THE PHASE TRANSITION OF DISCREPANCY IN RANDOM HYPERGRAPHS

Authors

• CALUM MACRURY, University of Toronto,
• TOMÁŠ MASARÍK, Charles University, University of Warsaw, Simon Fraser University
• LEILANI PAI, University of Nebraska-LincolnFollow
• Xavier Perez Gimenez, University of Nebraska - LincolnFollow

Document Type

Article

Date of this Version

10-23-2021

Citation

Published (2023) SIAM Journal on Discrete Mathematics, 37 (3), pp. 1818-1841. DOI: 10.1137/21M1451427

Comments

Used by permission.

Abstract

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 H₁ 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.) H₁ has discrepancy at least Ω(2^-n/m √pn) when m = O(n), and at least Ω( √pnlogү) when m ≽ n, where ү= min{m/n; pn}. In the second (edge-dependent) model, d is fixed and each vertex of H₂ 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. H₂ has discrepancy at least Ω(2^-n/m √dn/m) when m = O(n), and at least Ω( √(dn/m) logү) when m ≽ n, where ү= 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. H₁ and H₂ each have discrepancy O( √dn/m log(m/n)), provided d ⟶ ∞, dn=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.