  On a finite graph, there is a natural family of Boltzmann probability
measures on cycle-rooted spanning forests, parametrized by weights on cycles.
For a certain subclass of those weights, we construct Gibbs measures in
infinite volume, as limits of probability measures on cycle-rooted spanning
forests of increasing sequences of finite graphs. Those probability measures
extend the family of already known random spanning forests and can be sampled
by a random walks algorithm which generalizes Wilson's algorithm. We show that,
unlike for uniform spanning forests, almost surely, all connected components
are finite and two-points correlations decrease exponentially fast with the
distance.
