Espuny Díaz, Lichev, and Wesolek's local-resilience conjecture for random geometric graphs

For integers $d\geq 1$, let $T_d(n,r)$ be the random graph obtained by sampling $n$ random points of the $d$-dimensional torus $(\mathds{R}/\mathds{Z})^d$ and joining two vertices whenever their Euclidean distance is at most $r$. For a graph $G$, an $\alpha$-subgraph is a subgraph $H\subseteq G$ satisfying $d_H(v)\geq \alpha d_G(v)$ for every vertex $v$. Espuny Díaz, Lichev, and Wesolek's conjecture. For every $\varepsilon\in(0,1/2]$ and integer $d\geq1$, there is a constant $C$ such that, when $r\geq C(\log n/n)^{1/d}$, $T_d(n,r)$ has, with high probability, the property that all of its $(1/2+\varepsilon)$-subgraphs are Hamiltonian. This is a random-geometric analogue of optimal local-resilience results for Hamiltonicity; the source does not specify a resolution.