Box-boundary connection probabilities for the wired q = 2 measure

2 fully proved

A. The box-boundary connection event

DefinitionEvent that the box origin connects to the box boundary#✓fully proved

The event that the origin of the box Λ_n is connected to the boundary ∂Λ_n by open edges of Λ_n.

def boxBdryConnEvent (d n : ℕ) : Set (ConfigSpace (Sym2 (Site d))) :=
  boxRestrict d n ⁻¹'
    {ω' : ConfigSpace (Sym2 (boxVerts d n)) |
      ConnToBdry (boxGraph d n) (boxBoundary d n) ω' (IsingFK.boxOrigin d n)}

Uses 8 · Used by 28

Statement usesConfiguration space of a statistical-mechanics model · Vertex boundary of a box · Nearest-neighbour graph on a box · Restriction of a configuration of \(\mathbb{Z}^d\) to a box · Vertices of the box \((-n, n)^d\) · Open connection of a vertex to the boundary · Origin as a vertex of the box graph · Site of the hypercubic lattice

Used byWired \(q = 2\) percolation probability at most the Bernoulli one · Monotonicity in \(p\) of the Bernoulli percolation probability · Wired \(q = 2\) origin-to-boundary probability at most Bernoulli · 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 · Wired box-boundary connection probability is monotone in \(p\) at \(q = 2\) · Monotonicity of \(\theta^1(p, 2)\) in \(p\) · Box-boundary profile tends to \(\theta(p,2)\) if wired probabilities do · Closed-annulus wired bound on origin-to-\(\partial\Lambda_n\), \(q = 2\) · Box-boundary connection probability tends to percolation probability · A connection to outside the box passes through the box boundary · Uniform decay of two-point connectivity without percolation · Wired \(q = 2\) percolation probability as the limit of \(0 \leftrightarrow \partial\Lambda_n\) in \(\Lambda_n\) · Boundary connection of \(\Lambda_n\) is a \(\Lambda_m\) cylinder event · Boundary connection inside \(\Lambda_n\) reaches outside \(\Lambda_{n-1}\) · Events \(0 \leftrightarrow \partial\Lambda_n\) decrease in \(n \ge 1\) · One-step decrease of the events \(0 \leftrightarrow \partial\Lambda_n\) · Leaving \(\Lambda_n\) implies connecting to \(\partial\Lambda_n\) · Cylinder form of the \(\Lambda_n\) boundary connection is measurable · Wired \(q = 2\) probability of \(0 \leftrightarrow \partial\Lambda_k\) in \(\Lambda_{k+j}\) is nonincreasing in \(j\) · Wired \(q = 2\) probability of \(0 \leftrightarrow \partial\Lambda_n\) in \(\Lambda_n\) is nonincreasing in \(n\) · Wired \(q = 2\) probability of \(0 \leftrightarrow \partial\Lambda_n\) in \(\Lambda_{n+k+1}\) at most in \(\Lambda_{n+k}\) · Infinite-volume limit of the wired \(q = 2\) probability of \(0 \leftrightarrow \partial\Lambda_k\) · Wired \(q = 2\) probability of \(0 \leftrightarrow \partial\Lambda_n\) in \(\Lambda_{n+1}\) at most in \(\Lambda_n\) · Wired \(q = 2\) probability of \(0 \leftrightarrow \partial\Lambda_n\) in \(\Lambda_{m+1}\) at most in \(\Lambda_m\), \(m \ge n\) · Wired infinite-volume boundary connection tends to \(\theta^1(p, q)\)

B. Wired probabilities and the Ising profile

LemmaBox-boundary profile tends to \(\theta(p,2)\) if wired probabilities do#✓fully proved

If the wired q = 2 probabilities of {0 ↔ ∂Λ_n} converge to θ(p, 2), then so does the Ising box-boundary connection profile.

lemma boxBoundaryConnProfile_tendsto_of_diag (d : ℕ) {p : ℝ} (hp : 0 < p) (hp1 : p < 1)
    (hdiag : Tendsto
      (fun n => (wiredFiniteMeasure d n hp hp1 (by norm_num : (0 : ℝ) < 2)
          : Measure (ConfigSpace (Sym2 (Site d)))).real (boxBdryConnEvent d n))
      atTop (𝓝 (fkTheta d hp hp1 (by norm_num : (0 : ℝ) < 2) (q := 2)))) :
    Tendsto (fun n => IsingFK.boxBoundaryConnProfile d hp hp1 (by norm_num : (0 : ℝ) < 2) n)
      atTop (𝓝 (fkTheta d hp hp1 (by norm_num : (0 : ℝ) < 2) (q := 2)))

Uses 34 · Used by 1

Statement usesConfiguration space of a statistical-mechanics model · Event that the box origin connects to the box boundary · Wired random-cluster percolation probability \(\theta(p, q)\) · Wired random-cluster measure on a box · Wired probability that the origin reaches the box boundary · Site of the hypercubic lattice

Proof usesOpen connection between two vertices · Random-cluster probability with a boundary wiring · Random-cluster weight with a boundary wiring · Partition function with a boundary wiring · Vertex boundary of a box · Event that two sites are joined by open edges inside a box · Nearest-neighbour graph on a box · Restriction of a configuration of \(\mathbb{Z}^d\) to a box · Vertices of the box \((-n, n)^d\) · Event that two vertices are joined by open edges · Inclusion of box edges into \(\mathbb{Z}^d\) · Bernoulli edge product · Extension of a box configuration by closed edges · Number of clusters under a boundary wiring · Open subgraph of an edge configuration · Wired random-cluster measure as a probability mass function · Wired random-cluster probability of a configuration · Wired random-cluster weight · Wired random-cluster partition function · Open connection of a vertex to the boundary · Origin as a vertex of the box graph · Wired random-cluster probability of connecting to the boundary · Complete graph on the boundary vertices · Box \((-n, n)^d\) in the hypercubic lattice · Number of open clusters with wired boundary conditions · Open subgraph of a bond configuration · Boundary wiring relation · Wired open graph

Used byIsing FK box-boundary connection profile tends to \(\theta^1(p, 2)\)

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