The critical point of the random-cluster model

3 fully proved

A. The percolation probability

DefinitionWired random-cluster percolation probability \(\theta(p, q)\)#✓fully proved

The percolation probability θ(p, q) = φ^1_{p,q}(0 ↔ ∞) of the wired infinite-volume random-cluster measure on ℤ^d.

noncomputable def fkTheta (d : ℕ) {p q : ℝ} (hp : 0 < p) (hp1 : p < 1) (hq : 0 < q) : ℝ :=
  ((wiredInfiniteVolume d hp hp1 hq : ProbabilityMeasure (ConfigSpace (Sym2 (Site d)))) :
      Measure (ConfigSpace (Sym2 (Site d)))).real (percolationEvent d)

Uses 4 · Used by 35

Statement usesConfiguration space of a statistical-mechanics model · Wired infinite-volume random-cluster measure · Site of the hypercubic lattice · Percolation event for the origin

Used byPositive planar percolation at some \(p < 1\) via the Ising model · Wired \(q = 2\) percolation probability at most the Bernoulli one · FK critical point is less than one for \(d \ge 2\), \(q \ge 1\) · Inf-based and sup-based Ising critical points agree for \(d \ge 2\) · Sup-based \(\beta_c\) equals \(-\tfrac{1}{2}\log(1 - p_c)\) for \(d \ge 2\) · Existence of the Ising phase transition · First-order transition from \(\theta^0 = 0\) and \(\theta^1 > 0\) · \(\theta^0 \ne \theta^1\) at a first-order transition · Free and wired measures differ at a first-order transition · First-order transition of the random-cluster model · Free 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 · Wired percolation above the random-cluster critical point · Free percolation from domination and free–wired agreement at \(p'\) · Free percolation above \(p_c\) from monotonicity and countable agreement · No percolation for \(p \le p_c\) at \(q = 2\), given \(\theta^1(p_c, 2) = 0\) · No percolation below the critical point at \(q = 2\) · Monotonicity of \(\theta^1(p, 2)\) in \(p\) · Positivity of the random-cluster critical point · Subcritical set is bounded above by one · No wired percolation for \(2dp < 1\) and \(q \ge 1\) · Wired percolation bound \(\theta^1(p, q) \le (2dp)^m\) for \(q \ge 1\) · Beffara–Duminil-Copin theorem, conditional form · \(p_c(q)\) equals the self-dual point if percolation starts there, \(q \ge 1\) · Non-percolating parameters of the wired random-cluster measure · Peierls bound hypothesis for the wired measure on \(\mathbb{Z}^2\) · \(p_c(q) < 1\) on \(\mathbb{Z}^2\) for \(q \ge 1\) under a Peierls bound · Nonempty subcritical set on \(\mathbb{Z}^2\) for \(q \ge 1\) · Percolation from a Peierls bound with series below 1 · Positive percolation probability rules out subcriticality · Box-boundary profile tends to \(\theta(p,2)\) if wired probabilities do · Wired crossing probabilities converge to the percolation probability · Wired \(q = 2\) percolation probability as the limit of \(0 \leftrightarrow \partial\Lambda_n\) in \(\Lambda_n\) · Ising FK box-boundary connection profile tends to \(\theta^1(p, 2)\) · Wired infinite-volume boundary connection tends to \(\theta^1(p, q)\)

B. The critical parameter

DefinitionNon-percolating parameters of the wired random-cluster measure#✓fully proved

The parameters p ∈ (0, 1) at which the wired measure does not percolate (θ(p, q) = 0).

def fkSubcriticalSet (d : ℕ) (q : ℝ) : Set ℝ :=
  {p : ℝ | ∃ (hp : 0 < p) (hp1 : p < 1) (hq : 0 < q), fkTheta d hp hp1 hq (q := q) = 0}

Uses 1 · Used by 14

Statement usesWired random-cluster percolation probability \(\theta(p, q)\)

Used byFK critical point is less than one for \(d \ge 2\), \(q \ge 1\) · Inf-based and sup-based Ising critical points agree for \(d \ge 2\) · Sup-based \(\beta_c\) equals \(-\tfrac{1}{2}\log(1 - p_c)\) for \(d \ge 2\) · Existence of the Ising phase transition · Wired percolation above the random-cluster critical point · No percolation below the critical point at \(q = 2\) · Positivity of the random-cluster critical point · Subcritical set is bounded above by one · Subcritical parameters are at most the critical point · \(p_c(q)\) equals the self-dual point if percolation starts there, \(q \ge 1\) · Critical parameter \(p_c(q)\) of the random-cluster model · \(p_c(q) < 1\) on \(\mathbb{Z}^2\) for \(q \ge 1\) under a Peierls bound · Nonempty subcritical set on \(\mathbb{Z}^2\) for \(q \ge 1\) · Positive percolation probability rules out subcriticality

DefinitionCritical parameter \(p_c(q)\) of the random-cluster model#✓fully proved

The critical parameter p_c(q) = sup {p ∈ (0, 1) : θ(p, q) = 0} of the random-cluster model on ℤ^d (equal to 0 when the set is empty, e.g. for q ≤ 0).

noncomputable def fkPc (d : ℕ) (q : ℝ) : ℝ :=
  sSup (fkSubcriticalSet d q)

Uses 1 · Used by 17

Statement usesNon-percolating parameters of the wired random-cluster measure

Used byFK critical point is less than one for \(d \ge 2\), \(q \ge 1\) · Inf-based \(\beta_c\) equals \(-\tfrac{1}{2}\log(1 - p_c)\) for \(d \ge 2\) · Inf-based and sup-based Ising critical points agree for \(d \ge 2\) · Sup-based \(\beta_c\) equals \(-\tfrac{1}{2}\log(1 - p_c)\) for \(d \ge 2\) · Existence of the Ising phase transition · Wired percolation above the random-cluster critical point · Free percolation from domination and free–wired agreement at \(p'\) · Free percolation above \(p_c\) from monotonicity and countable agreement · No percolation for \(p \le p_c\) at \(q = 2\), given \(\theta^1(p_c, 2) = 0\) · No percolation below the critical point at \(q = 2\) · Positivity of the random-cluster critical point · Positive random-cluster critical point for \(d \ge 2\), \(q \ge 1\) · Subcritical parameters are at most the critical point · Beffara–Duminil-Copin theorem, conditional form · \(p_c(q)\) equals the self-dual point if percolation starts there, \(q \ge 1\) · \(p_c(q) < 1\) on \(\mathbb{Z}^2\) for \(q \ge 1\) under a Peierls bound · \(0 < p_c(q) < 1\) on \(\mathbb{Z}^2\) for \(q \ge 1\) under a Peierls bound

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