| import OAI.Probability.FreeIsing.Gaussian.TreeResponseObservedPosterior |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped NNReal Topology |
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| namespace Problem367 |
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| open PlantedSymmetry |
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| theorem HasFreeIsingLaw.finiteObservation_posteriorResponse_mem_Icc |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) |
| (D : Finset (TreeVertex d)) (hD : ConnectedFinset D) (t : ℝ≥0) (u v : D) : |
| ∀ᵐ p ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t), |
| posteriorResponse (μ.prod (gaussianNoiseLaw (TreeVertex d) t)) |
| (MeasurableSpace.comap (finiteObservation D t) inferInstance) |
| (fun q => spinSign (q.1 u.1)) (fun q => spinSign (q.1 v.1)) p ∈ |
| Set.Icc 0 (Real.tanh beta ^ reducedWordDistance u.1.1 v.1.1) := by |
| filter_upwards [hμ.finiteObservation_normalizedResponse D hD t u v] with p hp |
| change posteriorResponse (μ.prod (gaussianNoiseLaw (TreeVertex d) t)) |
| (MeasurableSpace.comap (finiteObservation D t) inferInstance) |
| (fun q => spinSign (q.1 u.1)) (fun q => spinSign (q.1 v.1)) p = _ at hp |
| rw [hp] |
| exact TreePathResponse.response_mem_Icc hbeta (observation t p) hD u v |
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| theorem HasFreeIsingLaw.gaussian_posteriorResponse_mem_Icc |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) |
| (t : ℝ≥0) (u v : TreeVertex d) : |
| ∀ᵐ p ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t), |
| gaussianPosteriorResponse μ t u v p ∈ |
| Set.Icc 0 (Real.tanh beta ^ reducedWordDistance u.1 v.1) := by |
| let D := connectedExhaustion u |
| have hfin : ∀ n, ∀ᵐ p ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t), |
| v ∈ D n → posteriorResponse (μ.prod (gaussianNoiseLaw (TreeVertex d) t)) |
| (MeasurableSpace.comap (finiteObservation (D n) t) inferInstance) |
| (fun q => spinSign (q.1 u)) (fun q => spinSign (q.1 v)) p ∈ |
| Set.Icc 0 (Real.tanh beta ^ reducedWordDistance u.1 v.1) := by |
| intro n |
| by_cases hv : v ∈ D n |
| · filter_upwards [hμ.finiteObservation_posteriorResponse_mem_Icc hbeta (D n) (connectedExhaustion_connected u n) t ⟨u, mem_connectedExhaustion u n⟩ ⟨v, hv⟩] with p hp |
| exact fun _ => hp |
| · exact Filter.Eventually.of_forall (fun p hp => False.elim (hv hp)) |
| have hlim := hμ.gaussian_posteriorResponse_tendsto_ae hbeta D |
| (connectedExhaustion_monotone u) |
| (fun w => (eventually_mem_connectedExhaustion u w).exists) t u v |
| have hall := (ae_all_iff).2 hfin |
| filter_upwards [hall, hlim] with p hp hlim |
| apply isClosed_Icc.mem_of_tendsto hlim |
| filter_upwards [eventually_mem_connectedExhaustion u v] with n hn |
| exact hp n hn |
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| theorem HasFreeIsingLaw.gaussian_posteriorResponse_row_bound_of_strict |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) |
| (hstrict : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 < 1) |
| (t : ℝ≥0) (u : TreeVertex d) : |
| ∀ᵐ p ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t), |
| Summable (fun v => gaussianPosteriorResponse μ t u v p ^ 2) ∧ |
| (∑' v, gaussianPosteriorResponse μ t u v p ^ 2) ≤ |
| 1 + (d : ℝ) * Real.tanh beta ^ 2 / |
| (1 - ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2) ∧ |
