Local Search for Almost-Spanning Square Grids in Erdős--Rényi Random Graphs

RESUMO

Finding large lattice subgraphs in sparse Erdős--Rényi random graphs is a classical problem at the interface of random graph theory and algorithms. General bounded-degree embedding and universality theorems give powerful results for broad graph families, but when specialized to square grids they operate at densities substantially larger than the grid-emergence scale. In this paper we exploit the specific geometry of the square grid. We introduce the Quarantined Local Search, a three-phase local algorithm that separates the construction of an initial boundary from the later corner-closure process and controls adaptive negative exposure through bounded pair-test histories. We prove that, for every fixed $\delta\in (0,1)$, there exists $C_\delta>0$ such that the algorithm embeds a $k\times k$ square grid with $k^2\le (1-\delta)n$ in $G(n,p)$ with high probability whenever $p\ge C_\delta\sqrt{\ln k/n}$. Thus, for $k^2=\Theta(n)$, a density of order $\sqrt{\log n/n}$ is sufficient, a factor of order $\sqrt{\log n}$ above the corresponding $n^{-1/2}$ emergence scale.