Single-threshold separation of DFRσ(k,w) above and below 2 for arbitrary alphabets

Establish whether, for every alphabet size σ, there exists a single integer threshold wσ such that for all integers k ≥ wσ and 2 ≤ w ≤ k, the density factor DFRσ(k,w) of random minimizers satisfies DFRσ(k,w) ≥ 2 for w < wσ and DFRσ(k,w) < 2 for w ≥ wσ; equivalently, prove that DRσ(k,w) crosses the limit value 2/(w+1) only once as w increases (for w ≤ k) for every σ.

Background

The paper analyzes the expected density DRσ(k,w) of random minimizers and its normalized form DFRσ(k,w)=(w+1)DRσ(k,w) for the regime w ≤ k. They show DFRσ(k,w)>2 for small w and DFRσ(k,w)<2 for sufficiently large w, and present numerical evidence suggesting a single transition threshold in w for fixed σ.

In Remark r:transit the authors note that experiments for small alphabets indicate a single constant separating the zones with DFRσ(k,w)>2 and DFRσ(k,w)<2, but they lack a proof for arbitrary σ. Proving the existence of such a unique threshold would sharpen their qualitative description of the density’s horizontal behavior.