Anticoncentration of independent complex Gaussian hafnians
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Problem
Do independent complex Gaussian hafnians satisfy a polynomial lower-tail bound at their root-mean-square scale? For each integer \(n\geq1\), let \(X=X^T\in\mathbb C^{2n\times2n}\) have zero diagonal. The entries above the diagonal are independent with density \(\pi^{-1}e^{-|z|^2}\). Define
where \(\mathcal M_{2n}\) is the set of perfect matchings of \(\{1,\ldots,2n\}\). Does there exist a polynomial \(p\), positive on \([1,\infty)^2\), such that
The definitions in Eq. (1) fix the ensemble and normalization. One polynomial must satisfy Eq. (2) for every \(n\) and \(\delta\).
Source
This is a precise lower-tail formulation motivated by the hafnian anticoncentration discussion following Eq. (11) of Hamilton et al. [HKS+17]. Its explicitly normalized independent-entry ensemble is an editorial refinement, rather than a verbatim numbered conjecture.
Progress
The matching expansion gives \(\mathbb E|\operatorname{Haf}X|^2=h_n\) because only equal matchings survive the expectation. It fixes the scale in Eq. (2), but supplies no lower-tail estimate.
For a different ensemble, Zhao’s September 2026 preprint proves a polynomial small-ball bound. If \(S\) is real symmetric with independent standard real Gaussian entries above the diagonal, Theorem 2.3 gives
\begin{equation} \sup_{z\in\mathbb R}\Pr\!\left[ |\operatorname{Haf}S-z|\leq t\sqrt{h_n}\right] \leq\min\!\left\{1,\frac{2}{\sqrt\pi}n^{3/8}t\right\}. \tag{3} \end{equation}Equation (3) settles the real symmetric analogue. It does not state a theorem for the circular complex ensemble in Eq. (1) [Zha26].
Comment
The complex independent-entry lower tail remains the archived gap. Real symmetric matrices, finite-rank Gaussian products \(YY^T\), and the permanent’s special bipartite block ensemble have different laws. Their results cannot be transferred by substituting the matrix name. Average-case computational hardness is also a separate question.