems such as spin-glass complexity and neural-network loss surfaces.

What carries the argument

The central object is the real $R$-transform, defined as the unique maximal real-analytic extension $R_{\mu}$ of $\widehat R_{\mu}(t)=G_{\mu}^{-1}(-t)-t^{-1}$ on a maximal interval $D_{\mu}$; the free additive convolution is the law whose real $R$-transform equals $R_{\mu}+R_{\nu}$ on $D_{\mu}\cap D_{\nu}$. The carrying identity is the derivative formula $$E'_{\mu,\nu,z}(g)=R'_{\mu}(-g)\bigl(g-G_{\nu}(z-R_{\mu}(-g))\bigr),$$ whose fixed-point equation is equivalent to $z=G_{\mu\boxplus\nu}^{-1}(g)$ for $g\in(0,g^*)$. Because $R'_{\mu}(-g)>0$ on that interval, the sign of the derivative is controlled by whether $g$ lies below or above $G_{\mu\boxplus\nu}(z)$, making the unique critical point the global minimum inside $(0,g^*)$. The companion function $F_{\mu,\nu}(h)=R_{\mu}(-G_{\nu}(h))+h$ encodes the edges of the support through $g^*=G_{\nu}(h^*)$ and $z^*=F_{\mu,\nu}(h^*)$, with $h^*$ the leftmost critical point of $F$ or the right endpoint of its domain.

What would settle it

Take an explicit pair with known R-transform and a two-point $\nu$, for instance the Figure 1c example $R_\mu(t)=t+t^2/6$, $\nu=(\delta_{-0.17}+\delta_{-1.17})/2$, $z=-2.77$; compute $U_{\mu\boxplus\nu}(z)$ from the Stieltjes transform of the convolution via the identity chain (1.18), and compare with the infimum of $E_{\mu,\nu,z}$ over $E_{\mu,\nu,z}\cap(0,g^*)$ using high-precision quadrature. Any disagreement beyond solver tolerance would falsify Theorem 1.1.

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Extended reading notes

Core claim

Theorem 1.1 states that for compactly supported probability measures $\mu,\nu$ on $\mathbb{R}$ with $\mu$ non-degenerate and for every real $z<z^*_{\mu\boxplus\nu}$, $$U_{\mu\boxplus\nu}(z)=\inf_{g\in\widehat E_{\mu,\nu,z}\cap(0,g^*) }E_{\mu,\nu,z}(g)=\inf_{g\in E_{\mu,\nu,z}\cap(0,g^*) }E_{\mu,\nu,z}(g),$$ where $E_{\mu,\nu,z}(g)=\int_0^g sR'_{\mu}(-s)\,ds+\int\log(\lambda-z+R_{\mu}(-g))\,\nu(d\lambda)$; here $\widehat E$ uses the real-analytic $R$-transform on its natural domain and $E$ uses its maximal real-analytic extension. The unique minimizer of both infima is $g=G_{\mu\boxplus\nu}(z)$. The proof identifies this point as the unique critical point of $E_{\mu,\nu,z}$ in $(0,g^*)$ by showing that $g=G_{\mu\boxplus\nu}(z)$ solves the fixed-point equation $G_{\nu}(z-R_{\mu}(-g))=g$, and that the derivative changes sign from negative to positive at this point. A companion characterization (Theorem 1.3) expresses the edge parameters as $g^*=G_{\nu}(h^*)$ and $z^*=F_{\mu,\nu}(h^*)$ with $F_{\mu,\nu}(h)=R_{\mu}(-G_{\nu}(h))+h$, so the formula can be used without first constructing the full convolution.

Load-bearing premise

The load-bearing premise is the real-analytic bridge: every compactly supported non-degenerate probability measure has a real R-transform with a unique maximal real-analytic extension, and the free additive convolution is exactly represented by additivity of these real R-transforms on the overlap of their domains; if that bridge failed for some measure class, the variational domain $E_{\mu,\nu,z}$ and the fixed-point identification of the minimizer with $G_{\mu\boxplus\nu}(z)$ would collapse.

