Anticoncentration of independent complex Gaussian hafnians
An all-dimension inverse-variance bound gives polynomial lower-tail control for the independent complex symmetric ensemble.
Yuxuan Zhang
Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
Full manuscript (PDF, 5 pages) · LaTeX source, proof and checks (ZIP)
Status: a complete proof candidate, internally checked with AI assistance. External expert review and publication priority remain unconfirmed. This is the round-17 result, prepared for publication on 20 September 2026. It has not been formally verified by a proof assistant.
The question
Let \(X\) be a symmetric \(2n\times2n\) matrix with zero diagonal and independent \(\mathcal{CN}(0,1)\) entries above it, normalized by \(\mathbb E\lvert X_{ij}\rvert^2=1\). The QIQCOP problem asks for one polynomial \(p\) such that, for every \(n\geq1\) and \(0<\delta<1\),
\[\Pr\!\left(|\operatorname{Haf}(X)|<\frac{\sqrt{(2n-1)!!}}{p(n,1/\delta)}\right)<\delta.\]The candidate theorem
Write \(H_n=\operatorname{Haf}(X)\) and
\[V_n=\sum_{j=2}^{2n}|\operatorname{Haf}(X\text{ with vertices }1,j\text{ removed})|^2.\]The manuscript proves
\[\mathbb E[V_n^{-1}]\leq\frac1{(2n-2)!!},\qquad 0!!=1.\]Consequently, for every complex center \(z\) and every \(\varepsilon\geq0\),
\[\Pr\bigl(|H_n-z|\leq\varepsilon\sqrt{(2n-1)!!}\bigr) \leq\min\left\{1,\frac2{\sqrt\pi}\sqrt n\,\varepsilon^2\right\}.\]In particular, \(p(n,u)=2nu\) satisfies the exact catalog question, including strict inequality for all \(0<\delta<1\).
Why the correlated minors can be handled
Conditioning on all edges outside the first row makes the hafnian a circular Gaussian with variance \(V_n\). The problem is therefore to control an inverse moment of that variance.
An exact partition of perfect matchings exposes two vertices and expresses the hafnian as a complex Gaussian bilinear form plus linear terms. After realification, a Gaussian integral has positive endpoint values and a purely imaginary mixed term. Hölder’s inequality then permits two first-row coordinates to be compressed into one without increasing the comparison bound.
Repeated compression reduces the dimension. With an independent \(G\sim\operatorname{Gamma}(2n-1,1)\), the resulting comparison is
\[\mathbb E e^{-sV_n}\leq\mathbb E e^{-sGV_{n-1}}\qquad(s\geq0).\]Integrating this inequality gives the recurrence \(\mathbb E V_n^{-1}\leq(2n-2)^{-1}\mathbb E V_{n-1}^{-1}\). The base case is \(V_1=1\). At no step are different hafnian minors treated as independent.
Verification and attribution
Separate internal reviews checked the phase reduction, exact matching partition, conditional endpoint positivity, Laplace-order direction and all-dimension induction. The archive also checks 60 matching decompositions and 60 vertex-phase identities exactly over Gaussian integers for \(n=2,\ldots,6\). These finite checks support the algebra; the theorem rests on the analytic proof.
The Gaussian-kernel and coordinate-compression method is adapted from Koehler–Leung, arXiv:2607.20329v1, on permanents, and Zhao, arXiv:2609.06526v1, whose Theorem 2.3 concerns independent real symmetric Gaussian hafnians. Zhao, arXiv:2608.17065, treats a different complex Gram ensemble. The manuscript derives the independent-complex adaptation explicitly.
This resolves the literal lower-tail statement if the proof is confirmed. It does not transfer a bound to correlated Gram matrices or prove the separate average-case hardness assumptions motivated by Gaussian boson sampling.
Authorship and preparation
The proof, checking and exposition were developed with substantial AI assistance under the author’s direction. Internal AI reviews do not constitute independent human peer review. Scientific Agent Skills informed manuscript preparation and evidence tracking and is cited in the PDF. Primary papers retain their own terms and are not bundled in the source archive.