Conjecture on Airy point-process fluctuations of higher eigenvalues
Let be the hexagonal-lattice rhombus and let be its weighted adjacency operator in either the i.i.d. or mixed model. Let and be the constants from the mixed-model theorem, and let be the th smallest positive eigenvalue of . Define
Higher-eigenvalue Airy-process conjecture. For any fixed , as , the tuple converges in distribution to the top points of the Airy point process. This conjecture extends the proved Tracy–Widom fluctuation result for the smallest positive eigenvalue to any fixed number of higher positive eigenvalues in both models.
References
Primary source
Marcin Kotowski and Bálint Virág, “Tracy-Widom fluctuations in 2D random Schrodinger operators”, arXiv:1803.11208 (2018).
Progress summary
The conjecture remains unproved beyond the first eigenvalue, with only a partial estimate for several eigenvalues.
A 2018 paper formulates the conjecture that, for each fixed , the normalized first positive eigenvalues converge jointly to the top points of the Airy point process in both models. No later proof, counterexample, or verification was found.
Known results
- In the mixed model, the fluctuation theorem is proved with Tracy–Widom GUE behavior.
- In the mixed model, the product of the lowest eigenvalues is related, up to order , to a nonintersecting-path partition function.
- The corresponding result for the i.i.d. model is described as requiring additional polymer-fluctuation results, rather than as proved.
- Joint Airy point-process convergence for any is not established.
Current status (as of August 2026): The higher-eigenvalue Airy-process conjecture remains open in both models; only the mixed-model first-eigenvalue theorem and a partial multi-eigenvalue estimate are recorded.
Sources
Solutions 0
No solutions have been posted yet.