Let X_n be distributed according to the Laguerre ensemble with parameter alpha=n, and define the rescaled inverse power traces tau_k(n)=n^k-1operatornameTrX_n^-kqquad (k≥ 0). Their expectations have the expansion Etau_k(n)=∑_g=0^∞kappa_g(k)n^-g. Positive-integrality conjecture. For beta∈1,2, the expansion coefficients satisfy kappa_g(k)∈N. This conjecture concerns the 1/n-expansion of cumulants and moments of the time-delay matrix for ballistic chaotic cavities, where the inverse power traces are singular spectral statistics of the Laguerre ensemble. The source gives no resolution, so the conjecture remains open.

Positive-integrality conjecture for time-delay matrix cumulants

About 16 years old · traced to 2010

Let $X_n$ be distributed according to the Laguerre ensemble with parameter $\alpha=n$ , and define the rescaled inverse power traces

\tau_k(n)=n^{k-1}\operatorname{Tr}X_n^{-k}\qquad (k\geq 0).

Their expectations have the expansion

\mathbb{E}\tau_k(n)=\sum_{g=0}^{\infty}\kappa_g(k)n^{-g}.

Positive-integrality conjecture. For $\beta\in\{1,2\}$ , the expansion coefficients satisfy

\kappa_g(k)\in\mathbb{N}.

This conjecture concerns the $1/n$ -expansion of cumulants and moments of the time-delay matrix for ballistic chaotic cavities, where the inverse power traces are singular spectral statistics of the Laguerre ensemble. The source gives no resolution, so the conjecture remains open.

References

Primary source

Fabio Deelan Cunden, Francesco Mezzadri, Neil O'Connell and Nick Simm, “Moments of random matrices and hypergeometric orthogonal polynomials”, arXiv:1805.08760 (2019).

Additional references

2 papers in this index state this conjecture (2010–2018). The statement above is taken from the most recent of them; the others are arXiv:1007.4181.

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed progress

The complex case is claimed solved, while the real case remains open.

Formulated in 2016, the conjecture asserts positive integer coefficients throughout the inverse-trace expansion for both symmetry classes $\beta=1$ and $\beta=2$ . The 2018 literature reports a resolution only for the complex class.

Known results

For $\beta=2$ , finite computations established the claim through $k\leq 10000$ and $g\leq 80$ (2016).

2018 complex-case resolution

The 2018 paper reports a proof for $\beta=2$ : the expansion is in $n^{-2}$ with positive integer coefficients, using reciprocity and a positive-integer permutation formula. It does not resolve $\beta=1$ .

Current status (as of August 2026): The $\beta=2$ case is reported as resolved, but remains unverified here; the $\beta=1$ case remains open.

Sources

No solutions have been posted yet.