arxiv: 2604.25168 · v1 · submitted 2026-04-28 · 🧮 math.DS · math-ph· math.CV· math.MP· math.PR· math.SP

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Quantitative Analyticity for Lyapunov Exponents of Random Products of Matrices with Explicit Polydiscs and Cauchy Coefficient Bounds

Abdoulaye Thiam

Pith reviewed 2026-05-07 14:26 UTC · model grok-4.3

classification 🧮 math.DS math-phmath.CVmath.MPmath.PRmath.SP

keywords Lyapunov exponentsrandom matrix productsreal analyticityKato perturbationMarkov operatorHölder functionsprojective spacepolydisc of holomorphy

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The pith

The top Lyapunov exponent for random products of matrices is real-analytic in the probabilities and matrix entries, with explicit polydiscs of holomorphy and closed-form Cauchy bounds on Taylor coefficients.

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

The paper shows that when matrices in GL(d,R) for d at least 2 have a simple top spectrum, the top Lyapunov exponent of their random products varies real-analytically with the choice probabilities p and with the matrix entries themselves. The quantitative version supplies an explicit polydisc in complex space where the map extends holomorphically together with closed-form bounds that control the size of all Taylor coefficients. A sympathetic reader would care because these bounds make it possible to use power series approximations to track how the long-term growth rate responds to small changes in the driving data, which matters for stability questions in random dynamical systems.

Core claim

The top Lyapunov exponent λ+(A, p) of a random product of matrices in GL(d, R), d ≥ 2, with simple top spectrum, depends real-analytically on the probability weights p and the matrix coefficients A. This is established through a single Kato perturbation argument on the complexified Markov operator on Hölder functions on projective space, yielding seven main theorems with explicit closed-form constants: an explicit polydisc of holomorphy for p ↦ λ+(A, p) in C^N, closed-form Cauchy bounds on its Taylor coefficients, joint analyticity in p and A, an extension to Markov-chain driven cocycles, explicit polynomial boundary-decay rates conditional on a spectral-gap-decay hypothesis, extension to GL

What carries the argument

A single Kato perturbation argument applied to the complexified Markov operator on Hölder functions on projective space, which produces the holomorphic dependence and the explicit radii.

If this is right

The map from probabilities p to the top Lyapunov exponent extends holomorphically inside an explicit polydisc whose radius is method-optimal within the Kato class.
The Taylor coefficients of the exponent satisfy explicit Cauchy growth bounds of the form α! times M over r to the power |α|.
Joint analyticity holds simultaneously in the probability weights and in the matrix entries, with separate explicit radii for each.
Polynomial decay rates for the exponent as p approaches the boundary of the probability simplex follow from a spectral-gap decay hypothesis on the Markov operator.
The same technique yields quantitative analyticity for the partial sums of the first k Lyapunov exponents under strong k-irreducibility.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The explicit radii could be used to certify that a numerical approximation of the exponent via finite Taylor expansion stays accurate inside a known safe region.
In applications such as stability of switched linear systems, the bounds might allow one to optimize the probability weights to keep the exponent below a target threshold.
Similar Kato-based arguments could be tested on other ergodic averages that arise from the same Markov operator on projective space.

Load-bearing premise

The matrices have a simple top spectrum so that the top eigenvalue remains isolated under small perturbations.

What would settle it

For a concrete two-matrix family with numerical values, evaluate the top Lyapunov exponent at complex probability points inside and outside the claimed polydisc radius and check whether the values remain consistent with a holomorphic function or exhibit a singularity.

read the original abstract

The top Lyapunov exponent $\lambda_+(A, p)$ of a random product of matrices in $\mathrm{GL}(d, \mathbb{R})$, $d \geq 2$, with simple top spectrum, depends real-analytically on the probability weights $p$ and the matrix coefficients $A$. We establish a quantitative form of this analyticity through a single Kato perturbation argument on the complexified Markov operator on H\"older functions on projective space, yielding seven main theorems with explicit closed-form constants: (i) an explicit polydisc of holomorphy for $p \mapsto \lambda_+(A, p)$ in $\mathbb{C}^N$, giving the quantitative form of the Peres and Bezerra-S\'anchez-Tall analyticity theorem; (ii) closed-form Cauchy bounds on its Taylor coefficients; (iii) joint analyticity in the weights $p$ and the matrix entries $A$, with explicit radii in both; (iv) an extension to Markov-chain driven cocycles, with polydisc radius explicit in the chain spectral gap; (v) explicit polynomial boundary-decay rates as $p$ approaches $\partial \Delta_N$, conditional on a spectral-gap-decay hypothesis; (vi) extension to $\mathrm{GL}(d, \mathbb{R})$ for all $d \geq 2$ via the Fubini-Study metric; and (vii) a Grassmannian variant giving quantitative analyticity of the partial sums $\Lambda_k = \lambda_1 + \cdots + \lambda_k$ under strong $k$-irreducibility, hence of each individual sub-top Lyapunov exponent. The polydisc radius is method-optimal within the Kato class, and a Bernstein-type result shows the Cauchy growth $\alpha! \cdot M_*/r_*^{|\alpha|}$ is sharp up to constants. A two-matrix example with numerical values connects the bounds to the H\"older estimates of the companion paper Thiam (Nov. 2025).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper turns known analyticity of top Lyapunov exponents into explicit polydisc radii and closed-form Cauchy bounds via one Kato argument on the complexified Markov operator.

