Asymptotic local resilience of binomial random graphs for Hamiltonicity

Let $\mathcal{G}(n,p)$ be the binomial random graph on $n$ vertices, let $\mathcal{HAM}$ denote the property of being Hamiltonian, and let $r_\ell(G,\mathcal{HAM})$ denote the local resilience of $G$ with respect to Hamiltonicity. Binomial Hamiltonicity resilience conjecture. For every $\varepsilon>0$ there exists an integer $K(\varepsilon)>0$ such that for every $p \geq \frac{K\ln n}{n}$, w.h.p.

$$\left|r_\ell(\mathcal{G}(n,p),\mathcal{HAM})-\frac{np}{2}\right|\leq\varepsilon np.$$

This predicts that throughout the logarithmic-degree range and above, the local resilience of a typical binomial random graph is asymptotic to half its expected degree. The paper notes that the corresponding conjectural order was previously suggested by Sudakov and Vu.