Probabilistic Analysis of the Random Spectral Radius for a Matrix Family

Anastasiia Trofimova; Francesco Paolo Maiale; Nicola Guglielmi

arxiv: 2511.16271 · v2 · submitted 2025-11-20 · 🧮 math.DS

Probabilistic Analysis of the Random Spectral Radius for a Matrix Family

Francesco Paolo Maiale , Anastasiia Trofimova , Nicola Guglielmi This is my paper

Pith reviewed 2026-05-17 20:44 UTC · model grok-4.3

classification 🧮 math.DS

keywords random spectral radiusmatrix productslaw of large numberscentral limit theoremswitching systemsasymptoticsspectral radius

0 comments

The pith

The random spectral radius of products from diagonal and triangular matrix families satisfies a law of large numbers and central limit theorem with closed-form expressions.

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

This paper focuses on the typical, probabilistic growth rate of random products of matrices from a family, rather than the extreme cases captured by the joint spectral radius. It proves that for diagonal, commuting, triangular, and small perturbation cases, the logarithm of the spectral radius of length-n products converges almost surely to a specific limit and is asymptotically normal with explicit variance. These results are relevant for understanding average behavior in switching dynamical systems where matrices are chosen randomly at each step. The analysis also covers the coalescence case where the limiting distribution is the maximum of correlated Gaussians.

Core claim

We establish a law of large numbers and a central limit theorem for the random spectral radius of length-n products sampled according to a probability measure on families of diagonal or commuting matrices, yielding the exact limit as the expected logarithm of the absolute value of the dominant eigenvalue and an explicit variance formula. The same holds for upper or lower triangular matrices and small perturbations thereof. When leading eigenvalues coalesce, the limit law is the maximum of a vector of correlated Gaussians with explicit covariance.

What carries the argument

Asymptotic analysis of the random spectral radius via the spectral radius of randomly selected matrix products in commuting and nearly commuting families.

If this is right

If the claim holds, the typical growth rate in random switching systems can be predicted exactly without Monte Carlo simulation for these matrix classes.
The explicit variance allows precise confidence intervals for finite-n approximations in applications.
Phase transitions in growth rates are determined by eigenvalue coalescence, leading to non-Gaussian limits.
Closed-form constants enable accurate numerical computations in control and stability analysis.

Where Pith is reading between the lines

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

The approach might generalize to other matrix families if they can be approximated by diagonal ones.
Similar probabilistic tools could apply to non-commuting cases by considering Lyapunov exponents.
These results suggest testable predictions for the distribution of growth rates in simulated switching systems.

Load-bearing premise

The results require the matrices to be diagonal, triangular, or small perturbations of diagonal ones, with the probability measure satisfying suitable regularity conditions.

What would settle it

Generate many random products of length 1000 from a diagonal matrix family with known eigenvalues and verify if the sample mean and variance of log spectral radii match the paper's predicted values within statistical error.

Figures

Figures reproduced from arXiv: 2511.16271 by Anastasiia Trofimova, Francesco Paolo Maiale, Nicola Guglielmi.

**Figure 1.** Figure 1: Histogram of ρ800(T , PT ) compared with the Gaussian pdf. Here, the family consists of two 3 × 3 real-valued matrices with J = {1}, ρ∞ ≈ 1.323194 and σ∞ ≈ 0.086861. Remark 3. When the covariance matrix ΣT ,PT is diagonal, the components of the Gaussian vector G = (Gj )j∈J are independent. In this case, the limiting distribution takes a factorized form. To be more precise, writing J = {j1, . . . , js}, we… view at source ↗

**Figure 2.** Figure 2: The histogram represents the empirical distribution, and its kernel density estimate (KDE) is shown as the green curve. These are compared against the theoretical limiting probability density function (pdf, orange curve) for ξ (3) = ρ∞ · maxj∈{j1,j2,j3} Gj . (left) The components of G have equal variances and correlations. (right) The components of G have unequal variances and are uncorrelated. These cons… view at source ↗

**Figure 3.** Figure 3: Univariate Edgeworth expansions: relative errors of E[ρn]and n Var(ρn) for 1) (top) and 2) (bottom) models. All errors decay at the theoretically predicted rates or better [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗

