Critical point of the random-cluster model

• The critical point of the random-cluster model
• The self-dual point of the planar random-cluster model
• First-order transitions of the random-cluster model
• Positivity of the random-cluster critical point
• Free percolation above the random-cluster critical point
• The random-cluster critical point on ℤ² is below one, given a Peierls bound
• The critical point of the planar random-cluster model equals the self-dual point
• No percolation below the q = 2 random-cluster critical point

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