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Garland's Technique for Posets and High Dimensional Grassmannian Expanders [article]

Tali Kaufman, Ran J. Tessler
2022 arXiv   pre-print
In this work we develop a local to global machinery for general posets. We show that the high dimensional expansion notions and many recent expansion results have a generalization to posets.  ...  Examples are fast convergence of high dimensional random walks generalizing [KO,AL], an equivalence with a global random walk definition, generalizing [DDFH] and a trickling down theorem, generalizing  ...  The focus of our work is to define high dimensional expansion for general posets as a local to global property.  ... 
arXiv:2101.12621v3 fatcat:xru7b4ytunhobmaa7mfmvwc6dq

Hypercontractivity on High Dimensional Expanders: a Local-to-Global Approach for Higher Moments [article]

Mitali Bafna, Max Hopkins, Tali Kaufman, Shachar Lovett
2021 arXiv   pre-print
We handle these barriers with the introduction of two new tools of independent interest: a new explicit combinatorial Fourier basis for HDX that behaves well under restriction, and a new local-to-global  ...  method for analyzing higher moments.  ...  Most recent work on high dimensional expansion is based on the local-to-global paradigm, in which local properties of a complex are lifted to a desired global property (e.g. mixing or agreement testing  ... 
arXiv:2111.09444v2 fatcat:glm7cfkvgrbn7ck7knabem7fgq

Near Coverings and Cosystolic Expansion – an example of topological property testing [article]

Irit Dinur, Roy Meshulam
2019 arXiv   pre-print
This gives a new combinatorial-topological interpretation to cosystolic expansion which is a well studied notion of high dimensional expansion.  ...  Given a map f:Y→ X that satisfies almost all of the local conditions of being a cover, is it close to being a genuine cover of X? Complexes X for which this holds are called cover-stable.  ...  Motivation Expansion and stable local to global phenomena Garland's method [7] is a general way to deduce global information about a complex by looking at the local views, more specifically, at the local  ... 
arXiv:1909.08507v1 fatcat:l4csd67gyndhnflnw2rc2miln4

Agreement testing theorems on layered set systems [article]

Yotam Dikstein, Irit Dinur
2019 arXiv   pre-print
Previous work has shown that high dimensional expansion is useful for agreement tests. We extend these results to more general families of subsets, beyond simplicial complexes.  ...  These include - Agreement tests for set systems whose sets are faces of high dimensional expanders.  ...  Acknowledgement We wish to thank Prahladh Harsha for many helpful discussions.  ... 
arXiv:1909.00638v1 fatcat:gngutr7ywvatlgmi2gxzc4rzhi

Topology of random simplicial complexes: a survey [article]

Matthew Kahle
2013 arXiv   pre-print
This expository article is based on a lecture from the Stanford Symposium on Algebraic Topology: Application and New Directions, held in honor of Gunnar Carlsson, Ralph Cohen, and Ib Madsen.  ...  For a deeper discussion of Garland's method, see A. Borel's account in Séminaire Bourbaki [13] .  ...  Then the Nerve Lemma allows one to bootstrap local information about connectivity of a large number of random graphs into global information about cohomology vanishing.  ... 
arXiv:1301.7165v2 fatcat:kwn722crdranldqdjqd3dmsq7m

Sharp vanishing thresholds for cohomology of random flag complexes [article]

Matthew Kahle
2013 arXiv   pre-print
We describe the vanishing threshold and show that it is sharp. Using the same spectral methods, we also find a sharp threshold for the fundamental group π_1(X) to have Kazhdan's property (T).  ...  Combining with earlier results, we obtain as a corollary that for every k > 3 there is a regime in which the random flag complex is rationally homotopy equivalent to a bouquet of k-dimensional spheres.  ...  I first learned of applications of Garland's method in topological combinatorics from [1] , where Aharoni, Berger, and Meshulam establish a global analogue of the cohomology vanishing theorem for flag  ... 
arXiv:1207.0149v3 fatcat:gz2tk43b25gmvmb2kmyzwy2liu