Random Spanning Forests and Hyperbolic Symmetry
Roland Bauerschmidt, Nicholas Crawford, Tyler Helmuth, Andrew Swan
Abstract
Abstract We study (unrooted) random forests on a graph where the probability of a forest is multiplicatively weighted by a parameter $$\beta >0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> per edge. This is called the arboreal gas model, and the special case when $$\beta =1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> is the uniform forest model. The arboreal gas can equivalently be defined to be Bernoulli bond percolation with parameter $$p=\beta /(1+\beta )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mi>β</mml:mi> <mml:mo>/</mml:mo> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>β</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> conditioned to be acyclic, or as the limit $$q\rightarrow 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>→</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> with $$p=\beta q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mi>β</mml:mi> <mml:mi>q</mml:mi> </mml:mrow> </mml:math> of the random cluster model. It is known that on the complete graph $$K_{N}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:math> with $$\beta =\alpha /N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>=</mml:mo> <mml:mi>α</mml:mi> <mml:mo>/</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> there is a phase transition similar to that of the Erdős–Rényi random graph: a giant tree percolates for $$\alpha > 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> and all trees have bounded size for $$\alpha <1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo><</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . 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 $${\mathbb {Z}}^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>Z</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> for any finite $$\beta >0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . 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.