**Figure 4.** Figure 4: Validation of multivariate Edgeworth expansions via Monte Carlo simulation with NMC = 108 samples per n-value. Note that, for n > 103 , entries of matrix products become more and more unreliable due to accumulation of errors. 2. Perturbation theory In this section, we present a perturbation theory for families of matrices perturbed by a sufficiently small perturbation. Let the family of matrices Fε := A1… view at source ↗

**Figure 5.** Figure 5: Histogram of ρn(F, PF ) for n ∈ {100, 400}. The family consists of three 3 × 3 matrices with probabilities {0.3, 0.3, 0.4} and µ = [PITH_FULL_IMAGE:figures/full_fig_p027_5.png] view at source ↗

**Figure 6.** Figure 6: Histogram of ρn(F, PF ) for n ∈ {100, 400}. The family consists of three 3 × 3 matrices with probabilities {0.5, 0.4, 0.1} and µ = [PITH_FULL_IMAGE:figures/full_fig_p028_6.png] view at source ↗

**Figure 7.** Figure 7: Histogram of ρn(F, PF ) for n ∈ {100, 400}. The family consists of three 3 × 3 matrices with probabilities {0.2, 0.2, 0.6} and µ = [PITH_FULL_IMAGE:figures/full_fig_p028_7.png] view at source ↗

read the original abstract

We investigate joint spectral characteristics of a family of matrices $\mathcal F $, associated with products in the semigroup generated by $\mathcal F$. In the literature, extremal measures such as the well-known joint spectral radius and the lower spectral radius have been extensively studied. However, these measures fail to capture the typical growth rate of matrix products, focusing instead on the worst and best-case scenarios. Nevertheless, when examining, for instance, a switching dynamical system, a probabilistic rate of growth, which characterizes typical trajectories, emerges as a highly intriguing and significant measure. In this article, we study the random spectral radius, defined as the spectral radius of a length-$n$ product sampled at random from the semigroup according to a given probability measure. We establish asymptotic results, namely a Law of Large Numbers and a Central Limit Theorem, for diagonal (equivalently, commuting), upper- or lower-triangular, and small perturbations of diagonal matrices. Beyond recovering the correct scaling, we obtain exact closed-form expressions for the limiting value and variance, which may be useful in numerical applications requiring precise constants. In the coalescence regime, where the leading eigenvalues merge, the limiting distribution is non-Gaussian: it is given by the maximum of a correlated Gaussian vector with explicit covariance structure. This phenomenon governs phase transitions between distinct growth regimes in switching systems.

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 gives explicit LLN, CLT, and closed-form limits for the random spectral radius on diagonal, triangular, and small-perturbation matrix families, with a max-of-Gaussians limit in coalescence.

read the letter

The main takeaway is that this work supplies concrete expressions for the typical growth rate of random matrix products in commuting and triangular families, plus a non-Gaussian limit when leading eigenvalues coalesce. That combination of closed forms and the phase-transition description looks new relative to the extremal joint spectral radius results cited in the abstract. The authors recover the right scaling and then give exact limiting value and variance for the diagonal case, extend it to triangular matrices, and handle small perturbations of diagonals. The coalescence regime gets an explicit covariance structure for the max of correlated Gaussians. These constants could be directly useful in numerical work on switching systems where one wants typical rather than worst-case behavior. The derivations appear to rest on standard limit theorems applied to the logs of the eigenvalues, which is a reasonable route. The soft spot is the small-perturbation transfer. The stress-test note is on point: without an almost-sure lower bound on the gap between the largest and second-largest |eigenvalue| or a quantitative condition tying perturbation size to that random gap, the claimed closed forms for the perturbed case can fail for large n on sets of positive measure. The abstract does not spell out such a bound, so the proofs need to be checked for whether they supply it or restrict to a measure-zero set. Minor issues aside, the paper is aimed at people working on probabilistic aspects of switched dynamical systems and control. A reader who needs explicit constants for simulation or analysis of typical trajectories will find it worth reading. It is coherent on its own terms and shows honest engagement with the literature, so it deserves a serious referee even if the perturbation section requires tightening.

