| import OAI.Probability.BenjaminiSchramm.HalfspaceRootComparison | |
| namespace OAI | |
| noncomputable section | |
| open Set MeasureTheory | |
| open scoped Classical ENNReal | |
| namespace Percolation.BenjaminiSchramm | |
| variable {V E : Type*} [Encodable V] [Countable E] [Nontrivial V] | |
| (G : BondGraph V E) [Fintype (criticalModel G).VertexOrbit] | |
| theorem critical_normalized_halfspace_uniform (hc : Connected G) (hl : LocallyFinite G) | |
| (hq : QuasiTransitive G) (hu : ¬ (criticalModel G).CombinatoriallyUnimodular) | |
| (μ : (criticalModel G).VertexOrbit → ℝ) (hμ : ∀ i, 0 < μ i) | |
| (hstat : orbitChainStep ((criticalModel G).orbitWalkKernel hl) μ = μ) : | |
| ∃ A B : ℝ, 0 < A ∧ 0 ≤ B ∧ ∀ r : ℝ, 0 ≤ r → ∀ (x : V) (t : ℝ), 0 ≤ t → | |
| (law (pc G)).real ((criticalModel G).adjustedHalfspaceHit | |
| ((criticalModel G).stationaryOrbitWeight μ) x (-r) t) ≤ | |
| (pc G : ℝ)^(-(A*r+B)) * Real.exp (-t) := by | |
| let H := criticalModel G | |
| let a := H.stationaryOrbitWeight μ | |
| have hHc := criticalModel_connected G hc | |
| have hHq := criticalModel_quasiTransitive G hq | |
| obtain ⟨A,B,hA,hB,hpath⟩ := H.stationaryOrbitHeight_orbit_above hHc hl hHq μ hμ hstat hu | |
| obtain ⟨m,M,hm,hbounds⟩ := H.stationaryOrbitWeight_uniform_bounds hHc hl μ hμ | |
| have hauto := H.stationaryOrbitWeight_aut μ | |
| have hp : (0 : ℝ) < pc G := by exact_mod_cast pc_pos G hl hq | |
| refine ⟨A,B,hA,hB,fun r hr x t ht => ?_⟩ | |
| let α : ℝ := (pc G : ℝ)^(A*r+B) | |
| have hα : 0 < α := Real.rpow_pos_of_pos hp _ | |
| have hroot (u : V) : | |
| α*(law (pc G)).real (H.adjustedHalfspaceHit a x (-r) t) ≤ | |
| (law (pc G)).real (H.adjustedHalfspaceHit a u 0 t) := by | |
| obtain ⟨y,w,horb,hlen,hw,hy⟩ := hpath r hr u (H.vertexOrbit x) | |
| obtain ⟨φ,hφ⟩ := MulAction.mem_orbit_iff.mp (Quotient.exact horb) | |
| change φ.vertex x = y at hφ | |
| have hcomp := H.adjustedHalfspaceHit_path_comparison hHc hl a (pc G) w r t hr hw hy | |
| have he := CriticalPercolation.BondGraph.Aut.adjustedHalfspaceHit_measure H a hauto φ (pc G) x (-r) t | |
| rw [hφ] at he | |
| have her := congrArg ENNReal.toReal he | |
| change (H.law (pc G)).real (H.adjustedHalfspaceHit a y (-r) t) = | |
| (H.law (pc G)).real (H.adjustedHalfspaceHit a x (-r) t) at her | |
| rw [her] at hcomp | |
| have hcoeff : α ≤ (pc G : ℝ)^w.length := by | |
| simpa only [Real.rpow_natCast] using | |
| Real.rpow_le_rpow_of_exponent_ge hp (pc G).2.2 hlen | |
| exact (mul_le_mul_of_nonneg_right hcoeff ENNReal.toReal_nonneg).trans hcomp | |
| have hmin : α*(law (pc G)).real (H.adjustedHalfspaceHit a x (-r) t) ≤ | |
| (H.infimalAdjustedCrossing a (pc G) t).toReal := by | |
| have he : ENNReal.ofReal (α*(law (pc G)).real (H.adjustedHalfspaceHit a x (-r) t)) ≤ | |
| H.infimalAdjustedCrossing a (pc G) t := by | |
| apply le_iInf | |
| intro u | |
| exact ENNReal.ofReal_le_of_le_toReal (hroot u) | |
| have hf := ENNReal.toReal_mono | |
| (ne_top_of_le_ne_top ENNReal.one_ne_top (H.infimalAdjustedCrossing_le_one a (pc G) t)) he | |
| have hnonneg : 0 ≤ α*(law (pc G)).real (H.adjustedHalfspaceHit a x (-r) t) := | |
| mul_nonneg hα.le measureReal_nonneg | |
| rwa [ENNReal.toReal_ofReal hnonneg] at hf | |
| have hh := hmin.trans (critical_infimal_adjusted_crossing_bound G hc hl hq a hauto | |
| m M hm hbounds t ht) | |
| have hh' := mul_le_mul_of_nonneg_left hh (inv_nonneg.mpr hα.le) | |
| rw [← mul_assoc,inv_mul_cancel₀ hα.ne',one_mul] at hh' | |
| have he : α⁻¹ = (pc G : ℝ)^(-(A*r+B)) := by | |
| exact (Real.rpow_neg (pc G).2.1 _).symm | |
| rwa [he] at hh' | |
| end Percolation.BenjaminiSchramm | |
| end | |
| end OAI | |