Sharpness of the logarithmic correction in the large-γ diameter bound

Determine whether the typical diameter of the two-weight uniform spanning tree on the complete graph with γ≥5 is of order n^{1/3} log n, thereby eliminating the additional log log n factor in the proved upper bound.

Background

For γ≥5, the paper proves that the diameter is bounded below by a constant multiple of n{1/3} log n and above by a constant multiple of n{1/3} log n log log n. The authors attribute the extra log log n factor in the upper bound to a technical limitation arising from a union bound over several random walks. They expect the true order to match the lower bound.

References

We suspect that this term arises from technical limitations of our approach, and that the typical diameter of the tree should be of order $n{1/3} \log n$.

— Repeat times and a two-weight UST model  (2512.21977 - Ambroggio et al., 26 Dec 2025) in Section 1, immediately after Theorem 1 (Theorem~\ref{T:diam_extreme})