For every fixed integer d≥ 3, there exists a constant c_d>0 such that, for every connected d-regular graph G on n vertices and every tree T on n vertices, if T_G is a uniformly random spanning tree of G, then P(T_G cong T)≤ exp(-c_d n).

H. Lee’s spanning-tree anticoncentration conjecture

H. Lee’s conjecture predicts that no particular tree can occur too often as the uniformly random spanning tree of a sufficiently dense connected graph, implying many non-isomorphic spanning trees. The new work also claims the sharp asymptotic estimate underlying this concentration statement.

Known results

Almost-regular graphs: Lee’s setting was previously known to satisfy exponential anticoncentration, with $\mathbb{P}(\mathcal{T}\cong T)\le e^{-\Omega(n)}$ for every $n$ -vertex tree $T$ (2026).

October 2026 claimed solution

On October 6, 2026, Beke, Bošković, Kamčev, and Wang reported an unrefereed preprint proving, for $\Delta(G)\le d$ , $\operatorname{inj}(T,G)\le (d/e)^n e^{o_d(1)n}$ , and deriving Lee’s conjecture for connected $d$ -regular graphs. A separate 2026 preprint claims the stronger minimum-degree conclusion $\mathbb{P}(\mathcal{T}\cong T)\le n^{-(1/2-o_n(1))(d-1)}$ .

Current status (as of October 2026): The conjecture has claimed proofs in unrefereed preprints, but the resolution remains unverified.

H. Lee’s spanning-tree anticoncentration conjecture

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