Editorial extensions

If this is right

For $\mu=\mu_{sc,\beta}$, Theorem 1.1 yields (1.6): $U_{\mu\boxplus\nu}(z)=\inf_{g\in(0,g^*)}\{\beta^2g^2/2+\int\log(\lambda-z-\beta^2g)\,\nu(d\lambda)\}$, restricted to $\beta^2g<\operatorname{supp}_-\nu-z$, valid for $z<z^*$.
For $\mu=\mu_{MP,\beta}$, the same theorem yields (1.7): $U_{\mu\boxplus\nu}(z)=\inf_{g\in(0,g^*)}\{\beta\log(1+g)-\beta g/(1+g)+\int\log(\lambda-z+\beta/(1+g))\,\nu(d\lambda)\}$, with constraint $z-\operatorname{supp}_-\nu<\beta/(1+g)$.
In both special cases $F_{\mu,\nu}$ is strictly concave, so $h^*$ is the global maximum and (1.9) gives explicit formulas for $z^*$ and $g^*$ in terms of $G_{\nu}$; the critical-point trichotomy (1.8) lets one locate the edge by inspecting $E_{\mu,\nu,z}$.
For general $\mu,\nu$, Theorems 1.1 and 1.3 together give a way to compute $U_{\mu\boxplus\nu}(z)$ below the edge by solving a one-dimensional fixed-point equation rather than by first constructing the density of $\mu\boxplus\nu$.
The identity $U_{\mu\boxplus\nu}(z)=E_{\mu,\nu,z}(G_{\mu\boxplus\nu}(z))$ combined with matching derivatives gives a direct variational proof of the logarithmic-potential formula, the input used for determinant asymptotics of sums of independent random matrices in the companion work.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Going beyond the paper, the one-dimensional nature of the minimization suggests a numerical scheme for determinants of sums of random matrices: evaluate $U_{\mu\boxplus\nu}(z)$ by solving the scalar fixed-point equation, avoiding the cost of constructing the full convolution; the paper does not propose this algorithm.
The real-$R$-transform bridge used in Section 5 requires only finite variance for part of the construction, so the variational formula may extend to non-compactly supported measures with finite variance if the domain inclusions (2.18) can be re-established in that class; the paper does not address this extension.
The edge formula $z^*=\sup_h F_{\mu,\nu}(h)$ reads like a variational characterization of a phase boundary: changes in the number of solutions of $F_{\mu,\nu}(h)=z$ as $z$ crosses $z^*$ could serve as a stability criterion in random-matrix determinant problems; this dynamical reading is not made by the authors.
Theorem 6.1, the stronger invertibility of $G_{\mu}$ on $\{z:\operatorname{Re}z\notin\operatorname{supp}\mu\}$, is stated as independently useful; a concrete test would be to see whether it simplifies other free-probability arguments that currently rely on large-disk invertibility.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

I read this with some care and came out more confident than I expected. The paper proves Theorem 1.1, a genuine variational formula for U_{mu push nu}(z) as a one-dimensional infimum built from R_mu and U_nu, with the unique minimizer identified as G_{mu push nu}(z). The semicircle and Marchenko-Pastur special cases (1.6)-(1.7) are explicit and ready for use, and Theorem 1.3 pins down the left edge via F(h). None of that is in the cited literature: [6, Lemma 4.3] is a special upper bound, and [11] uses the fixed-point equation without a variational characterization.

The proof is structurally sound. The derivative identity (4.1), the fixed-point equivalence (1.18), the sign bookkeeping in (4.6), and the uniqueness in (0,g*) via monotonicity all hang together. The way the paper handles spurious critical points, by explicitly restricting the infimum to (0,g*), is honest and necessary. There are no fitted parameters and no circular dependence on the authors’ own work; their [4] appears only as an application. I traced the algebra in Lemnas 4.2 and 4.4, and the derivation of Corollary 4.6, and could not find a concrete failure.

The soft spots are real but not damning. The bridge between the real-analytic R-transform and the Mingo–Speicher formalism (Lemmas 2.1–2.2, Section 5) is terse and load-bearing, and it imports (2.18) from [8, Lemma 6.1]. A careful referee should check the domain inclusions and the analytic-continuation steps there; that is the part I would focus on. The auxiliary Theorem 6.1 is explicitly not used in the main proof, so it adds length without adding support; it is a genuine result, but its presence is a slight distraction. Minor typos in inline math do not impede reading.

Who gets value: random-matrix and probability people working on determinant asymptotics, Kac–Rice landscapes, spin glass complexity, and empirical-risk surfaces. The explicit formulas for semicircle and Marchenko-Pastur are likely to be cited. The paper deserves a serious referee—the proof is long and uses nontrivial external results, so it needs verification, not a desk rejection. My recommendation: send it to peer review with a referee who knows free probability and will spend time on Section 5.

