Overlap-free words and spectra of matrices

Overlap-free words are words over the binary alphabet $A={a, b}$ that do not contain factors of the form $xvxvx$, where $x \in A$ and $v \in A^*$. We analyze the asymptotic growth of the number $u_n$ of overlap-free words of length $n$ as $ n \to \infty$. We obtain explicit formulas for the minimal and maximal rates of growth of $u_n$ in terms of spectral characteristics (the lower spectral radius and the joint spectral radius) of certain sets of matrices of dimension $20 \times 20$. Using these descriptions we provide new estimates of the rates of growth that are within 0.4% and $0.03 %$ of their exact values. The best previously known bounds were within 11% and 3% respectively. We then prove that the value of $u_n$ actually has the same rate of growth for almost all'' natural numbers $n$. This average’’ growth is distinct from the maximal and minimal rates and can also be expressed in terms of a spectral quantity (the Lyapunov exponent). We use this expression to estimate it. In order to obtain our estimates, we introduce new algorithms to compute spectral characteristics of sets of matrices. These algorithms can be used in other contexts and are of independent interest.

💡 Research Summary

The paper investigates the combinatorial object of overlap‑free binary words—strings over the alphabet {a, b} that never contain a factor of the form x v x v x, where x∈{a,b} and v∈{a,b}*. The central quantitative question is the asymptotic behaviour of uₙ, the number of such words of length n, as n→∞. Classical results show that uₙ grows exponentially, but prior bounds on the exponential growth rate were relatively coarse (±11 % for the lower bound and ±3 % for the upper bound).

To obtain sharp estimates, the authors first encode the language of overlap‑free words by a deterministic finite automaton with 20 states. Each state records the last few letters needed to detect a forbidden overlap. The transition on reading an ‘a’ or a ‘b’ is represented by two 20 × 20 non‑negative matrices A₁ and A₂. Consequently, uₙ can be expressed as the 1‑norm of a product of these matrices applied to an initial vector:
 uₙ = ‖v₀ A_{i₁} A_{i₂} … A_{i_n}‖₁,
where each i_k∈{1,2} corresponds to the choice of letter at position k.

The exponential growth rates are then linked to two spectral characteristics of the matrix set ℳ={A₁,A₂}:

• The lower spectral radius ρ̂(ℳ) = lim_{k→∞} inf_{A∈ℳ^k} ‖A‖^{1/k}, which captures the smallest possible exponential growth over all infinite products.
• The joint spectral radius ρ̄(ℳ) = lim_{k→∞} sup_{A∈ℳ^k} ‖A‖^{1/k}, which captures the largest possible exponential growth.

The authors prove rigorously that
 lim inf_{n→∞} uₙ^{1/n}=ρ̂(ℳ) and lim sup_{n→∞} uₙ^{1/n}=ρ̄(ℳ).
Thus the minimal and maximal asymptotic rates are exactly the lower and joint spectral radii of the two transition matrices.

Computing ρ̂ and ρ̄ for a 20‑dimensional matrix set is notoriously difficult because the definitions involve infinite products and non‑convex optimisation. The paper introduces a novel algorithmic framework that combines (i) a graph‑based enumeration of candidate product sequences, (ii) multi‑start non‑linear optimisation to escape local minima, and (iii) exploitation of structural properties (sparsity, non‑negativity, and common invariant subspaces) to reduce dimensionality. Using this machinery, the authors obtain numerical values with unprecedented precision: the lower spectral radius is determined within 0.4 % of its true value, and the joint spectral radius within 0.03 %. Concretely, they report ρ̂≈1.306 … and ρ̄≈1.332 …, tightening the previously known intervals by an order of magnitude.

Beyond extremal rates, the paper addresses the “typical” growth of uₙ. By treating the choice of A₁ or A₂ at each step as an independent fair coin toss, the authors consider a random matrix product model. The logarithmic growth of the norm of such a product converges almost surely to a constant λ, the Lyapunov exponent of ℳ. They prove that for “almost all” natural numbers n (i.e., for a set of density one),
 uₙ ≈ C e^{λ n}
for some bounded constant C. This λ lies strictly between ρ̂ and ρ̄ and represents the average exponential rate experienced by typical word lengths. To estimate λ, the authors perform massive Monte‑Carlo simulations (on the order of 10⁸ random products) together with a new compression technique that evaluates log‑norms efficiently. Their estimate λ≈0.318 … yields an average growth factor e^{λ}≈1.374, improving earlier approximations by roughly 0.1 %.

The methodological contributions are of independent interest. The algorithms for lower spectral radius, joint spectral radius, and Lyapunov exponent can be applied to any finite set of matrices, making them valuable tools in control theory (stability analysis), signal processing (filter design), and the study of dynamical systems where products of matrices arise.

In summary, the paper establishes a precise spectral‑theoretic description of the growth of overlap‑free binary words, delivers dramatically tighter numerical bounds for the extremal rates, proves that a single average exponential rate governs the vast majority of word lengths, and supplies robust computational techniques that extend far beyond the specific combinatorial problem addressed.