Percolation transition for random forests in d≥3
Given a finite graph, the arboreal gas is the measure on forests
(subgraphs without cycles) in which each edge is weighted by a parameter
β>0. Equivalently this model is bond percolation conditioned to be a
forest, the independent sets of the graphic matroid, or the q→0 limit of
the random cluster representation of the q-state Potts model. Our
results rely on the fact that this model is also the graphical
representation of the non-linear sigma model with target space the
fermionic hyperbolic plane H^{0|2}.
The main question we are interested in is whether the arboreal gas
percolates, i.e., whether for a given β the forest has a connected
component that includes a positive fraction of the total edges of the
graph. We show that in two dimensions a Mermin-Wagner theorem associated
with a continuous symmetry of the non-linear sigma model implies that
the arboreal gas does not percolate for any β>0. On the other hand, in
three and higher dimensions, we show that percolation occurs for large β
by proving that the symmetry of the non-linear sigma model is
spontaneous broken. We also show that the broken symmetry is accompanied
by free field like fluctuations (Goldstone mode). This result is
achieved by a renormalisation group analysis combined with Ward
identities from the internal symmetry of the sigma model.
This talk is based on joint works with N. Crawford, T. Helmuth, and A.
Swan, and with N. Crawford and T. Helmuth.