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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 | import Mathlib
import OAI.Probability.CriticalPercolation.ProductMeasure.FiniteMarkovAvg
namespace OAI
section
section
open Set MeasureTheory ProbabilityTheory
open Set
open scoped BigOperators
namespace CriticalPercolation.FiniteMarkov
open Filter
open scoped Topology
variable {X : Type*} [Fintype X] [Preorder X]
def Associated (μ : X → ℝ) : Prop :=
∀ f g : X → ℝ, Monotone f → Monotone g →
avg μ f * avg μ g ≤ avg μ (fun x => f x * g x)
omit [Preorder X] in
lemma avg_neg (μ : X → ℝ) (f : X → ℝ) :
avg μ (fun x => -f x) = -avg μ f := by
simp [avg, Finset.sum_neg_distrib]
lemma Associated.antitone {μ : X → ℝ} (h : Associated μ)
(f g : X → ℝ) (hf : Antitone f) (hg : Antitone g) :
avg μ f * avg μ g ≤ avg μ (fun x => f x * g x) := by
have hh := h (fun x => -f x) (fun x => -g x) hf.neg hg.neg
simpa only [avg_neg, neg_mul_neg] using hh
lemma Associated.cross {μ : X → ℝ} (h : Associated μ)
(f g : X → ℝ) (hf : Monotone f) (hg : Antitone g) :
avg μ (fun x => f x * g x) ≤ avg μ f * avg μ g := by
have hh := h f (fun x => -g x) hf hg.neg
simp only [avg_neg, mul_neg] at hh
linarith
omit [Preorder X] in
lemma step_mono {K : X → X → ℝ} (hK : ∀ x y, 0 ≤ K x y)
{f g : X → ℝ} (hfg : ∀ x, f x ≤ g x) (x : X) :
step K f x ≤ step K g x := avg_mono (hK x) hfg
lemma association_halfstep {K L : X → X → ℝ}
(hK : ∀ x y, 0 ≤ K x y)
(hKa : ∀ x, Associated (K x)) (hLa : ∀ x, Associated (L x))
(hLa' : ∀ f : X → ℝ, Monotone f → Antitone (step L f))
(f g : X → ℝ) (hf : Monotone f) (hg : Monotone g) (x : X) :
step K (step L f) x * step K (step L g) x ≤
step K (step L (fun y => f y * g y)) x := by
calc
_ ≤ step K (fun y => step L f y * step L g y) x :=
(hKa x).antitone _ _ (hLa' f hf) (hLa' g hg)
_ ≤ _ := step_mono hK (fun y => hLa y f g hf hg) x
lemma step_antitone_of_neg {K : X → X → ℝ}
(hK : ∀ f : X → ℝ, Monotone f → Antitone (step K f))
{f : X → ℝ} (hf : Antitone f) : Monotone (step K f) := by
have hh := (hK (fun x => -f x) hf.neg).neg
change Monotone (fun x => avg (K x) f)
simpa only [step, avg_neg, neg_neg] using hh
lemma iterate_association {K : X → X → ℝ}
(hK : ∀ x y, 0 ≤ K x y)
(hKa : ∀ x, Associated (K x))
(hmono : ∀ f : X → ℝ, Monotone f → Monotone (step K f))
(f g : X → ℝ) (hf : Monotone f) (hg : Monotone g) (n : ℕ) (x : X) :
(step K)^[n] f x * (step K)^[n] g x ≤
(step K)^[n] (fun y => f y * g y) x := by
have hit (f : X → ℝ) (hf : Monotone f) (n : ℕ) : Monotone ((step K)^[n] f) := by
induction n with
| zero => exact hf
| succ n ih => simpa only [Function.iterate_succ_apply'] using hmono _ ih
induction n generalizing x with
| zero => exact le_rfl
| succ n ih =>
rw [Function.iterate_succ_apply', Function.iterate_succ_apply',
Function.iterate_succ_apply']
calc
