Diameter exponent in the small nonnegative-γ regime

Establish constants c_1, c_2, and γ^* such that, for every ε>0, all sufficiently large n, and γ≤γ^*, the two-weight uniform spanning tree on the complete graph satisfies n^{1/2-c_1γ}≤diam(𝕋)≤n^{1/2-c_2γ} with averaged probability at least 1−ε.

Background

The paper proves that negative γ yields the uniform spanning tree diameter scale of order n{1/2}, while γ≥5 yields a diameter of order n{1/3} log n up to a log log n factor. The intermediate interval γ∈[0,5) remains unresolved. The authors conjecture that, for sufficiently small γ, the diameter follows a power law whose exponent decreases linearly with γ. The subsequent remark explains that existing bottleneck estimates already provide a lower bound of the conjectured form, while a suitable upper bound remains the main difficulty.

References

There exists constants $c_1, c_2, \gamma*$ such that for any $\epsilon > 0$, $\gamma \leq \gamma*$ and $n \geq n_0(\varepsilon)$, we have \begin{align} E\Big[ _ \big( n{1/2 - c_1 \gamma} \leq diam() \leq n{1/2 - c_2 \gamma} \big) \Big] \geq 1 - \varepsilon. \end{align}

— Repeat times and a two-weight UST model  (2512.21977 - Ambroggio et al., 26 Dec 2025) in Conjecture~\ref{C:small_gamma}, Section 5, “Small and intermediate γ”