Referee Report

1 major / 2 minor

Summary. The manuscript studies the random spectral radius of length-n products drawn randomly from a matrix family F according to a probability measure. It establishes a law of large numbers and central limit theorem for the logarithm of this radius, together with exact closed-form expressions for the limiting value and variance, in the cases of diagonal (commuting) matrices, upper- or lower-triangular matrices, and small perturbations of diagonal matrices. In the coalescence regime where leading eigenvalues merge, the limiting distribution is shown to be the maximum of a correlated Gaussian vector with explicit covariance structure. The results are motivated by applications to switching dynamical systems and typical growth rates.

Significance. If the derivations hold, the work supplies precise, usable constants for the typical asymptotic growth of random matrix products, which is a natural complement to the extremal joint spectral radius and lower spectral radius. The explicit closed forms and the non-Gaussian coalescence limit provide concrete tools for numerical analysis and for identifying phase transitions between growth regimes in switched systems.

major comments (1)

[small-perturbation results] The section on small perturbations of diagonal matrices claims that the LLN and CLT, together with the exact closed-form limiting value and variance, transfer from the commuting diagonal case via eigenvalue perturbation theory. For this transfer to preserve the exact expressions, the perturbation norm must remain smaller than the gap between the largest and second-largest |eigenvalue| of the unperturbed length-n product. Because this gap is random and can be arbitrarily small with positive probability under any measure that places positive mass on distinct diagonal entries, a fixed perturbation size violates the required condition for large n on a set of positive measure. The manuscript must supply either an almost-sure quantitative lower bound on the gap or an explicit smallness condition (e.g., ||E|| < c · min-gap) that holds with probability 1; without it the claimed closed forms are,

minor comments (2)

[Introduction and main results] Notation for the probability measure on the matrix family and the precise regularity conditions (moment assumptions, support restrictions) should be stated uniformly at the beginning of the main theorems rather than introduced piecemeal.
[coalescence regime] The coalescence-regime covariance structure is described as “explicit,” but the explicit matrix entries or generating function are not displayed; adding a short displayed formula would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of significance, and constructive major comment. We address the concern on small perturbations point by point below and will incorporate revisions to clarify the conditions under which the transfer of exact closed forms holds.

read point-by-point responses

Referee: [small-perturbation results] The section on small perturbations of diagonal matrices claims that the LLN and CLT, together with the exact closed-form limiting value and variance, transfer from the commuting diagonal case via eigenvalue perturbation theory. For this transfer to preserve the exact expressions, the perturbation norm must remain smaller than the gap between the largest and second-largest |eigenvalue| of the unperturbed length-n product. Because this gap is random and can be arbitrarily small with positive probability under any measure that places positive mass on distinct diagonal entries, a fixed perturbation size violates the required condition for large n on a set of positive measure. The manuscript must supply either an almost-sure quantitative lower bound on the gap or an explicit smallness condition (e.g., ||E|| < c · min-gap) that holds with probability 1; without it

Authors: We agree that a fixed perturbation size does not automatically satisfy the gap condition uniformly for all realizations and all large n. When the top Lyapunov exponent is strictly dominant (the regime treated in this section, distinct from the coalescence case handled separately), large-deviation estimates imply that the probability of the gap falling below any fixed positive threshold decays exponentially in n. Thus the perturbation condition holds with probability tending to 1. Moreover, on the almost-sure set where the empirical occupation measures converge to the maximizing measure, the gap grows exponentially almost surely. We will revise the manuscript to (i) explicitly assume strict dominance of the leading exponent, (ii) state the smallness condition in terms of the random gap, and (iii) add a remark that the closed-form expressions therefore hold in probability and almost surely under this assumption. This preserves the claimed transfer while making the probabilistic nature of the bound transparent. revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivations apply standard limit theorems to explicit matrix classes

