| 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) |
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|
| 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 |
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|
| 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 |
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|
| 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) |
|
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| end CriticalPercolation.FiniteMarkov |
|
|
| end |
| end |
|
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| end OAI |
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