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f61e547 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 | import OAI.Probability.FreeIsing.Cavity.Homeomorph
import OAI.Probability.FreeIsing.Energy.CoupledAttenuation
import OAI.Probability.FreeIsing.Estimates.RelabelMeasure
import OAI.Probability.FreeIsing.Trees.Transitivity
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
namespace Problem367.CavityArray
variable {d : ℕ} {theta : ℝ}
def field (h0 : 0 ≤ theta) (h1 : theta < 1)
(H : TreeVertex d → ℝ) (v w : TreeVertex d) : ℝ :=
(cavityFromSites h0 h1 (H v, H w)).1
theorem continuous_field (h0 : 0 ≤ theta) (h1 : theta < 1) :
Continuous (field (d := d) h0 h1) := by
apply continuous_pi
intro v
apply continuous_pi
intro w
exact continuous_fst.comp ((continuous_cavityFromSites h0 h1).comp
((continuous_apply v).prodMk (continuous_apply w)))
theorem measurable_field (h0 : 0 ≤ theta) (h1 : theta < 1) :
Measurable (field (d := d) h0 h1) :=
(continuous_field h0 h1).measurable
theorem measurable_field_apply (h0 : 0 ≤ theta) (h1 : theta < 1)
(v w : TreeVertex d) : Measurable (fun H => field h0 h1 H v w) :=
(measurable_pi_apply w).comp ((measurable_pi_apply v).comp
(measurable_field h0 h1))
theorem field_reverse (h0 : 0 ≤ theta) (h1 : theta < 1)
(H : TreeVertex d → ℝ) (v w : TreeVertex d) :
field h0 h1 H w v = (cavityFromSites h0 h1 (H v, H w)).2 := by
exact congrArg Prod.fst (cavityFromSites_swap h0 h1 (H v, H w))
theorem field_spec (h0 : 0 ≤ theta) (h1 : theta < 1)
(H : TreeVertex d → ℝ) (v w : TreeVertex d) :
field h0 h1 H v w + cavityMessage theta (field h0 h1 H w v) = H v := by
rw [field_reverse h0 h1 H v w]
exact (cavityFromSites_spec h0 h1 (H v, H w)).1
theorem field_relabel (h0 : 0 ≤ theta) (h1 : theta < 1)
(g : TreeAut d) (H : TreeVertex d → ℝ) (v w : TreeVertex d) :
field h0 h1 (relabel g H) v w =
field h0 h1 H (g.toEquiv.symm v) (g.toEquiv.symm w) := rfl
theorem field_neg (h0 : 0 ≤ theta) (h1 : theta < 1)
(H : TreeVertex d → ℝ) (v w : TreeVertex d) :
field h0 h1 (-H) v w = -field h0 h1 H v w := by
exact congrArg Prod.fst (cavityFromSites_neg h0 h1 (H v, H w))
theorem field_zero (h0 : 0 ≤ theta) (h1 : theta < 1)
(v w : TreeVertex d) : field h0 h1 0 v w = 0 := by
have h := cavityFromSites_eq h0 h1 (z := (0, 0)) (p := (0, 0))
(by simp)
exact congrArg Prod.fst h
theorem field_site_bound (h0 : 0 ≤ theta) (h1 : theta < 1)
(H : TreeVertex d → ℝ) (v w : TreeVertex d) :
|field h0 h1 H v w - H v| ≤ Real.artanh theta :=
(cavityFromSites_bound h0 h1 (H v, H w)).1
theorem field_sub_abs_le (h0 : 0 ≤ theta) (h1 : theta < 1)
{H H' : TreeVertex d → ℝ} {M : ℝ} (v w : TreeVertex d)
(hv : |H v - H' v| ≤ M) (hw : |H w - H' w| ≤ M) :
|field h0 h1 H v w - field h0 h1 H' v w| ≤ M / (1 - theta) :=
cavity_pair_sub_abs_le h0 h1
(field_spec h0 h1 H v w) (field_spec h0 h1 H w v)
(field_spec h0 h1 H' v w) (field_spec h0 h1 H' w v) hv hw
section Joining
abbrev SitePair (d : ℕ) := (TreeVertex d → ℝ) × (TreeVertex d → ℝ)
def pairRelabel (g : TreeAut d) : SitePair d → SitePair d :=
Prod.map (relabel g) (relabel g)
theorem measurable_pairRelabel (g : TreeAut d) : Measurable (pairRelabel g) :=
(measurable_relabel g).prodMap (measurable_relabel g)
theorem site_sub_identDistrib {ν : Measure (SitePair d)}
(hinv : ∀ g : TreeAut d, MeasurePreserving (pairRelabel g) ν ν)
(u v : TreeVertex d) :
IdentDistrib (fun p : SitePair d => p.1 u - p.2 u)
(fun p : SitePair d => p.1 v - p.2 v) ν ν := by
obtain ⟨g, hg⟩ := exists_treeAut_map u v
have hm (w : TreeVertex d) : Measurable (fun p : SitePair d => p.1 w - p.2 w) :=
((measurable_pi_apply w).comp measurable_fst).sub
((measurable_pi_apply w).comp measurable_snd)
refine ⟨(hm u).aemeasurable, (hm v).aemeasurable, ?_⟩
calc
ν.map (fun p : SitePair d => p.1 u - p.2 u) =
ν.map ((fun p : SitePair d => p.1 v - p.2 v) ∘ pairRelabel g) := by
congr 1
funext p
simp [pairRelabel, relabel, ← hg]
_ = (ν.map (pairRelabel g)).map (fun p : SitePair d => p.1 v - p.2 v) :=
(Measure.map_map (hm v) (measurable_pairRelabel g)).symm
_ = ν.map (fun p : SitePair d => p.1 v - p.2 v) := by rw [(hinv g).map_eq]
theorem integral_field_sub_sq_le {ν : Measure (SitePair d)}
(h0 : 0 ≤ theta) (h1 : theta < 1)
(hinv : ∀ g : TreeAut d, MeasurePreserving (pairRelabel g) ν ν)
(o v w : TreeVertex d)
(hInt : Integrable (fun p : SitePair d => (p.1 o - p.2 o) ^ 2) ν) :
(∫ p, (field h0 h1 p.1 v w - field h0 h1 p.2 v w) ^ 2 ∂ν) ≤
(∫ p : SitePair d, (p.1 o - p.2 o) ^ 2 ∂ν) / (1 - theta) ^ 2 := by
have hv := (site_sub_identDistrib hinv o v).sq
have hw := (site_sub_identDistrib hinv o w).sq
exact integral_cavity_pair_sub_sq_le h0 h1
((measurable_field_apply h0 h1 v w).comp measurable_fst)
((measurable_field_apply h0 h1 v w).comp measurable_snd)
(hv.integrable_snd hInt) (hw.integrable_snd hInt)
(Filter.Eventually.of_forall (fun p =>
⟨field_spec h0 h1 p.1 v w, field_spec h0 h1 p.1 w v,
field_spec h0 h1 p.2 v w, field_spec h0 h1 p.2 w v⟩))
hv.integral_eq.symm.le hw.integral_eq.symm.le
end Joining
end Problem367.CavityArray
end
end OAI
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