full rationale

The paper derives LLN and CLT for the random spectral radius on diagonal/commuting, triangular, and small-perturbation matrices by direct computation of the spectral radius for the first two classes (products reduce to entrywise products or maxima) followed by classical ergodic theorems and CLT for i.i.d. sums or maxima; closed-form limits and variances are obtained from explicit moment calculations on the eigenvalue logs rather than any fitted parameter or self-referential definition. The perturbation transfer invokes standard eigenvalue perturbation bounds but does not redefine the target quantity in terms of itself or rely on a self-citation chain for the core result. No load-bearing step reduces by construction to an input that is defined using the output, and the work remains self-contained against external benchmarks such as the Furstenberg-Kesten theorem and standard perturbation theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; the paper relies on standard assumptions from probability theory (LLN and CLT for dependent random variables) and matrix analysis (spectral radius properties for triangular and commuting matrices). No free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

axioms (1)

standard math Standard law of large numbers and central limit theorem apply to the logarithm of the spectral radius of random matrix products under suitable moment conditions.
Invoked implicitly when stating LLN and CLT for the random spectral radius.

pith-pipeline@v0.9.0 · 5540 in / 1366 out tokens · 32093 ms · 2026-05-17T20:44:38.827747+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We establish asymptotic results, namely a Law of Large Numbers and a Central Limit Theorem, for diagonal (equivalently, commuting), upper- or lower-triangular, and small perturbations of diagonal matrices. Beyond recovering the correct scaling, we obtain exact closed-form expressions for the limiting value and variance.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

[1]

Billingsley,Probability and Measure, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, New York, 3rd ed., 1995

2 [3]P. Billingsley,Probability and Measure, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, New York, 3rd ed., 1995. 9 [4]V. D. Blondel, R. Jungers, and V. Protasov,On the complexity of computing the capacity of codes that avoid forbidden difference patterns, IEEE Trans. Inform. Theory, 52 (2006), pp. 5122–5127. 2 [5]Y. Chitou...

work page 1995
[2]

9 [17]J. E. Kolassa,Series Approximation Methods in Statistics, vol. 88 of Lecture Notes in Statistics, Springer, New York, NY, 3 ed., 2006. 19 [18]D. Kruyswijk,On some well-known properties of the partition functionp(n)and Euler’s infinite product, Nieuw Arch. Wiskunde, 23 (1950), pp. 97–107. 2 [19]B. E. Moision, A. Orlitsky, and P. H. Siegel,On codes th...

work page 2006
[3]

2 Francesco Paolo Maiale Gran Sasso Science Institute Viale Rendina 26–28, L’Aquila, 67100, Italy Email address:francescopaolo.maiale@gssi.it Anastasiia Trofimova Gran Sasso Science Institute Viale Rendina 26–28, L’Aquila, 67100, Italy Email address:anastasiia.trofimova@gssi.it Nicola Guglielmi Gran Sasso Science Institute Viale Rendina 26–28, L’Aquila, 6...

work page

[1] [1]

Billingsley,Probability and Measure, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, New York, 3rd ed., 1995

2 [3]P. Billingsley,Probability and Measure, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, New York, 3rd ed., 1995. 9 [4]V. D. Blondel, R. Jungers, and V. Protasov,On the complexity of computing the capacity of codes that avoid forbidden difference patterns, IEEE Trans. Inform. Theory, 52 (2006), pp. 5122–5127. 2 [5]Y. Chitou...

work page 1995

[2] [2]

9 [17]J. E. Kolassa,Series Approximation Methods in Statistics, vol. 88 of Lecture Notes in Statistics, Springer, New York, NY, 3 ed., 2006. 19 [18]D. Kruyswijk,On some well-known properties of the partition functionp(n)and Euler’s infinite product, Nieuw Arch. Wiskunde, 23 (1950), pp. 97–107. 2 [19]B. E. Moision, A. Orlitsky, and P. H. Siegel,On codes th...

work page 2006

[3] [3]

2 Francesco Paolo Maiale Gran Sasso Science Institute Viale Rendina 26–28, L’Aquila, 67100, Italy Email address:francescopaolo.maiale@gssi.it Anastasiia Trofimova Gran Sasso Science Institute Viale Rendina 26–28, L’Aquila, 67100, Italy Email address:anastasiia.trofimova@gssi.it Nicola Guglielmi Gran Sasso Science Institute Viale Rendina 26–28, L’Aquila, 6...

work page