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Ding-Ding Dong

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Preprint Sep 2026

Optimal girth-dependent bounds for the Bethe approximation of the permanent

For an $n\times n$ nonnegative matrix $A$, the Bethe permanent, which is computable in deterministic polynomial time, satisfies the tight universal comparison \[\operatorname{Bethe}(A) \leq \operatorname{per}(A) \leq 2^{n/2}\operatorname{Bethe}(A).\] The lower bound, due to Gurvits, is attained on forests. The upper bo...

Ding-Ding Dong, Vishesh Jain · 2 citations · ⚡1

Preprint Aug 2026

The Bethe-Hessian down to the Percolation Threshold

The Bethe-Hessian is a symmetric matrix for which the negative spectrum has been observed to encode the informative structure of sparse stochastic block models. We prove that, in the stochastic block model where all vertices have expected degree $d>1$, the number of negative eigenvalues of the Bethe-Hessian is exactly...

Ding-Ding Dong, Theo McKenzie · 0 citations