Percolation for 2D Classical Heisenberg Model and Exit Sets of Vector Valued GFF

Our motivation in this paper is twofold. First, we study the geometry of a class of exploration sets, called <i>exit sets</i>, which are naturally associated with a 2D vector-valued Gaussian Free Field : <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>ϕ</mml:mi> <mml:mo>:</mml:mo> <mml:msup><mml:mrow><mml:mi>Z</mml:mi></mml:mrow> <mml:mn>2</mml:mn></mml:msup> <mml:mo>→</mml:mo> <mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow> <mml:mi>N</mml:mi></mml:msup> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn></mml:mrow> </mml:math> . We prove that, somewhat surprisingly, these sets are a.s. degenerate as long as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>N</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn></mml:mrow> </mml:math> , while they are conjectured to be macroscopic and fractal when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn></mml:mrow> </mml:math> . This analysis allows us, when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>N</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn></mml:mrow> </mml:math> , to understand the percolation properties of the level sets of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>{</mml:mo> <mml:mo>‖</mml:mo> <mml:mi>ϕ</mml:mi> <mml:mrow><mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo></mml:mrow> <mml:msub><mml:mo>‖</mml:mo> <mml:mn>2</mml:mn></mml:msub> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:msup><mml:mrow><mml:mi>Z</mml:mi></mml:mrow> <mml:mn>2</mml:mn></mml:msup> <mml:mo>}</mml:mo></mml:mrow> </mml:math> and leads us to our second main motivation in this work: if one projects a spin <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo></mml:mrow> </mml:math> model (the case <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn></mml:mrow> </mml:math> corresponds to the classical Heisenberg model) down to a spin <i>O</i>(<i>N</i>) model, we end up with a spin <i>O</i>(<i>N</i>) in a quenched disorder given by random conductances on <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> . Using the <i>exit sets</i> of the <i>N</i>-vector-valued GFF, we obtain a local and geometric description of this random disorder in the limit <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>β</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi></mml:mrow> </mml:math> . This allows us in particular to revisit a series of celebrated works by Patrascioiu and Seiler (J Stat Phys 69(3):573-595, 1992, Nucl Phys B Proc Suppl 30:184-191, 1993, J Stat Phys 106(3):811-826, 2002) which argued against Polyakov's prediction that spin <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo></mml:mrow> </mml:math> model is massive at all temperatures as long as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>N</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn></mml:mrow> </mml:math> (Polyakov in Phys Lett B 59(1):79-81, 1975). We make part of their arguments rigorous and more importantly we provide the following counter-example: we build ergodic environments of (arbitrary) high conductances with (arbitrary) small and disconnected regions of low conductances in which, despite the predominance of high conductances, the <i>XY</i> model remains massive. Of independent interest, we prove that at high <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>β</mml:mi></mml:math> , the fluctuations of a classical Heisenberg model near a north pointing spin are given by a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn></mml:mrow> </mml:math> vectorial GFF. This is implicit for example in Polyakov (1975) but we give here the first (non-trivial) rigorous proof. Also, independently of the recent work Dubédat and Falconet (Random clusters in the villain and xy models, arXiv preprint arXiv:2210.03620, 2022), we show that two-point correlation functions of the spin <i>O</i>(<i>N</i>) model can be given in terms of certain percolation events in the <i>cable graph</i> for any <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>N</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn></mml:mrow> </mml:math> .