Degree fluctuations in 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 asked for the limiting distribution of the standardized degree $(D-\mathbb E D)/\sqrt{\operatorname{Var} D}$ of a vertex. We show that if $k\to\infty$ and $n-k\to\infty$, then $(k-D)/s_n$, where $s_n^2=k(n-k)/n$, converges in distribution and in its first two moments to $\phi(Z)+Z\Phi(Z)$ with $Z$ standard normal. Consequently $\mathbb E D=k-s_n/\sqrt\pi+o(s_n)$, $\operatorname{Var} D=(1/3+(\sqrt3-2)/(2\pi))s_n^2+o(s_n^2)$, and the standardized degree converges to an explicit law that is bounded above, hence not normal. If instead $r=n-1-k\ge1$ is fixed, then $k-D$ converges in distribution and in all moments to a mixed Poisson variable whose mean is an explicit function of a gamma variable with shape $r$. These limits depend on $r$ and, after standardization, converge to the interior law as $r\to\infty$. Together with the Erlang weighted tree of Moharrami, Subramanian, Liu and Sundaresan for fixed $k$, this identifies every subsequential limit of the standardized degree; no single law serves for all sequences $k(n)$. We also show that $\mathbb E(k-D)$ takes the same value at the budgets $k$ and $n-1-k$ for every $n$.