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