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Random forests and hyperbolic symmetry

Roland Bauerschmidt

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 H0|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 massless 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.