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b317718 | 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 | import OAI.Probability.GammaPassage.ActualNoCorner
import OAI.Probability.GammaPassage.AnalyticSpecialization
import OAI.Probability.GammaPassage.ActualTimeConstant
import OAI.Probability.GammaPassage.Target
namespace OAI
/-!
# Differentiability for Gamma weights
Theorem `thm:gamma-differentiability` of
*Strict convexity and differentiability of the planar exponential first-passage limit shape*.
The almost-sure passage-time limit is asserted for each fixed real direction.
-/
namespace GammaFPP
open PlanarFPP MeasureTheory
/-- Differentiability and regular C¹ boundary charts for planar Gamma first-passage percolation. -/
theorem gamma_differentiability : GammaDifferentiabilityTarget := by
intro shape rate hs hr Ω _ P τ h
have hnc : Analytic.NoCornerWedge (gammaMeanNorm h hs hr) := by
intro b hb
simpa [PlanarFPP.extremeSlope] using
CrossingBridge.gamma_no_corner h hs hr b hb
exact Analytic.gammaMeanNorm_target_conjunction_of_inputs h hs hr
(h.isTimeConstant hs hr) hnc
end GammaFPP
end OAI
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