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f61e547 | 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 | import OAI.Probability.FirstPassage.EndpointProbability
import OAI.Probability.FirstPassage.EndpointAsymptotics
namespace OAI
namespace PlanarFPP
open Set Filter MeasureTheory ProbabilityTheory
open scoped Topology
variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {τ : Edge → Ω → ℝ}
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
end PlanarFPP
end OAI
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