Infinite-volume random-cluster measures

• Finite- and infinite-volume FK measures on ℤ^d
• The wired box measure as a conditional measure on the next box
• Boundary conditions induced on a box by the outer annulus
• Ordering of free and wired subsequential limits
• The FK percolation probability as a limit of crossing probabilities
• Two-point functions of infinite-volume FK measures
• Box-boundary connection probabilities for the wired q = 2 measure
• One-step comparison of wired boundary-connection probabilities
• Wired random-cluster measures decrease with the box
• The free q = 2 random-cluster measures increase with the box
• The wired infinite-volume limit at q = 2
• FKG inequality and ordering for infinite-volume FK measures at q = 2
• Free and wired infinite-volume limits at q = 2
• Multi-edge events and translation invariance of q = 2 FK measures
• The off-centre carrier for wired q = 2 FK marginals
• Domination of an inner wired FK measure by an outer one
• Two-point connectivity with and without percolation

DefinitionNumber of open edges in a finite edge set#✓fully proved

The number of open edges of ω in the finite set E, as a real number.

noncomputable def dfi_openEdgeCount {V : Type*} [Fintype V] [DecidableEq V]
    (E : Finset (Sym2 V)) (ω : ConfigSpace (Sym2 V)) : ℝ :=
  ∑ e ∈ E, (if ω e then (1 : ℝ) else 0)

Uses 1 · Used by 7

Statement usesConfiguration space of a statistical-mechanics model

Used byExpected wired open-edge count is the sum of edge marginals · Per-box \(q = 2\) uniqueness off a countable set of tilts · Per-box uniqueness of the random-cluster measures off a countable set · Constant free density gives an upper bound on expected open edges · Free upper bound on expected open edges along a box-symmetry orbit · Constant wired density gives a lower bound on expected open edges · Wired lower bound on expected open edges along a box-symmetry orbit

DefinitionEvent that a box edge is open#✓fully proved

The event that the box edge eb of Λ_N is open.

def boxEdgeOpenEvent (d N : ℕ) (eb : Sym2 (boxVerts d N)) :
    Set (ConfigSpace (Sym2 (boxVerts d N))) :=
  {σ | σ eb = true}

Uses 2 · Used by 30

Statement usesConfiguration space of a statistical-mechanics model · Vertices of the box \((-n, n)^d\)

Used byBox-edge marginal uniqueness off a countable set under tilt data · Edge-marginal uniqueness off a countable set from convex free energy · Left-continuity of the free \(q = 2\) edge density in \(t\) · Free \(q = 2\) edge density is monotone in \(p\) on \((0, 1)\) · Wired \(q = 2\) edge density is monotone in \(t\) · Right-continuity of the wired \(q = 2\) edge density in \(t\) · Secant bound gives free–wired density agreement off a countable set · Free density is at most each translated wired marginal · Translated wired marginals converge to the wired density · Equal wired densities from equal translated marginals · Per-box \(q = 2\) uniqueness off a countable set of tilts · Secant bound gives \(q = 2\) edge-event uniqueness off a countable set · Per-box uniqueness of the random-cluster measures off a countable set · Countable-exception \(q = 2\) uniqueness from one-sided derivative data · Free-energy data give \(q = 2\) box uniqueness off countably many \(t\) · Free unit-edge marginals converge to the free density · Wired unit-edge marginals converge to the wired density · Free \(q = 2\) edge density lies in \((0, 1)\) · Wired \(q = 2\) edge density lies in \((0, 1)\) · Box symmetries preserve the free \(q = 2\) density of box edges · Box symmetries preserve the wired \(q = 2\) density of box edges · Free \(q = 2\) box edge marginals are at most the free density · Wired \(q = 2\) box edge marginals are at least the wired density · Constant free density gives an upper bound on expected open edges · Equal free densities from merging free marginals · Constant wired density gives a lower bound on expected open edges · Equal wired densities from merging wired marginals · Free box integral of an edge indicator is the free edge marginal · Wired box integral of an edge indicator is the wired edge marginal · Decay of the edge-indicator functional from equal wired densities

2 fully proved

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