Sampling biased spanning forests through marked vertices

Abstract

We study biased spanning forests on the complete graph \(K_N\), obtained by assigning weight proportional to \(κ^q\) to spanning trees on \(K_N\cup\{Δ\}\), where \(q\) is the degree of a distinguished root \(Δ\). Fixing a finite set \(L\) of marked vertices, we analyze the minimal \(Δ\)-rooted subtree connecting \(L\) to \(Δ\). In this sense, we investigate the effect of partially sampling a large random spanning forest through finitely many vertices. For fixed \(κ\), the reduced subtree is asymptotically a uniformly distributed binary tree, and graph distances rescaled by \(\sqrt N\) converge jointly in distribution to the explicit limit introduced by Aldous in his study of the Brownian CRT. We show that the scale \(κ\asymp\sqrt N\) is critical: if \(κ=o(\sqrt N)\), the marked vertices lie asymptotically in a single component; if \(κ\gg\sqrt N\), the induced partition is asymptotically discrete and the distances to \(Δ\) are negligible on the \(\sqrt N\) scale. In the critical regime \(κ=c\sqrt N\), the induced partition of the marked vertices converges to a non-degenerate limit law, namely the \(\tfrac12\)-stable Poisson--Kingman partition studied by Pitman. We also describe a continuous-time edge-cutting dynamics: under the critical scaling, its fixed-time marginals are obtained by the parameter shift \(c\mapsto c+a\). Our approach remains entirely discrete and combinatorial: the asymptotics are obtained from explicit determinant formulas and elementary expansions, without invoking continuum limits.