| import Mathlib |
| import OAI.Probability.SKBarriers.Coverage.EnergyTail |
| import OAI.Probability.SKBarriers.Coverage.FiniteCoverage |
| import OAI.Probability.SKBarriers.Coverage.WeightedCoverage |
|
|
| namespace OAI |
|
|
| section |
|
|
| section |
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| theorem finiteOverlapMass_mono {n : ℕ} (ν : Config n → ℝ) |
| (hν : ∀ x, 0 ≤ ν x) {T S : Finset (Config n)} (hTS : T ⊆ S) (q : ℝ) : |
| finiteOverlapMass ν T q ≤ finiteOverlapMass ν S q := by |
| unfold finiteOverlapMass |
| apply le_trans (Finset.sum_le_sum (g := fun x => ∑ y ∈ S, |
| if q ≤ |overlap x y| then ν x*ν y else 0) (fun x _ => ?_)) |
| · exact Finset.sum_le_sum_of_subset_of_nonneg hTS (fun x _ _ => |
| Finset.sum_nonneg (fun y _ => by split_ifs; exact mul_nonneg (hν x) (hν y); rfl)) |
| · exact Finset.sum_le_sum_of_subset_of_nonneg hTS (fun y _ _ => by split_ifs; exact mul_nonneg (hν x) (hν y); rfl) |
|
|
| theorem weightedOverlapMass_conditional_mono {n : ℕ} (ν : Config n → ℝ) |
| (hν : ∀ x, 0 ≤ ν x) {T S : Finset (Config n)} (hTS : T ⊆ S) |
| (hT : 0 < finiteMass ν T) (q : ℝ) : |
| weightedOverlapMass (conditionalWeights ν T) q ≤ |
| (finiteMass ν S/finiteMass ν T)^2*weightedOverlapMass (conditionalWeights ν S) q := by |
| have hS : 0 < finiteMass ν S := hT.trans_le (finiteMass_mono ν hν hTS) |
| rw [weightedOverlapMass_conditional,weightedOverlapMass_conditional] |
| have HE : (finiteMass ν S/finiteMass ν T)^2*(finiteOverlapMass ν S q/(finiteMass ν S)^2) = |
| finiteOverlapMass ν S q/(finiteMass ν T)^2 := by |
| field_simp |
| rw [HE] |
| exact div_le_div_of_nonneg_right (finiteOverlapMass_mono ν hν hTS q) (sq_nonneg _) |
|
|
| def energyTrim {n : ℕ} (J : Disorder n) (e : ℝ) (S : Finset (Config n)) : Finset (Config n) := |
| S.filter (fun x => hamiltonian J x ≤ (n:ℝ)*e) |
|
|
| theorem gibbsMass_energyTrim {n : ℕ} (β e : ℝ) (J : Disorder n) (S : Finset (Config n)) : |
| finiteMass (gibbs β J) S-gibbsEnergyTail β e J ≤ |
| finiteMass (gibbs β J) (energyTrim J e S) := by |
| have Hadd := Finset.sum_filter_add_sum_filter_not S |
| (fun x => hamiltonian J x ≤ (n:ℝ)*e) (gibbs β J) |
| have Hle : (∑ x ∈ S.filter (fun x => ¬hamiltonian J x ≤ (n:ℝ)*e), gibbs β J x) ≤ |
| gibbsEnergyTail β e J := by |
| unfold gibbsEnergyTail |
| simp only [not_le] |
| rw [Finset.sum_filter] |
| exact Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ S) |
| (fun x _ _ => by split_ifs; exact le_of_lt (gibbs_pos β J x); rfl) |
| change (∑ x ∈ S, gibbs β J x)-gibbsEnergyTail β e J ≤ |
| ∑ x ∈ S.filter (fun x => hamiltonian J x ≤ (n:ℝ)*e), gibbs β J x |
| linarith |
|
|
| theorem partition_ratio_mix {n : ℕ} (β a b : ℝ) (J G : Disorder n) : |
| partition β (mixDisorder a b (J,G))/partition β J = |
| ∑ x : Config n, gibbs β J x* |
| Real.exp (β*(a-1)*hamiltonian J x+β*b*hamiltonian G x) := by |
| unfold partition gibbs |
| rw [Finset.sum_div] |
| apply Finset.sum_congr rfl |
| intro x _ |
| rw [hamiltonian_mix] |
| rw [div_mul_eq_mul_div,← Real.exp_add] |
| congr 2 |
| ring |
|
|