Referee Report

0 major / 4 minor

Summary. The paper establishes a variational formula for the logarithmic potential U_{μ⊞ν}(z) when μ,ν are compactly supported probability measures on R, μ non-degenerate, for all real z below the left edge z*_{μ⊞ν} of the support of μ⊞ν. The formula expresses U_{μ⊞ν}(z) as the infimum, over g in (0,g*_{μ⊞ν}), of an energy E_{μ,ν,z}(g) built from the real R-transform of μ and the logarithmic potential of ν; the unique minimizer is identified as g = G_{μ⊞ν}(z). Explicit semicircle and Marchenko-Pastur special cases are derived, and Theorem 1.3 gives formulas for z*_{μ⊞ν} and g*_{μ⊞ν} in terms of the auxiliary function F_{μ,ν}(h) = R_μ(-G_ν(h)) + h. The proof combines a derivative computation for E_{μ,ν,z}, a sign analysis of the fixed-point equation g = G_ν(z - R_μ(-g)), monotonicity of the inverse Stieltjes transform, and a comparison of derivatives and limits to identify E_{μ,ν,z}(G_{μ⊞ν}(z)) with U_{μ⊞ν}(z). An auxiliary complex invertibility theorem for Stieltjes transforms is included but explicitly not used in the proofs of the main results.

Significance. If the main theorem holds, this is a clean and useful addition to the free-probability toolkit: it gives a variational description of the logarithmic potential of free additive convolutions, with direct applications to determinant asymptotics for sums of independent random matrices and to random landscape problems. The paper is honest about the presence of spurious critical points outside (0,g*_{μ⊞ν}), and it carefully restricts the infimum to the interval where the minimizer is unique. The explicit semicircle and Marchenko-Pastur formulas, together with the edge-characterization Theorem 1.3, are concrete and likely to be used. The real-analytic R-transform framework is developed carefully, and the connection to the standard formal/holomorphic theory via Mingo-Speicher is handled in a self-contained section. The auxiliary Theorem 6.1 is a bonus that strengthens a known invertibility result; the manuscript correctly states that it is not load-bearing for the main theorems.

minor comments (4)

[Section 4, Lemma 4.4(b), Eq. (4.20)] Equation (4.20) as printed states sign(E'_{μ,ν,z}(g)) = sign(G_{μ⊞ν}(z) - g), which is opposite to the sign obtained from Lemma 4.2(a)-(b) together with R'_μ(-g)>0 on (0,g*_{μ⊞ν}); the correct identity is sign(E'_{μ,ν,z}(g)) = sign(g - G_{μ⊞ν}(z)). The subsequent display (4.21) is consistent with the corrected sign, so this appears to be a sign typo that should be fixed.
[Section 2.3, Eq. (2.18)] The displayed inequality (2.17) only bounds the left-edge Stieltjes value G_{μ⊞ν}(supp_- μ⊞ν), but the claimed inclusion Dhat_{μ⊞ν} ⊂ Dhat_μ ∩ Dhat_ν also requires the corresponding right-edge inequality. That right-edge half follows by applying the same bound to reflected measures, or by citing the symmetric statement in [8, Lemma 6.1]; since (2.18) is used in Lemma 4.2(c) and Lemma 4.4, this one-line justification should be added.
[Section 6, proof of Theorem 6.1(b)] The sentence 'G_μ(z)≠0 for any z such that |z|<∞, since G_μ is injective' is not by itself a complete proof of non-vanishing; injectivity on B does not rule out a zero unless 0 is known to occur elsewhere in the image. The conclusion is true because Re G_μ(z) has a definite sign for Re z outside the support, so the sentence should be rephrased accordingly.
[Abstract and Section 1] There are a few minor grammatical slips, for example 'two compactly supported probability measure' in the abstract and 'the inf. resp. sup.' in Section 1; these should be corrected in a final polish.

Circularity Check

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No significant circularity: Theorem 1.1 is derived from independent standard inputs, not from its own conclusion.

full rationale

The central claim is not circular. Theorem 1.1 is proved by differentiating E_{mu,nu,z} (Lemma 4.1), reducing critical points to the fixed-point equation (1.17), and using the standard additivity identity R_{mu push nu} = R_mu + R_nu together with monotonicity of Stieltjes transforms to identify the unique minimizer as G_{mu push nu}(z) (Lemma 4.4). The value identity U_{mu push nu}(z) = E_{mu,nu,z}(G_{mu push nu}(z)) is then obtained by matching derivatives in z and the same limit at -infinity (Lemma 4.5 and Corollary 4.6), not by assuming the desired equality. The inputs R_mu and G_nu are genuinely different from the output U_{mu push nu}; no fitted parameter is renamed as a prediction. The only self-citation, [4], is an application of the formulas, not a premise. External results used (Mingo-Speicher Theorems 17, 26, 28; Guionnet-Maida [8, Lemma 6.1]) are standard and do not restate the theorem; the auxiliary Theorem 6.1 is explicitly not used in the main proofs. I find no equation that reduces by construction to its own input.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central result uses no fitted parameters and no ad hoc constants. It relies on standard free probability facts from [10], on [8, Lemma 6.1] for the domain inclusion, and on a textbook global inverse function theorem in Section 6. No new physical or mathematical entities are postulated.

