We prove a strengthening of the trickle down theorem for partite complexes. Given a (d+1)-partite d-dimensional simplicial complex, we show that if "on average" the links of faces of co-dimension 2 are (1-δ)/d-(one-sided) spectral expanders, then the link of any face of co-dimension k is an O((1-δ)/(kδ))-(one-sided) spectral expander, for all 3 ≤ k ≤ d+1. For an application, using our theorem as a black-box, we show that links of faces of co-dimension k in recent constructions of bounded degree high dimensional expanders have spectral expansion at most O(1/k) fraction of the spectral expansion of the links of the worst faces of co-dimension 2.
@InProceedings{abdolazimi_et_al:LIPIcs.CCC.2023.10, author = {Abdolazimi, Dorna and Oveis Gharan, Shayan}, title = {{An Improved Trickle down Theorem for Partite Complexes}}, booktitle = {38th Computational Complexity Conference (CCC 2023)}, pages = {10:1--10:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-282-2}, ISSN = {1868-8969}, year = {2023}, volume = {264}, editor = {Ta-Shma, Amnon}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.10}, URN = {urn:nbn:de:0030-drops-182807}, doi = {10.4230/LIPIcs.CCC.2023.10}, annote = {Keywords: Simplicial complexes, High dimensional expanders, Trickle down theorem, Bounded degree high dimensional expanders, Locally testable codes, Random walks} }
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