The self-dual point of the planar random-cluster model

DefinitionPlanar dual parameter \(p^{\ast}\)#✓fully proved

The planar dual parameter p* = (1 - p) q / ((1 - p) q + p), characterized by p* p / ((1 - p*)(1 - p)) = q.

noncomputable def dualParam (p q : ℝ) : ℝ := (1 - p) * q / ((1 - p) * q + p)

DefinitionSelf-dual point \(p_{sd}(q) = \sqrt q/(1 + \sqrt q)\)#✓fully proved

The self-dual point p_sd(q) = √q / (1 + √q).

noncomputable def selfDualPoint (q : ℝ) : ℝ := Real.sqrt q / (1 + Real.sqrt q)

Used by 4

Used byFree and wired measures differ at self-dual \(p\) for \(q > 4\) if first order · Free and wired measures differ at \(\sqrt{q}/(1+\sqrt{q})\) for \(q > 4\) if first order · Beffara–Duminil-Copin theorem, conditional form · \(p_c(q)\) equals the self-dual point if percolation starts there, \(q \ge 1\)

2 fully proved

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