Open quantitative question of Leake and Oveis Gharan
Determine the sharp quantitative local-to-global spectral bound for down-up walks on the faces of a pure simplicial complex, assuming sufficiently strong spectral expansion of the codimension- links. In particular, for the notation of the Leake–Oveis Gharan trickle-down theorem, establish or disprove the claimed sharpening , where is the relevant down-up walk, is the dimension parameter, is the face level, and is the local spectral-expansion parameter.
References
Primary source
Additional references
- Spectral Gap of Down-Up Walks via Trickle-Down: A Simplified and Sharpened Analysis — arXiv — Xiaoyu Chen, Kuikui Liu
Progress summary
A new paper claims to settle the open quantitative question, but its resolution has not been independently verified.
The question asks for sharper quantitative local-to-global spectral bounds in trickle-down theorems, with consequences for mixing and high-dimensional expanders.
September 2026 claimed resolution
On September 17, 2026, Xiaoyu Chen and Kuikui Liu reported that their paper Spectral Gap of Down-Up Walks via Trickle-Down: A Simplified and Sharpened Analysis gives a sharper dependence, including estimates such as , and thereby resolves the question. A related paper claims optimal trickle-down constants for path complexes; these are claimed results, not independently verified here. The authors also report similar strengthening by Guo and Zhang, but no source for that work was supplied.
Current status (as of September 2026): Chen and Liu claim a quantitative resolution, while independent verification of the proof remains outstanding.
Trickle-down spectral-gap question receives a sharpened analysis
Solutions 0
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