The Phase Transition of Discrepancy in Random Hypergraphs

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 $H_1$ 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 $pn \rightarrow \infty$ and $pm \rightarrow \infty$, we show that with high probability (w.h.p.) $H_1$ has discrepancy at least $\Omega(2^{-n/m} \sqrt{pn})$ when $m = O(n)$, and at least $\Omega(\sqrt{pn \log\gamma })$ when $m \gg n$, where $\gamma = \min\{ m/n, pn\}$. 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 $d \rightarrow \infty$ and $dn/m \rightarrow \infty$, we prove that w.h.p. $H_{2}$ has discrepancy at least $\Omega(2^{-n/m} \sqrt{dn/m})$ when $m = O(n)$, and at least $\Omega(\sqrt{(dn/m) \log\gamma})$ when $m \gg n$, where $\gamma =\min\{m/n, dn/m\}$. Furthermore, we obtain nearly matching asymptotic upper bounds on the discrepancy in both models (when $p=d/m$), in the dense regime of $m \gg n$. 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 $O( \sqrt{dn/m} \log(m/n))$, provided $d \rightarrow \infty$, $d n/m \rightarrow \infty$ and $m \gg n$. This result is algorithmic, and together with the work of Bansal and Meka characterizes how the discrepancy of each random hypergraph model transitions from $\Theta(\sqrt{d})$ to $o(\sqrt{d})$ as $m$ varies from $m=\Theta(n)$ to $m \gg n$.