Some Properties of the Plaquette Random-Cluster Model

We show that the <i>i</i>-dimensional plaquette random-cluster model with coefficients in <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> is dual to a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>-</mml:mo> <mml:mi>i</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:math> -dimensional plaquette random cluster model. In addition, we explore boundary conditions, infinite volume limits, and uniqueness for these models. For previously known results, we provide new proofs that rely more on the tools of algebraic topology.