| import OAI.Probability.FirstPassage.EndpointProbability |
| import OAI.Probability.FirstPassage.EndpointAsymptotics |
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| namespace OAI |
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| namespace PlanarFPP |
| open Set Filter MeasureTheory ProbabilityTheory |
| open scoped Topology |
| variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {τ : Edge → Ω → ℝ} |
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| theorem exponential_arbitraryEndpoint_tendsto_of_grids {ι : Type*} {l : Filter ι} |
| (henv : ExponentialEnvironment P τ) |
| (μ : Plane → ℝ) (hμ : TimeConstantNorm μ) (ht : IsTimeConstant P τ μ) |
| (b K A R ε q₀ : ℝ) (hb : |b| ≤ 1) (hK : 0 ≤ K) |
| (hA : 0 < A) (hR : 0 < R) (hε : 0 < ε) (hq : 0 < q₀) (plus : Bool) |
| (k m : ι → ℕ) (hk : Tendsto k l atTop) |
| (hratio : Tendsto (fun i => (2 : ℝ)^(m i)/(2 : ℝ)^(k i)) l atTop) |
| (hm : ∀ᶠ i in l, m i ≤ 3*k i) |
| (hgrid : ∀ J : ℕ, 1 ≤ J → ∀ᶠ i in l, |
| P.real {ω | (fun e => τ e ω) ∉ gridConnectionsEvent b (μ (direction b)) K A R (k i) (m i) J} ≤ |
| gridFailureTail (gridIndexConstant A R) q₀ J) : |
| Tendsto (fun i => P.real {ω | (fun e => τ e ω) ∈ arbitraryEndpointEvent b (μ (direction b)) |
| (extremeSlope μ b plus) A R ε (k i) (m i)}) l (𝓝 0) := by |
| apply tendsto_order.2 |
| constructor |
| · intro a ha |
| exact Eventually.of_forall (fun _ => ha.trans_le measureReal_nonneg) |
| · intro δ hδ |
| have htE : Tendsto (fun J => 32*(K+200)*endpointErrorTail J) atTop (𝓝 0) := by |
| simpa using endpointErrorTail_tendsto.const_mul (32*(K+200)) |
| have htG := gridFailureTail_tendsto (gridIndexConstant A R) q₀ hq |
| obtain ⟨J,hJE,hJG,hJ⟩ := ((htE.eventually (gt_mem_nhds (by linarith : 0 < ε/4))).and |
| ((htG.eventually (gt_mem_nhds (by linarith : 0 < δ/2))).and (eventually_ge_atTop 1))).exists |
| let M : ℝ := (coarsePairIndices A J).card |
| have hM : 0 ≤ M := by positivity |
| let κ : ℝ := δ/(2*(M+1)) |
| have hκ : 0 < κ := by dsimp [κ]; positivity |
| have hκM : M*κ < δ/2 := by |
| have hh : κ*(2*(M+1))=δ := by dsimp [κ]; field_simp |
| nlinarith |
| obtain ⟨S,D,_hS,_hD,hprob⟩ := exponential_arbitraryEndpoint_probability_bound henv μ hμ ht |
| b K A R ε hb hK (norm_direction_pos μ hμ b) hA hR hε plus J hJ κ hκ |
| have hs := (tendsto_pow_atTop_atTop_of_one_lt (by norm_num : (1 : ℝ) < 2)).comp hk |
| have hd := endpoint_mesh_ratio_tendsto k m hratio J |
| let C := 2*R*|extremeSlope μ b plus| /(μ (direction b)) |
| filter_upwards [endpoint_levels_eventually k m hk hratio hm J, |
| endpoint_error_eventually k hk K (extremeSlope μ b plus) ε hε J hJE.le, |
| hs.eventually (eventually_ge_atTop S),hd.eventually (eventually_ge_atTop D), |
| hd.eventually (eventually_ge_atTop C),hgrid J hJ] with i hlev herr hsi hdi hci hgi |
| have hs0 : (0 : ℝ) < 2^(k i) := by positivity |
| have hp := hprob (k i) (m i) hlev.1 hlev.2 hsi ((le_div_iff₀ hs0).mp hdi) herr |
| ((le_div_iff₀ hs0).mp hci) |
| have hsmall : P.real {ω | (fun e => τ e ω) ∉ |
| gridConnectionsEvent b (μ (direction b)) K A R (k i) (m i) J} < δ/2 := hgi.trans_lt hJG |
| change M*κ < δ/2 at hκM |
| change _ ≤ _+M*κ at hp |
| linarith |
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| end PlanarFPP |
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| end OAI |
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