Long twins in random words

TL;DR

This paper studies the maximum length of identical disjoint subwords (twins) in random words over finite alphabets, providing improved probabilistic bounds that surpass previous deterministic estimates for small alphabet sizes.

Contribution

It establishes new probabilistic lower bounds for the length of twins in random words, improving upon known deterministic bounds especially for small alphabet sizes.

Findings

In a random ternary word, twins of length at least 0.41n exist with high probability.

For alphabets of size k ≥ 3, lower bounds of approximately 1.64/(k+1) n are achieved.

Results extend to multiple twins, showing similar probabilistic bounds.

Abstract

Twins in a finite word are formed by a pair of identical subwords placed at disjoint sets of positions. We investigate the maximum length of twins in a random word over a $k$-letter alphabet. The obtained lower bounds for small values of $k$ significantly improve the best estimates known in the deterministic case. Bukh and Zhou in 2016 showed that every ternary word of length $n$ contains twins of length at least $0.34n$. Our main result states that in a random ternary word of length $n$, with high probability, one can find twins of length at least $0.41n$. In the general case of alphabets of size $k\geq 3$ we obtain analogous lower bounds of the form $\frac{1.64}{k+1}n$ which are better than the known deterministic bounds for $k\leq 354$. In addition, we present similar results for multiple twins in random words.