The connectivity window of bilateral preference random graphs

In the bilateral preference random graph $LK(n,k)$ of La and Kabkab, independent uniform scores are placed on the edges of $K_n$, each vertex prefers its $k$ incident edges of smallest score, and an edge is kept when both of its endpoints prefer it. Dabirian and Subramanian showed that $LK(n,k)$ is connected with high probability when $k\ge\log n+t\log\log n\sqrt{\log n}$ with $t>1/2$ and disconnected when $t<-1/2$, and left the window between these two conditions open. We show that for every fixed $C$, uniformly over integers $k$ with $|k-\log n|\le C$, the number of isolated vertices is asymptotically Poisson with mean $ne^{-k}$ in total variation and the probability that $LK(n,k)$ is connected is $\exp(-ne^{-k})+o(1)$. Consequently $LK(n,k)$ is connected with high probability if and only if $k-\log n\to\infty$, and disconnected with high probability if and only if $k-\log n\to-\infty$. The main estimate is that a fixed set of $r$ vertices is isolated with probability $e^{-rk}(1+o(1))$; it comes from an exact integral formula for the rejections at one vertex and a direct count of the few edges through which the vertices of the set can interact.