The sharp anticoncentration conjecture for random spanning trees

Let $d$ be sufficiently large, and let $n$ be sufficiently large relative to $d$. Suppose that $G$ is a connected graph with $n$ vertices and minimum degree at least $d$, and let $\mathcal{T}$ be a uniformly random spanning tree of $G$.

Sharp anticoncentration conjecture. For every tree $T$,

$$\mathbb{P}(\mathcal{T} \cong T) \le n^{-(1/2-o_n(1))(d-1)}.$$

The conjecture proposes the optimal constant factor in the exponent and matches the anticoncentration scale suggested by the complete bipartite graph $K_{d,n-d}$. The paper proves a weaker anticoncentration bound, so this sharper estimate remains open.