  We give estimates of the critical parameter for random loop models that are
related to quantum spin systems. A special case of the model that we consider
is the interchange- or random-stirring process. We consider here the model
defined on regular trees of large degrees, which are expected to approximate
high spatial dimensions. We find a critical parameter that indeed shares
similarity with existing numerical results for the cubic lattice. In the case
of the interchange process our results improve on earlier work by Angel and by
Hammond, in that we determine the second-order term of the critical parameter.