assumptions (4)

standard math Free additive convolution exists uniquely and its R-transform satisfies R_{mu push nu} = R_mu + R_nu as formal power series (Mingo-Speicher, Theorem 18, Chapter 2).
Used at definition (1.4) and in the equivalence chain (1.18) to identify the fixed-point solution g with G_{mu push nu}(z).
standard math There exists a holomorphic R-transform function for compactly supported measures, and the inverse Stieltjes transform is defined on an exterior domain (Mingo-Speicher, Theorems 17 and 28, Chapter 3).
Section 5 uses these results to prove Lemmas 2.1 and 2.2, which connect the real-analytic R-transform to the standard R-transform and justify the domain inclusion used throughout Sections 3 and 4.
domain assumption G_{mu push nu}(supp_- mu push nu) <= min(G_mu(supp_- mu), G_nu(supp_- nu)), from [8, Lemma 6.1].
Used in (2.17)-(2.18) to show Dhat_{mu push nu} is contained in Dhat_mu and Dhat_nu, which forces the minimizer G_{mu push nu}(z) to lie in the variational domains Ehat and E.
standard math Global inverse function theorem for C^1 maps on connected bounded domains, from [7].
Used only in the proof of Theorem 6.1, the auxiliary Stieltjes-transform invertibility result. The main variational theorem does not depend on this in the current version.

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Pith. "Pith review of Variational formula for the logarithmic potential of free additive convolutions." pith.science (2026). https://pith.science/paper/RLT4PFRW

@misc{pith2026250619064,
  author       = {Pith},
  title        = {Pith review of: Variational formula for the logarithmic potential of free additive convolutions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RLT4PFRW}},
  note         = {Machine review of arXiv:2506.19064}
}

abstract

We establish a general variational formula for the logarithmic potential of the free additive convolution of two compactly supported probability measure on $\R$. The formula is given in terms of the $R$-transform of the first measure, and the logarithmic potential of second measure. The result applies in particular to the additive convolution with the semicircle or Marchenko-Pastur laws, for which the formula simplifies. The logarithmic potential of additive convolutions appears for instance in estimates of the determinant of sums of independent random matrices.

Figures

Figures reproduced from arXiv: 2506.19064 by the authors.

**Figure 1.** Plots of Eµ,ν,z, G⟨−1⟩ µ⊞ν , G[−1] µ⊞ν and Fµ,ν for four combinations of µ and ν. See Remark 1.11 for additional discussion. a) Rµ(t) = 1 2(1−t) ν = 1 2 δ−1 + 1 2 δ1 z = −1.5 b) Rµ(t) = t ν = 1 2 δ−1 + 1 2 δ1 z = −3 c) Rµ(t) = t + t 2 6 ν = 1 2 δ−0.17 + 1 2 δ−1.17 z = −2.77 d) Rµ(t) = t + t 2 4.5 + t 3 80 ν = 1 2 δ−0.17 + 1 2 δ−1.17 z = −2.53725 > z∗ µ⊞ν = −2.63725 0 1 2 3 4 0.4 0.5 0.6 0.7 g c) Eµ,ν,z −1 0 1 2 0.8 … view at source ↗

**Figure 2.** Examples of different possibilities for the “shapes” of G ⟨−1⟩ ρ and G [−1] ρ , cf. Lemma 3.1. Under the change of variables z = G ⟨−1⟩ ρ (−t) ⇐⇒ Gρ(z) = −t for z ∈ (supp ρ) c (which is valid by (2.10)), the expression (2.32) turns into ˆ ρ(dλ) (λ − z) 2 − ˆ ρ(dλ) λ − z 2 . (2.33) Finally (2.33) is non-negative for any ρ, with equality only if ρ is degenerate. This proves that R′ ρ (t) > 0 for all t ∈ Dˆ ρ \ {0}, … view at source ↗

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

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work page Pith review arXiv 2024

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