| 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 | |