| import OAI.Probability.ContinuumTransition.FreeGeometry |
|
|
| namespace OAI |
|
|
| section |
| open scoped BigOperators |
| open Set |
| namespace ContinuumTemperature |
| open Classical |
|
|
| theorem freeConfig_roots {ι : Type} [Fintype ι] {t L : ℝ} {x : ι → Space} |
| (ht : 2*(1+shellDelta)*Real.exp (-t) < 1) (hx : FreeConfig t x) |
| (hL : ∀ i, x i ∈ cube L) : |
| ∃ s : Finset ι, s.card ≤ packingNumber L ∧ Fintype.card ι ≤ 4*s.card ∧ |
| ∀ j, ∃ i ∈ s, ‖x j-x i‖ ≤ (1+shellDelta)*Real.exp (-t) := by |
| let Q := Quotient (freeSetoid ht hx) |
| let s : Finset ι := Finset.univ.image (fun c : Q => c.out) |
| have ho : Function.Injective (fun c : Q => c.out) := by |
| intro a b hab |
| have he := congrArg (Quotient.mk (freeSetoid ht hx)) hab |
| simpa only [Quotient.out_eq] using he |
| have hs : s.card = Fintype.card Q := by |
| exact (Finset.card_image_of_injective _ ho).trans (Finset.card_univ) |
| have hc (j : ι) : ∃ i ∈ s, j ∈ freeCluster x i := by |
| refine ⟨(Quotient.mk (freeSetoid ht hx) j).out, ?_, ?_⟩ |
| · exact Finset.mem_image.mpr ⟨Quotient.mk (freeSetoid ht hx) j, Finset.mem_univ _, rfl⟩ |
| · apply mem_freeCluster.mpr |
| exact Quotient.exact (Quotient.out_eq (Quotient.mk (freeSetoid ht hx) j)) |
| refine ⟨s, ?_, ?_, ?_⟩ |
| · rw [hs] |
| exact free_component_card_le_packing ht hx hL |
| · have hu : Finset.univ ⊆ s.biUnion (freeCluster x) := by |
| intro j _ |
| obtain ⟨i,hi,hj⟩ := hc j |
| exact Finset.mem_biUnion.mpr ⟨i,hi,hj⟩ |
| calc |
| Fintype.card ι = (Finset.univ : Finset ι).card := Finset.card_univ.symm |
| _ ≤ (s.biUnion (freeCluster x)).card := Finset.card_le_card hu |
| _ ≤ ∑ i ∈ s, (freeCluster x i).card := Finset.card_biUnion_le |
| _ ≤ ∑ _i ∈ s, 4 := Finset.sum_le_sum (fun i _ => freeCluster_card_le_four ht hx i) |
| _ = 4*s.card := by simp only [Finset.sum_const, smul_eq_mul, Nat.mul_comm] |
| · intro j |
| obtain ⟨i,hi,hj⟩ := hc j |
| exact ⟨i,hi,by simpa only [norm_sub_rev] using freeCluster_close_root hx hj⟩ |
|
|
| end ContinuumTemperature |
| end |
|
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| end OAI |
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