A-Localized States for Clock Models on Trees and Their Extremal Decomposition into Glassy States

Abstract We consider $$\mathbb {Z}_q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Z</mml:mi> <mml:mi>q</mml:mi> </mml:msub> </mml:math> -valued clock models on a regular tree, for general classes of ferromagnetic nearest neighbor interactions which have a discrete rotational symmetry. It has been proved recently that, at strong enough coupling, families of homogeneous Markov chain Gibbs states $$\mu _A$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>μ</mml:mi> <mml:mi>A</mml:mi> </mml:msub> </mml:math> coexist whose single-site marginals concentrate on $$A\subset \mathbb {Z}_q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>⊂</mml:mo> <mml:msub> <mml:mi>Z</mml:mi> <mml:mi>q</mml:mi> </mml:msub> </mml:mrow> </mml:math> , and which are not convex combinations of each other [1]. In this note, we aim at a description of the extremal decomposition of $$\mu _A$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>μ</mml:mi> <mml:mi>A</mml:mi> </mml:msub> </mml:math> for $$|A|\ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>A</mml:mi> <mml:mo>|</mml:mo> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> into all extremal Gibbs measures, which may be spatially inhomogeneous. First, we show that in regimes of very strong coupling, $$\mu _A$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>μ</mml:mi> <mml:mi>A</mml:mi> </mml:msub> </mml:math> is not extremal. Moreover, $$\mu _A$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>μ</mml:mi> <mml:mi>A</mml:mi> </mml:msub> </mml:math> possesses a single-site reconstruction property which holds for spin values sent from the origin to infinity, when these initial values are chosen from A . As our main result, we show that $$\mu _A$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>μ</mml:mi> <mml:mi>A</mml:mi> </mml:msub> </mml:math> decomposes into uncountably many extremal inhomogeneous states. The proof is based on multi-site reconstruction which allows to derive concentration properties of branch overlaps. Our method is based on a new good site/bad site decomposition adapted to the A -localization property, together with a coarse graining argument in local state space.