An Improved Trickle down Theorem for Partite Complexes

Authors Dorna Abdolazimi, Shayan Oveis Gharan

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Dorna Abdolazimi
  • University of Washington, Seattle, WA, USA
Shayan Oveis Gharan
  • University of Washington, Seattle, WA, USA


The discussion that initiated this work took place at the DIMACS Workshop on Entropy and Maximization. We would like to thank the DIMACS center and the workhop organizers for making this happen. In particular, we would like to thank Tali Kaufman for raising the question of an improved trickle down theorem for sparse simplical complexes in that workshop. We also would like to thank Ryan O’Donnell and Kevin Pratt for helpful discussions on high dimensional expanders based on Chevalley groups.

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Dorna Abdolazimi and Shayan Oveis Gharan. An Improved Trickle down Theorem for Partite Complexes. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 10:1-10:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Expander graphs and randomness extractors
  • Theory of computation → Error-correcting codes
  • Theory of computation → Random walks and Markov chains
  • Simplicial complexes
  • High dimensional expanders
  • Trickle down theorem
  • Bounded degree high dimensional expanders
  • Locally testable codes
  • Random walks


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