2020/09/03 by Eyal Lubetzky, Lubetzky, Eyal, Yuval Peled +1 · 1 citation
Computer Science · Mathematics · #05C80 #60K35 #Combinatorics (math.CO) #FOS: Mathematics #Geometry and complex manifolds #Probability (math.PR) #Stochastic processes and statistical mechanics #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2009.01707
openalex publication_date 2020/09/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study noise sensitivity of properties of the largest components (\cal Cj)j≥ 1 of the random graph \cal G(n,p) in its critical window p=(1+λn-1/3)/n. For instance, is the property "|\cal C1| exceeds its median size" noise sensitive? Roberts and Şengül (2018) proved that the answer to this is yes if the noise ε is such that ε≫ n-1/6, and conjectured the correct threshold is ε≫ n-1/3. That is, the threshold for sensitivity should coincide with the critical window---as shown for the existence of long cycles by the first author and Steif (2015). We prove that for ε≫ n-1/3 the pair of vectors n-2/3(|\cal Cj|)j≥ 1 before and after the noise converges in distribution to a pair of i.i.d. random variables, whereas for ε≪ n-1/3 the ℓ2-distance between the two goes to 0 in probability. This confirms the above conjecture: any Boolean function of the vector of rescaled component sizes is sensitive in the former case and stable in the latter. We also look at the effect of the noise on the metric space n-1/3(\cal Cj)j≥ 1. E.g., for ε≥ n-1/3+o(1), we show that the joint law of the spaces before and after the noise converges to a product measure, implying noise sensitivity of any property seen in the limit, e.g., "the diameter of \cal C1 exceeds its median."