Topological Phases in the Plaquette Random-Cluster Model and Potts Lattice Gauge Theory

Abstract The i -dimensional plaquette random-cluster model on a finite cubical complex is the random complex of i -plaquettes with each configuration having probability proportional to $$p^{\#\text { of plaquettes}}\left( 1-p \right) ^{\#\text { of complementary plaquettes}}q^{\textbf{b}_{i-1}},$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>p</mml:mi> <mml:mrow> <mml:mo>#</mml:mo> <mml:mspace/> <mml:mtext>of plaquettes</mml:mtext> </mml:mrow> </mml:msup> <mml:msup> <mml:mfenced> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mi>p</mml:mi> </mml:mfenced> <mml:mrow> <mml:mo>#</mml:mo> <mml:mspace/> <mml:mtext>of complementary plaquettes</mml:mtext> </mml:mrow> </mml:msup> <mml:msup> <mml:mi>q</mml:mi> <mml:msub> <mml:mi>b</mml:mi> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:msup> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> where $$q\ge 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> is a real parameter and $$\textbf{b}_{i-1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>b</mml:mi> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:math> denotes the rank of the $$(i-1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>i</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -homology group with coefficients in a specified coefficient field. When q is prime and the coefficient field is $$\mathbb {F}_q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>q</mml:mi> </mml:msub> </mml:math> , this model is coupled with the $$(i-1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>i</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -dimensional q -state Potts lattice gauge theory. We prove that the probability that an $$(i-1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>i</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -cycle in $$\mathbb {Z}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>Z</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> is null-homologous in the plaquette random-cluster model equals the expectation of the corresponding generalized Wilson loop variable. This provides the first rigorous justification for a claim of Aizenman, Chayes, Chayes, Frölich, and Russo that there is an exact relationship between Wilson loop variables and the event that a loop is bounded by a surface in an interacting system of plaquettes. We also prove that the i -dimensional plaquette random-cluster model on the 2 i -dimensional torus exhibits a sharp phase transition at the self-dual point $$p_{\textrm{sd}} :=\frac{\sqrt{q}}{1+\sqrt{q}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>p</mml:mi> <mml:mtext>sd</mml:mtext> </mml:msub> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mfrac> <mml:msqrt> <mml:mi>q</mml:mi> </mml:msqrt> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:msqrt> <mml:mi>q</mml:mi> </mml:msqrt> </mml:mrow> </mml:mfrac> </mml:mrow> </mml:math> in the sense of homological percolation. This implies a qualitative change in the generalized Swendsen–Wang dynamics from local to non-local behavior.