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The Doob transform technique enables the study of a killed random walk (KRW)via a random walk (RW) with transition probabilities tilted by a discretemassive harmonic function. The main contribution of this paper is to transferthis powerful technique to statistical mechanics by relating two models, namelyrandom rooted spanning forests (RSF) and random spanning trees (RST), andprovide applications. More precisely, our first main theorem explicitly relatesmodels on the level of partition functions, and probability measures, in thecase of finite and infinite graphs. Then, in the planar case, we also rely onthe dimer model: we introduce a killed and a drifted dimer model, extending tothis general framework the models introduced in [Chh12,dT20]. Using Temperley'sbijection between RST and dimers, this allows us to relate RSF to dimers andthus extend partially this bijection to RSF. As immediate applications, we givea short and transparent proof of Kenyon's result stating that the spectralcurve of RSF is a Harnack curve, and provide a general setting to relatediscrete massive holomorphic and harmonic functions. The other importantapplication consists in proving universality of the convergence of thenear-critical loop-erased RW, RST and dimer models by extending the results of[Chh12,CW21,HSB22] from the square lattice to any isoradial graphs: weintroduce a loop erased RW, RST and dimer model on isoradial discretizations ofany simply connected domain and prove convergence in the massive scaling limittowards continuous objects described by a massive version of SLE2.

The Doob transform and the tree behind the forest, with application to near-critical dimers