We show that the local limit of uniform spanning tree on any finite, simple, connected, regular graph sequence with degree tending to $$\infty $$ is Poisson(1) branching process conditioned survive forever. An extension “almost” graphs and a quenched version are also given.

In this paper, we develop a general approach to proving global and local uniform limit theorems for the Horvitz–Thompson empirical process arising from complex sampling designs. Global such as Glivenko–Cantelli Donsker theorems, asymptotic modulus related ratio-type are proved both process, its calibrated version. Limit of other variants their conditional versions also established. Our reveals ...