File size: 3,112 Bytes
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 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 63 64 65 66 67 68 69 70 71 72 73 | import OAI.Probability.FreeIsing.Trees.PathResponse
namespace OAI
noncomputable section
namespace Problem367.TreePathResponse
theorem expectation_update_field {d : ℕ} (beta : ℝ) (h : TreeVertex d → ℝ)
{D : Finset (TreeVertex d)} {v : TreeVertex d} (hv : v ∈ D)
(f : (D → IsingSpin) → ℝ) (t : ℝ) :
LeafGibbs.expectation
(fun eta => GibbsLeaf.energy beta (Function.update h v (h v + t)) D
(extendFiniteSpin D eta)) f =
FinitePosterior.mean
(fun eta => GibbsLeaf.energy beta h D (extendFiniteSpin D eta))
(fun eta => spinSign (extendFiniteSpin D eta v)) f t := by
classical
simp only [LeafGibbs.expectation, GibbsLeaf.energy_update_field beta h D hv,
FinitePosterior.mean, FinitePosterior.weight, FinitePosterior.partition]
theorem hasDerivAt_mean_coordinate {d : ℕ} (beta : ℝ) (h : TreeVertex d → ℝ)
{D : Finset (TreeVertex d)} (u : TreeVertex d) {v : TreeVertex d}
(hv : v ∈ D) (t : ℝ) :
HasDerivAt (fun s => mean beta (Function.update h v (h v + s)) D u)
(covariance beta (Function.update h v (h v + t)) D u v) t := by
classical
let : Nonempty (D → IsingSpin) := ⟨fun _ => .pos⟩
simpa only [mean, covariance, LeafGibbs.covariance,
expectation_update_field beta h hv] using
FinitePosterior.hasDerivAt_mean
(fun eta => GibbsLeaf.energy beta h D (extendFiniteSpin D eta))
(fun eta => spinSign (extendFiniteSpin D eta v))
(fun eta => spinSign (extendFiniteSpin D eta u)) t
def field {d : ℕ} (beta : ℝ) (h : TreeVertex d → ℝ)
(D : Finset (TreeVertex d)) (u : TreeVertex d) : ℝ :=
Real.artanh (mean beta h D u)
theorem hasDerivAt_field_coordinate {d : ℕ} (beta : ℝ) (h : TreeVertex d → ℝ)
{D : Finset (TreeVertex d)} {u v : TreeVertex d}
(hu : u ∈ D) (hv : v ∈ D) (t : ℝ) :
HasDerivAt (fun s => field beta (Function.update h v (h v + s)) D u)
(response beta (Function.update h v (h v + t)) D u v) t := by
have ha := hasDerivAt_artanh_interval
(mean_mem_Ioo beta (Function.update h v (h v + t)) hu)
have hd := ha.comp t (hasDerivAt_mean_coordinate beta h u hv t)
convert! hd using 1
unfold response
ring
theorem lipschitzWith_root_field_coordinate {d : ℕ} {beta : ℝ} (hbeta : 0 ≤ beta)
(h : TreeVertex d → ℝ) {D : Finset (TreeVertex d)} (hD : ConnectedFinset D)
(hr : ConnectedHull.root d ∈ D) {v : TreeVertex d} (hv : v ∈ D) :
LipschitzWith
⟨Real.tanh beta ^ v.1.length, pow_nonneg (tanh_nonneg_of_beta_nonneg hbeta) _⟩
(fun t => field beta (Function.update h v (h v + t)) D (ConnectedHull.root d)) := by
apply lipschitzWith_of_nnnorm_deriv_le
· intro t
exact (hasDerivAt_field_coordinate beta h hr hv t).differentiableAt
· intro t
rw [(hasDerivAt_field_coordinate beta h hr hv t).deriv]
have hb := root_response_mem_Icc hbeta (Function.update h v (h v + t)) hD hr hv
change |response beta (Function.update h v (h v + t)) D (ConnectedHull.root d) v| ≤ _
rw [abs_of_nonneg hb.1]
exact hb.2
end Problem367.TreePathResponse
end
end OAI
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