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f205175 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 | import Mathlib
import OAI.Probability.CriticalPercolation.GraphGeometry.GraphBalls
import OAI.Probability.CriticalPercolation.Gluing.CorridorNormalization
import OAI.Probability.CriticalPercolation.Gluing.MoveWitness
namespace OAI
section
section
noncomputable section
open Set Function MeasureTheory ProbabilityTheory
open scoped Classical ENNReal
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E)
lemma boxMove_aut_subset (t z : V → ℝ) (m : ℕ) (q B : ℝ) (o : V) (s : ℝ)
(φ : G.Aut) (ht : ∀ x y, t (φ.vertex x)-t (φ.vertex y)=t x-t y)
(hz : ∀ x y, z (φ.vertex x)-z (φ.vertex y)=z x-z y) :
G.boxMove t z m q B o s ⊆ (fun ω => φ.edge '' ω) ⁻¹'
G.boxMove t z m q B (φ.vertex o) s := by
rintro ω ⟨a,ha,b,w,hw,hbt,hbz⟩
let w' := w.map (Aut.openGraphIso G φ ω).toHom
refine ⟨φ.vertex a,(Aut.vertexBall_mem G φ o a m).mpr ha,φ.vertex b,w',?_,?_,?_⟩
· intro x hx
change x∈(w.map (Aut.openGraphIso G φ ω).toHom).support at hx
rw [SimpleGraph.Walk.support_map] at hx
obtain ⟨v,hv,rfl⟩ := List.mem_map.mp hx
change |t (φ.vertex v)-t (φ.vertex o)|≤B ∧ |z (φ.vertex v)-z (φ.vertex o)|≤B
rw [ht,hz]
exact hw v hv
· have h : t (φ.vertex b)-(t (φ.vertex o)+1)=t b-(t o+1) := by linarith [ht b o]
rwa [h]
· have h : z (φ.vertex b)-(z (φ.vertex o)+s)=z b-(z o+s) := by linarith [hz b o]
rwa [h]
lemma boxMove_aut_probability [Countable V] [Countable E]
(t z : V → ℝ) (m : ℕ) (q B : ℝ) (o : V) (s : ℝ) (p : unitInterval)
(φ : G.Aut) (ht : ∀ x y, t (φ.vertex x)-t (φ.vertex y)=t x-t y)
(hz : ∀ x y, z (φ.vertex x)-z (φ.vertex y)=z x-z y) :
(G.law p).real (G.boxMove t z m q B o s) ≤
(G.law p).real (G.boxMove t z m q B (φ.vertex o) s) := by
have hm : G.law p (G.boxMove t z m q B o s) ≤
G.law p (G.boxMove t z m q B (φ.vertex o) s) := by
conv_rhs => rw [← G.law_map_aut p φ]
rw [Measure.map_apply (BernoulliProduct.measurable_image_equiv φ.edge)
(G.boxMove_measurable _ _ _ _ _ _ _)]
exact measure_mono (G.boxMove_aut_subset t z m q B o s φ ht hz)
exact ENNReal.toReal_mono (measure_ne_top _ _) hm
lemma boxMove_close_center (t z : V → ℝ) (m c : ℕ) (e s : ℝ) (he : e<1/10)
{x y : V} (hxy : x∈G.vertexBall y c)
(ht : |t x-t y|≤e/8) (hz : |z x-z y|≤e/8) :
G.boxMove t z m (e/8) 30 x s ⊆ G.coordinateMove t z (m+c) e y s := by
rintro ω ⟨a,ha,b,w,hw,hbt,hbz⟩
refine ⟨a,?_,b,w,?_,?_,?_⟩
· simpa only [Nat.add_comm c m] using G.vertexBall_add hxy m ha
· intro v hv
have hvb := hw v hv
constructor
· have := abs_sub_le (t v) (t x) (t y)
linarith [hvb.1]
· have := abs_sub_le (z v) (z x) (z y)
linarith [hvb.2]
· have hh := abs_sub_le (t b) (t x+1) (t y+1)
rw [add_sub_add_right_eq_sub] at hh
linarith
· have hh := abs_sub_le (z b) (z x+s) (z y+s)
rw [add_sub_add_right_eq_sub] at hh
linarith
theorem uniform_moves_of_origin [Countable V] [Countable E]
(t z : V → ℝ) (o : V) (m c : ℕ) (e eta rho : ℝ)
(he : e<1/10) (hrho : 0≤rho) (hmesh : c*rho≤e/8)
(ht : ∀ ⦃x y⦄, G.fullGraph.Adj x y → |t x-t y|≤rho)
(hz : ∀ ⦃x y⦄, G.fullGraph.Adj x y → |z x-z y|≤rho)
(hdense : ∀ y, ∃ φ : G.Aut, φ.vertex o∈G.vertexBall y c ∧
(∀ x y, t (φ.vertex x)-t (φ.vertex y)=t x-t y) ∧
(∀ x y, z (φ.vertex x)-z (φ.vertex y)=z x-z y))
(pc : unitInterval) (hpc : 0<pc)
(horigin : ∀ b : Bool, 1-eta<(BernoulliProduct.law (fun _ : E => pc)).real
(G.boxMove t z m (e/8) 30 o (Corridor.sign b))) :
∃ p : unitInterval, (pc : ℝ)/2≤p ∧ p<pc ∧ ∀ y s, |s|=1 →
1-eta≤(BernoulliProduct.law (fun _ : E => p)).real
(G.coordinateMove t z (m+c) e y s) := by
obtain ⟨p,hpl,hpu,hp⟩ := G.lower_two_boxMoves t z m (e/8) 30 o pc hpc eta horigin
refine ⟨p,hpl,hpu,?_⟩
intro y s hs
obtain ⟨b,hb⟩ : ∃ b : Bool, Corridor.sign b=s := by
rcases (abs_eq (by norm_num : (0 : ℝ)≤1)).mp hs with h|h
· exact ⟨true,by simpa [Corridor.sign] using h.symm⟩
· exact ⟨false,by simpa [Corridor.sign] using h.symm⟩
obtain ⟨φ,hφ,hφt,hφz⟩ := hdense y
have htt := (G.coordinate_ball_bound t rho hrho ht hφ).trans hmesh
have hzz := (G.coordinate_ball_bound z rho hrho hz hφ).trans hmesh
have hsub := G.boxMove_close_center t z m c e s he hφ htt hzz
have hm : (G.law p).real (G.boxMove t z m (e/8) 30 (φ.vertex o) s)≤
(G.law p).real (G.coordinateMove t z (m+c) e y s) :=
ENNReal.toReal_mono (measure_ne_top _ _) (measure_mono hsub)
have hp' := hp b
rw [hb] at hp'
have hp'' : 1-eta≤(G.law p).real (G.boxMove t z m (e/8) 30 o s) := by
simpa only [law,BernoulliProduct.law_const] using hp'.le
simpa only [law,BernoulliProduct.law_const] using
hp''.trans ((G.boxMove_aut_probability t z m (e/8) 30 o s p φ hφt hφz).trans hm)
end CriticalPercolation.BondGraph
end
end
end
end OAI
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