The giant component of bilateral preference graphs with random budgets

Place independent continuous costs on the edges of $K_n$, give each vertex $i$ an independent budget $D_i$ with law $P$ on the positive integers, let $i$ prefer its $D_i$ cheapest incident edges, and keep an edge when both endpoints prefer it. Moharrami, Subramanian, Liu and Sundaresan showed that the local limit of this graph is the Erlang weighted tree, so that the fraction of vertices in the largest component is asymptotically at most the survival probability $\rho(P)$ of that tree, and asked whether the reverse inequality holds. We prove that it does whenever $P$ has a finite mean: the largest component contains $(\rho(P)+o(1))n$ vertices with probability tending to one. The same holds for deterministic budget arrays whose empirical laws converge weakly to $P$. The proof runs the graph in cost time, shows that every large component of the graph built from edges of cost at most $t$ contains many vertices that can still accept edges, and joins these components by the edges arriving shortly after $t$. It remains to show that the survival probability of the tree with its edges above cost $t$ removed tends to $\rho(P)$. This is done through a return process to a compact set of types, whose mean operator is self-adjoint in a weighted space.