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2016

Gap Probabilities and Betti Numbers of a Random Intersection of Quadrics

2 years 11 months ago
Gap Probabilities and Betti Numbers of a Random Intersection of Quadrics
We consider the Betti numbers of an intersection of k random quadrics in RPn . Sampling the quadrics independently from the Kostlan ensemble, as n → ∞ we show that for each i ≥ 0 the expected ith Betti number satisfies: Ebi(X) = 1 + O(n−M ) for all M > 0. In other words, each fixed Betti number of X is asymptotically expected to be one; in fact as long as i = i(n) is sufficiently bounded away from n/2 the above rate of convergence is uniform (and in this range Betti numbers concentrate to their expected value). For the special case k = 2 we study the expectation of the sum of all Betti numbers of X. It was recently shown [27] that this expected sum equals n + o(n); here we sharpen this asymptotic, showing that: (1) n j=0 Ebj (X) = n + 2 √ π n1/2 + O(nc ) for any c > 0. (the term 2√ π n1/2 comes from contributions of middle Betti numbers). The proofs are based on a combination of techniques from random matrix theory and spectral sequences. In particular (1) is base...
Antonio Lerario, Erik Lundberg
Added 01 Apr 2016
Updated 01 Apr 2016
Type Journal
Year 2016
Where DCG
Authors Antonio Lerario, Erik Lundberg
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