Let H_N be an N× N random matrix from the standard Gaussian unitary ensemble or Gaussian orthogonal ensemble, and let p_N(z)=operatornamePer(zI_N-H_N). Define the normalized zero-counting measure, with zeros counted with multiplicity, by mu_N=frac1N∑_p_N(z)=0delta_z. The conjecture asserts that, almost surely as N→∞, mu_N converges weakly to a semicircular law supported on the imaginary axis: for the GUE, the limit is the pushforward under x mapsto ix of the standard Wigner semicircle law on [-2,2], with support [-2i,2i]; for the GOE, under the normalization used in the conjecture, the analogous limit has support [-2√2,i,2√2,i].

Fyodorov’s conjecture on permanental roots of Gaussian random matrices

Let $H_N$ be an $N\times N$ random matrix from the standard Gaussian unitary ensemble or Gaussian orthogonal ensemble, and let $p_N(z)=\operatorname{Per}(zI_N-H_N)$ . Define the normalized zero-counting measure, with zeros counted with multiplicity, by $\mu_N=\frac{1}{N}\sum_{p_N(z)=0}\delta_z$ . The conjecture asserts that, almost surely as $N\to\infty$ , $\mu_N$ converges weakly to a semicircular law supported on the imaginary axis: for the GUE, the limit is the pushforward under $x\mapsto ix$ of the standard Wigner semicircle law on $[-2,2]$ , with support $[-2i,2i]$ ; for the GOE, under the normalization used in the conjecture, the analogous limit has support $[-2\sqrt{2}\,i,2\sqrt{2}\,i]$ .

References

Primary source

arXiv

Additional references

Rotated semicircle laws for permanental roots of Gaussian random matrices — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A new unrefereed preprint claims to prove the conjecture for two major Gaussian matrix models, while broader cases remain unsettled.

Fyodorov and collaborators proposed asymptotic laws for where the zeros of permanental polynomials cluster in Gaussian random-matrix ensembles in 2006. For the unitary and orthogonal models, the conjecture predicts semicircular limiting distributions on imaginary-axis segments.

Known results

Fyodorov and collaborators, 2006: derived correlation formulas for several ensembles but left the limiting zero distributions conjectural.
For GUE, the conjectured support is $[-2i,2i]$ with a semicircular density.
For GOE, the analogous support is $[-2\sqrt{2}\,i,2\sqrt{2}\,i]$ .

August 2026 claimed proof

On August 26, 2026, Rotated semicircle laws for permanental roots of Gaussian random matrices reported a proof of the conjectured limiting distributions for GOE and GUE. The result is unrefereed, so the claimed resolution remains unverified; extensions to other ensembles are not established here.

Current status (as of August 2026): The GOE and GUE cases are claimed solved by an unrefereed preprint, pending verification; other ensemble cases remain open or conjectural.

Sources

No solutions have been posted yet.