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The lambda-spanning forest on a graph $G$ is a random subgraph of $G$ that is equal to a given forest with probability weighted by the number of components of the forest, and the sizes of these components. It is an extension of the classical model of the uniform spanning tree (UST), which corresponds to the case lambda=0.

It was shown by Grimmett in 1980 that the local limit of the UST of the complete graph is the Poisson(1) Bienaymé-Galton-Watson tree conditioned to survive. An analogous result was recently obtained for lambda-spanning forests on the complete graph by D'Achillle, Enriquez and Melotti (2026), who showed that the lambda-SF exhibits three types of limiting behaviour, depending on how the parameter lambda grows compared to $n$.

We will discuss a recent result with Breki Pálsson showing that the same result holds for the lambda spanning forests of any (almost) regular graph sequence with degrees tending to infinity; in this case the parameter must be tuned as a function of the degrees.