On the Asymptotic Growth of Constrained Random Matrix Products: A Structural Approach to Random Horadam-Type Recurrences and Their Associated Lyapunov Exponents
Abstract
This paper investigates Lyapunov exponents and asymptotic growth phenomena for a class of random Horadam-type recurrences constrained by quadratic manifold geometries. For a fixed structural parameter k>0, we introduce the \emph{random k-Horadam recurrence}, defined by the initial values G0,k=0, G1,k=1 and the non-deterministic relation Gn,k=αn,kGn−1,k+βn,kGn−2,k for n≥2, where the random companion coefficients are constrained to the \emph{k-Constrained Sum-Product} (k-CSP) variety satisfying αn,k+βn,k=k and αn,kβn,k=−k. Stochasticity is driven by an i.i.d.\ sign process selecting, at each discrete time step, one of the two admissible algebraic branches. This formulation induces a random matrix cocycle in GL2(ℝ) supported on a discrete non-commutative manifold. Using Kingman’s subadditive ergodic theorem embedded in a projective normalization framework, we establish the almost sure existence of the top Lyapunov exponent, λk. Crucially, by establishing the non-compactness and strong irreducibility of the normalized semigroup for k ≥ 2, we characterize the top Lyapunov exponent λk and derive strict, non-trivial analytical bounds satisfying 1/2log(k) < λk < logδk. Here, δk = (k+√(k2+4k))/2 is the positive root of t2 − kt − k = 0, the characteristic equation governing the algebraic structure of the coefficients αn,k and βn,k. To complement the structural analysis, we establish the upper bound using the Joint Spectral Radius and the Berger-Wang theorem, while the lower bound is rigorously derived via the expectation of a two-step product norm. Furthermore, under appropriate log-integrability conditions, we analyze the structural stability of the system under both additive and multiplicative perturbations. Employing weak convergence criteria for random measures, we prove that the perturbed exponents vary continuously and converge to the unperturbed attractor in the small-noise limit. We also examine a singular, degenerate boundary regime where the multi-dimensional non-commutativity collapses, freezing the coefficient βn,k to a constant parameter a∈(0,1]. This reduction isolates a scalar multiplicative process governed by a deterministic functional envelope k(a) = a2/(a+1). We evaluate this scalar system via the Strong Law of Large Numbers and derive explicit functional and integrated two-sided bounds.
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BibTeX
@misc{basse2026asymptotic,
title = {{On the Asymptotic Growth of Constrained Random Matrix Products: A Structural Approach to Random Horadam-Type Recurrences and Their Associated Lyapunov Exponents}},
author = {Moïse Basse and El Hadji Deme and Mohamed Ben Fraj Ben Maaouia},
howpublished = {HAL},
year = {2026},
month = aug,
url = {https://hal.science/hal-05731432v2},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - On the Asymptotic Growth of Constrained Random Matrix Products: A Structural Approach to Random Horadam-Type Recurrences and Their Associated Lyapunov Exponents
AU - Moïse Basse
AU - El Hadji Deme
AU - Mohamed Ben Fraj Ben Maaouia
PY - 2026
DA - 2026/08/30
PB - HAL
UR - https://hal.science/hal-05731432v2
AB - This paper investigates Lyapunov exponents and asymptotic growth phenomena for a class of random Horadam-type recurrences constrained by quadratic manifold geometries. For a fixed structural parameter k>0, we introduce the \emph{random k-Horadam recurrence}, defined by the initial values G_0,k=0, G_1,k=1 and the non-deterministic relation G_n,k=α_n,kG_n−1,k+β_n,kG_n−2,k for n≥2, where the random companion coefficients are constrained to the \emph{k-Constrained Sum-Product} (k-CSP) variety satisfying α_n,k+β_n,k=k and α_n,kβ_n,k=−k. Stochasticity is driven by an i.i.d.\ sign process selecting, at each discrete time step, one of the two admissible algebraic branches. This formulation induces a random matrix cocycle in GL_2(ℝ) supported on a discrete non-commutative manifold. Using Kingman’s subadditive ergodic theorem embedded in a projective normalization framework, we establish the almost sure existence of the top Lyapunov exponent, λ_k. Crucially, by establishing the non-compactness and strong irreducibility of the normalized semigroup for k ≥ 2, we characterize the top Lyapunov exponent λ_k and derive strict, non-trivial analytical bounds satisfying 1/2log(k) < λ_k < logδ_k. Here, δ_k = (k+√(k^2+4k))/2 is the positive root of t^2 − kt − k = 0, the characteristic equation governing the algebraic structure of the coefficients α_n,k and β_n,k. To complement the structural analysis, we establish the upper bound using the Joint Spectral Radius and the Berger-Wang theorem, while the lower bound is rigorously derived via the expectation of a two-step product norm. Furthermore, under appropriate log-integrability conditions, we analyze the structural stability of the system under both additive and multiplicative perturbations. Employing weak convergence criteria for random measures, we prove that the perturbed exponents vary continuously and converge to the unperturbed attractor in the small-noise limit. We also examine a singular, degenerate boundary regime where the multi-dimensional non-commutativity collapses, freezing the coefficient β_n,k to a constant parameter a∈(0,1]. This reduction isolates a scalar multiplicative process governed by a deterministic functional envelope k(a) = a^2/(a+1). We evaluate this scalar system via the Strong Law of Large Numbers and derive explicit functional and integrated two-sided bounds.
ER -