The Phase Transition of Discrepancy in Random Hypergraphs
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Motivated by the Beck-Fiala conjecture, we study the discrepancy problem in two related models of random hypergraphs on n vertices and m edges. In the first (edge-independent) model, a random hypergraph H_1 is constructed by fixing a parameter p and allowing each of the n vertices to join each of the m edges independently with probability p. In the parameter range in which pn →∞ and pm →∞, we show that with high probability (w.h.p.) H_1 has discrepancy at least Ω(2^-n/m√(pn)) when m = O(n), and at least Ω(√(pn logγ)) when m ≫ n, where γ = min{ m/n, pn}. In the second (edge-dependent) model, d is fixed and each vertex of H_2 independently joins exactly d edges uniformly at random. We obtain analogous results for this model by generalizing the techniques used for the edge-independent model with p=d/m. Namely, for d →∞ and dn/m →∞, we prove that w.h.p. H_2 has discrepancy at least Ω(2^-n/m√(dn/m)) when m = O(n), and at least Ω(√((dn/m) logγ)) when m ≫ n, where γ =min{m/n, dn/m}. Furthermore, we obtain nearly matching asymptotic upper bounds on the discrepancy in both models (when p=d/m), in the dense regime of m ≫ n. Specifically, we apply the partial colouring lemma of Lovett and Meka to show that w.h.p. H_1 and H_2 each have discrepancy O( √(dn/m)log(m/n)), provided d →∞, d n/m →∞ and m ≫ n. This result is algorithmic, and together with the work of Bansal and Meka characterizes how the discrepancy of each random hypergraph model transitions from Θ(√(d)) to o(√(d)) as m varies from m=Θ(n) to m ≫ n.
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