| theorem conditionalWeightedPartition {n : ℕ} (ν : Config n → ℝ) |
| (S : Finset (Config n)) (hS : 0 < finiteMass ν S) (b : ℝ) (G : Disorder n) : |
| finiteMass ν S*weightedPartition (conditionalWeights ν S) b G = |
| ∑ x ∈ S, ν x*Real.exp (b*hamiltonian G x) := by |
| unfold weightedPartition conditionalWeights |
| rw [Finset.mul_sum] |
| have HE (x : Config n) : finiteMass ν S*((if x ∈ S then ν x/finiteMass ν S else 0)* |
| Real.exp (b*hamiltonian G x)) = |
| if x ∈ S then ν x*Real.exp (b*hamiltonian G x) else 0 := by |
| split_ifs |
| · field_simp |
| · ring |
| simp_rw [HE] |
| rw [← Finset.sum_filter] |
| simp only [Finset.filter_mem_eq_inter,Finset.univ_inter] |
|
|
| theorem partition_ratio_mix_lower {n : ℕ} {β a : ℝ} (hβ : 0 ≤ β) (ha : a ≤ 1) |
| (b e : ℝ) (J G : Disorder n) (T : Finset (Config n)) |
| (hT : 0 < finiteMass (gibbs β J) T) |
| (henergy : ∀ x ∈ T, hamiltonian J x ≤ (n:ℝ)*e) : |
| finiteMass (gibbs β J) T*Real.exp (β*(a-1)*(n:ℝ)*e)* |
| weightedPartition (conditionalWeights (gibbs β J) T) (β*b) G ≤ |
| partition β (mixDisorder a b (J,G))/partition β J := by |
| rw [partition_ratio_mix] |
| calc |
| _ = Real.exp (β*(a-1)*(n:ℝ)*e)* |
| ∑ x ∈ T, gibbs β J x*Real.exp ((β*b)*hamiltonian G x) := by |
| rw [← conditionalWeightedPartition (gibbs β J) T hT] |
| ring |
| _ ≤ ∑ x ∈ T, gibbs β J x* |
| Real.exp (β*(a-1)*hamiltonian J x+β*b*hamiltonian G x) := by |
| rw [Finset.mul_sum] |
| apply Finset.sum_le_sum |
| intro x hx |
| rw [Real.exp_add] |
| have he : β*(a-1)*(n:ℝ)*e ≤ β*(a-1)*hamiltonian J x := by |
| have HH := mul_le_mul_of_nonpos_left (henergy x hx) |
| (mul_nonpos_of_nonneg_of_nonpos hβ (sub_nonpos.mpr ha)) |
| nlinarith |
| have HH := mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr he) |
| (mul_nonneg (gibbs_pos β J x).le (Real.exp_nonneg ((β*b)*hamiltonian G x))) |
| nlinarith |
| _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ T) |
| (fun x _ _ => mul_nonneg (gibbs_pos β J x).le (Real.exp_nonneg _)) |
|
|
| theorem logPartition_mix_lower {n : ℕ} {β a : ℝ} (hβ : 0 ≤ β) (ha : a ≤ 1) |
| (b e : ℝ) (J G : Disorder n) (T : Finset (Config n)) |
| (hT : 0 < finiteMass (gibbs β J) T) |
| (henergy : ∀ x ∈ T, hamiltonian J x ≤ (n:ℝ)*e) : |
| Real.log (finiteMass (gibbs β J) T)+β*(a-1)*(n:ℝ)*e+ |
| Real.log (weightedPartition (conditionalWeights (gibbs β J) T) (β*b) G) ≤ |
| logPartition β (mixDisorder a b (J,G))-logPartition β J := by |
| have hw := weightedPartition_pos (conditionalWeights (gibbs β J) T) |
| (conditionalWeights_nonneg (gibbs β J) (fun x => (gibbs_pos β J x).le) T) |
| (conditionalWeights_sum (gibbs β J) T hT) (β*b) G |
| have H := Real.log_le_log (mul_pos (mul_pos hT (Real.exp_pos _)) hw) |
| (partition_ratio_mix_lower hβ ha b e J G T hT henergy) |
| rw [Real.log_mul (mul_ne_zero hT.ne' (Real.exp_ne_zero _)) hw.ne', |
| Real.log_mul hT.ne' (Real.exp_ne_zero _),Real.log_exp, |
| Real.log_div (partition_pos β (mixDisorder a b (J,G))).ne' (partition_pos β J).ne'] at H |
| exact H |
| end SK.Analytic |
| |
| end |
| end |
| |
| end |
| |
| end OAI |
| |