_ ≤ step K (fun y => (step K)^[n] f y * (step K)^[n] g y) x :=
hKa x _ _ (hit f hf n) (hit g hg n)
_ ≤ _ := step_mono hK (fun y => ih y) x
noncomputable def compose (K L : X → X → ℝ) (x z : X) : ℝ :=
∑ y, K x y * L y z
omit [Preorder X] in
lemma step_compose (K L : X → X → ℝ) (f : X → ℝ) :
step (compose K L) f = step K (step L f) := by
funext x
simp only [step, avg, compose, Finset.sum_mul, Finset.mul_sum, mul_assoc]
exact Finset.sum_comm
omit [Preorder X] in
lemma compose_nonneg {K L : X → X → ℝ}
(hK : ∀ x y, 0 ≤ K x y) (hL : ∀ x y, 0 ≤ L x y) (x z : X) :
0 ≤ compose K L x z := Finset.sum_nonneg fun y _ => mul_nonneg (hK x y) (hL y z)
omit [Preorder X] in
lemma compose_mass {K L : X → X → ℝ}
(hK : ∀ x, ∑ y, K x y = 1) (hL : ∀ x, ∑ y, L x y = 1) (x : X) :
∑ z, compose K L x z = 1 := by
simp only [compose]
rw [Finset.sum_comm]
simp only [← Finset.mul_sum, hL, mul_one, hK]
omit [Preorder X] in
lemma compose_minorization {K L : X → X → ℝ}
(hK : ∀ x y, 0 ≤ K x y) (hKm : ∀ x, ∑ y, K x y = 1)
{a : ℝ} {z : X} (haL : ∀ x, a ≤ L x z) (x : X) :
a ≤ compose K L x z := by
calc
a = ∑ y, K x y * a := by rw [← Finset.sum_mul, hKm, one_mul]
_ ≤ _ := Finset.sum_le_sum fun y _ => mul_le_mul_of_nonneg_left (haL y) (hK x y)
lemma compose_associated {K L : X → X → ℝ}
(hK : ∀ x y, 0 ≤ K x y)
(hKa : ∀ x, Associated (K x)) (hLa : ∀ x, Associated (L x))
(hLa' : ∀ f : X → ℝ, Monotone f → Antitone (step L f)) (x : X) :
Associated (compose K L x) := by
intro f g hf hg
change step (compose K L) f x * step (compose K L) g x ≤
step (compose K L) (fun y => f y * g y) x
simp only [step_compose]
exact association_halfstep hK hKa hLa hLa' f g hf hg x
lemma compose_monotone {K L : X → X → ℝ}
(hKa' : ∀ f : X → ℝ, Monotone f → Antitone (step K f))
(hLa' : ∀ f : X → ℝ, Monotone f → Antitone (step L f))
{f : X → ℝ} (hf : Monotone f) : Monotone (step (compose K L) f) := by
rw [step_compose]
exact step_antitone_of_neg hKa' (hLa' f hf)
theorem stationary_associated {K : X → X → ℝ}
(hK : ∀ x y, 0 ≤ K x y) (hmass : ∀ x, ∑ y, K x y = 1)
(hKa : ∀ x, Associated (K x))
(hmono : ∀ f : X → ℝ, Monotone f → Monotone (step K f))
{a : ℝ} {z : X} (ha : 0 < a) (ha_one : a ≤ 1) (haK : ∀ x, a ≤ K x z)
(μ : X → ℝ) (hμ : ∀ x, 0 ≤ μ x) (hμmass : ∑ x, μ x = 1)
(hstat : ∀ y, ∑ x, μ x * K x y = μ y) : Associated μ := by
intro f g hf hg
have hf' := tendsto_iterate hK hmass ha ha_one haK μ hμ hμmass hstat f z
have hg' := tendsto_iterate hK hmass ha ha_one haK μ hμ hμmass hstat g z
have hfg' := tendsto_iterate hK hmass ha ha_one haK μ hμ hμmass hstat
(fun x => f x * g x) z
exact le_of_tendsto_of_tendsto (hf'.mul hg') hfg'
(Filter.Eventually.of_forall fun n => iterate_association hK hKa hmono f g hf hg n z)
end CriticalPercolation.FiniteMarkov
end
end
end OAI
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