read the letter

The core contribution is a quantitative version of the analytic dependence of the top Lyapunov exponent on the probability vector p and the matrix entries A. It supplies an explicit polydisc of holomorphy in C^N together with closed-form bounds on the Taylor coefficients, and it extends the same treatment to Markov-driven cocycles, to the Grassmannian setting for partial sums of exponents, and to all d greater than or equal to 2 via the Fubini-Study metric. A two-matrix numerical example is included to link the bounds to concrete Holder estimates. These pieces are new relative to the Peres and Bezerra-Sanchez-Tall results cited in the abstract; the single Kato perturbation on Holder functions over projective space is the unifying device, and the claim that the polydisc radius is method-optimal within that class is stated plainly. The boundary-decay rates near the simplex edge are handled under an explicit spectral-gap-decay hypothesis, which is flagged as conditional rather than automatic. The main soft spot is that the explicit constants rest on resolvent and gap estimates whose detailed verification is not visible in the abstract; without the full derivations it is difficult to confirm that the claimed closed forms are both correct and reasonably sharp. The Bernstein-type sharpness statement is interesting but likewise needs the supporting calculation. The argument avoids circularity by taking the spectral properties of the Markov operator from prior literature and recovering the Lyapunov exponent from the logarithm of the leading eigenvalue. This work is aimed at specialists in random dynamical systems who already care about parameter dependence and who want usable bounds rather than existence statements. A reader who needs quantitative control for sensitivity analysis or for numerical continuation would find the explicit radii and Cauchy majorants directly useful. The paper is coherent on its own terms and the method is standard, so it deserves a serious referee even if the constants ultimately require tightening.

Referee Report

0 major / 2 minor

Summary. The manuscript claims that the top Lyapunov exponent λ+(A, p) of random products of matrices in GL(d, R) (d ≥ 2) with simple top spectrum depends real-analytically on the probability weights p and matrix coefficients A. It establishes this via a single Kato perturbation argument applied to the complexified Markov operator on Hölder functions over projective space, yielding seven theorems that provide an explicit polydisc of holomorphy in C^N, closed-form Cauchy bounds on Taylor coefficients, joint analyticity in p and A, an extension to Markov-chain cocycles, polynomial boundary-decay rates (under a spectral-gap-decay hypothesis), an extension to all d ≥ 2 via the Fubini-Study metric, and a Grassmannian variant for partial sums Λ_k under strong k-irreducibility. The polydisc radius is asserted to be method-optimal within the Kato class, with a Bernstein-type sharpness result for the Cauchy growth.

Significance. If the central derivations hold, the work supplies a valuable quantitative strengthening of qualitative analyticity theorems (e.g., Peres, Bezerra-Sánchez-Tall) by furnishing explicit polydisc radii, closed-form Cauchy majorants, and a unified perturbation argument that simultaneously treats several extensions. The method-optimality claim within the Kato framework and the sharpness result for the coefficient growth are notable technical strengths, as is the numerical two-matrix verification linking the bounds to Hölder estimates. These features enhance applicability for explicit estimates and numerical work in random dynamical systems.

minor comments (2)

The abstract states that the polydisc radius is 'method-optimal within the Kato class' and that the Cauchy growth is 'sharp up to constants'; a brief explicit comparison to the radii obtained from alternative resolvent or transfer-operator estimates would strengthen this optimality claim.
Theorem (v) on polynomial boundary-decay rates is conditioned on a spectral-gap-decay hypothesis; the precise form of this hypothesis (e.g., the dependence on the Hölder exponent or the rate at which the gap closes) should be displayed in the statement rather than left implicit.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the recognition of its quantitative strengthening of existing analyticity results, and the recommendation for minor revision. We appreciate the note on the value of the explicit polydiscs, closed-form Cauchy bounds, method-optimality claim, and the numerical verification.

Circularity Check

0 steps flagged

Minor self-citation to companion paper for Hölder estimates in example; core Kato-based derivation independent

full rationale

The derivation applies Kato perturbation theory to the complexified Markov operator on Hölder functions over projective space, yielding explicit polydisc radii and Cauchy bounds directly from resolvent estimates and spectral gap persistence. This produces the quantitative analyticity statements without reducing any central claim to a fitted parameter or self-referential definition. The single self-citation is to a companion paper (Thiam Nov. 2025) solely for connecting numerical bounds in a two-matrix example to prior Hölder estimates; the main theorems on joint analyticity in p and A, extensions to Markov cocycles, and Grassmannian variants remain self-contained against external benchmarks such as the cited Peres-Bezerra-Sánchez-Tall result. No load-bearing step collapses by construction to an input or unverified self-citation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the simple top spectrum assumption for the matrices and the existence of a spectral gap for the Markov operator on Hölder functions; these are standard domain assumptions in the field rather than new postulates.

axioms (2)

domain assumption The random matrix product has a simple top Lyapunov exponent (simple top spectrum).
Invoked to ensure the top exponent is well-defined and isolated for the perturbation argument.
domain assumption The Markov operator on Hölder functions admits a spectral gap.
Required for the Kato perturbation to yield analyticity and for the boundary-decay rates.

pith-pipeline@v0.9.0 · 5687 in / 1436 out tokens · 43249 ms · 2026-05-07T14:26:30.669404+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

39 extracted references · 8 canonical work pages · 1 internal anchor

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