Characterization of DRσ(k,w) in the intermediate regime of window sizes

Characterize the behavior of the expected density DRσ(k,w) of random minimizers in the intermediate regime k < w < (σ/(σ−1))σ^k ln(σ^k), deriving tight asymptotic bounds or explicit formulas that bridge the gap between the small-w approximation DRσ(k,w) ≈ 2/(w+1) (for w ≤ k) and the very-large-w limit DRσ(k,w) ≈ σ^−k.

Background

The authors provide a detailed description of DRσ(k,w) for w ≤ k, and a near-tight characterization for very large w showing DRσ(k,w) ≈ σ^−k when w ≈ (σ/(σ−1))σ^k(ln σ^k + g(N)).

For medium values of w (between these regimes), the paper establishes monotonicity but lacks quantitative characterization. The authors explicitly state that filling this intermediate range is open, highlighting the need for asymptotics or bounds covering k < w < (σ/(σ−1))σ^k ln(σ^k).