Bernoulli percolation in the inverse-temperature parametrization

2 fully proved

A. The parametrization

DefinitionBond probability \(p(\beta) = 1 - e^{-\beta J}\)#✓fully proved

The bond-percolation edge probability p(β) = 1 - exp(-βJ) attached to coupling J at inverse temperature β.

noncomputable def pBeta (J β : ℝ) : ℝ := 1 - Real.exp (-β * J)

Used by 6

Used byAizenman–Barsky inequality on a finite graph · Probability of a physical configuration · Aizenman–Barsky inequality on infinite vertex-transitive graphs · Bond percolation measure on the ghost graph · Real edge parameters with a ghost field · Edge parameters with a field value on a subset

B. The critical point

DefinitionAuxiliary critical inverse temperature from \(\tilde p_c\)#✓fully proved

The auxiliary critical inverse temperature β̃_c = -log(1 - p̃_c)/J, so that p(β̃_c) = p̃_c.

noncomputable def betaTildeC (d : ℕ) (J : ℝ) : ℝ := -Real.log (1 - (tildePc d : ℝ)) / J

Uses 1

Statement usesDuminil-Copin–Tassion threshold

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