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تسجيل مستخدم جديدRandom spanning forests and hyperbolic symmetry
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We study (unrooted) random forests on a graph where the probability of a forest is multiplicatively weighted by a parameter $beta>0$ per edge. This is called the arboreal gas model, and the special case when $beta=1$ is the uniform forest model. The arboreal gas can equivalently be defined to be Bernoulli bond percolation with parameter $p=beta/(1+beta)$ conditioned to be acyclic, or as the limit $qto 0$ with $p=beta q$ of the random cluster model. It is known that on the complete graph $K_{N}$ with $beta=alpha/N$ there is a phase transition similar to that of the ErdH{o}s--Renyi random graph: a giant tree percolates for $alpha > 1$ and all trees have bounded size for $alpha<1$. In contrast to this, by exploiting an exact relationship between the arboreal gas and a supersymmetric sigma model with hyperbolic target space, we show that the forest constraint is significant in two dimensions: trees do not percolate on $mathbb{Z}^2$ for any finite $beta>0$. This result is a consequence of a Mermin--Wagner theorem associated to the hyperbolic symmetry of the sigma model. Our proof makes use of two main ingredients: techniques previously developed for hyperbolic sigma models related to linearly reinforced random walks and a version of the principle of dimensional reduction.
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The arboreal gas is the probability measure on (unrooted spanning) forests of a graph in which each forest is weighted by a factor $beta>0$ per edge. It arises as the $qto 0$ limit with $p=beta q$ of the $q$-state random cluster model. We prove that
in dimensions $dgeq 3$ the arboreal gas undergoes a percolation phase transition. This contrasts with the case of $d=2$ where all trees are finite for all $beta>0$. The starting point for our analysis is an exact relationship between the arboreal gas and a fermionic non-linear sigma model with target space $mathbb{H}^{0|2}$. This latter model can be thought of as the $0$-state Potts model, with the arboreal gas being its random cluster representation. Unlike the $q>0$ Potts models, the $mathbb{H}^{0|2}$ model has continuous symmetries. By combining a renormalisation group analysis with Ward identities we prove that this symmetry is spontaneously broken at low temperatures. In terms of the arboreal gas, this symmetry breaking translates into the existence of infinite trees in the thermodynamic limit. Our analysis also establishes massless free field correlations at low temperatures and the existence of a macroscopic tree on finite tori.
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