| ∀ L : ℕ, (∑' v : TreeVertex d, if L < reducedWordDistance u.1 v.1 |
| then gaussianPosteriorResponse μ t u v p ^ 2 else 0) ≤ |
| (d : ℝ) * Real.tanh beta ^ 2 * |
| (((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2) ^ L / |
| (1 - ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2) := by |
| have hall := (ae_all_iff).2 (fun v => hμ.gaussian_posteriorResponse_mem_Icc hbeta t u v) |
| obtain ⟨g, _, hd⟩ := exists_treeAut_to_root_distance u |
| have hdist (v : TreeVertex d) : |
| reducedWordDistance u.1 (g.toEquiv.symm v).1 = v.1.length := by |
| simpa only [Equiv.apply_symm_apply] using (hd (g.toEquiv.symm v)).symm |
| filter_upwards [hall] with p hp |
| have hnorm (v : TreeVertex d) : |
| ‖gaussianPosteriorResponse μ t u (g.toEquiv.symm v) p‖ ≤ Real.tanh beta ^ v.1.length := by |
| rw [Real.norm_eq_abs, abs_of_nonneg (hp _).1] |
| simpa only [hdist] using (hp (g.toEquiv.symm v)).2 |
| have hs := summable_norm_sq_of_radial_bound d hstrict |
| (fun v => gaussianPosteriorResponse μ t u (g.toEquiv.symm v) p) hnorm |
| have he := tsum_norm_sq_le_of_radial_bound d hstrict |
| (fun v => gaussianPosteriorResponse μ t u (g.toEquiv.symm v) p) hnorm |
| simp only [Real.norm_eq_abs, sq_abs] at hs he |
| rw [g.toEquiv.symm.tsum_eq (fun v => gaussianPosteriorResponse μ t u v p ^ 2)] at he |
| refine ⟨g.toEquiv.symm.summable_iff.mp hs, he, ?_⟩ |
| intro L |
| have hbound (v : TreeVertex d) : |
| gaussianPosteriorResponse μ t u (g.toEquiv.symm v) p ^ 2 ≤ |
| 1 * (Real.tanh beta ^ 2) ^ v.1.length := by |
| have hv := pow_le_pow_left₀ (norm_nonneg _) (hnorm v) 2 |
| simpa only [Real.norm_eq_abs, abs_pow, sq_abs, one_mul, ← pow_mul, Nat.mul_comm] using hv |
| have htail := tsum_cutoff_le_of_tree_radial_le |
| (fun v => sq_nonneg (gaussianPosteriorResponse μ t u (g.toEquiv.symm v) p)) |
| hbound (sq_nonneg _) hstrict L |
| rw [← g.toEquiv.symm.tsum_eq (fun v : TreeVertex d => if L < reducedWordDistance u.1 v.1 then gaussianPosteriorResponse μ t u v p ^ 2 else 0)] |
| simpa only [hdist, one_mul] using htail |
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| theorem HasFreeIsingLaw.gaussian_posteriorResponse_integrable_energy_of_strict |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) |
| (hstrict : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 < 1) |
| (t : ℝ≥0) (u : TreeVertex d) : |
| Integrable (fun p => ∑' v, gaussianPosteriorResponse μ t u v p ^ 2) |
| (μ.prod (gaussianNoiseLaw (TreeVertex d) t)) ∧ |
| (∫ p, ∑' v, gaussianPosteriorResponse μ t u v p ^ 2 |
| ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t)) ≤ |
| 1 + (d : ℝ) * Real.tanh beta ^ 2 / |
| (1 - ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2) := by |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure |
| have hm (v : TreeVertex d) : |
| Measurable (fun p => gaussianPosteriorResponse μ t u v p ^ 2) := |
| (((stronglyMeasurable_posteriorResponse _ _ _ _).mono |
| (observationSigma_le (TreeVertex d) t)).measurable).pow_const 2 |
| have hbound := hμ.gaussian_posteriorResponse_row_bound_of_strict hbeta hstrict t u |
| have hint : Integrable (fun p => ∑' v, gaussianPosteriorResponse μ t u v p ^ 2) |
| (μ.prod (gaussianNoiseLaw (TreeVertex d) t)) := by |
| apply (integrable_const (1 + (d : ℝ) * Real.tanh beta ^ 2 / |
| (1 - ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2))).mono' |
| (Measurable.tsum hm).aestronglyMeasurable |
| filter_upwards [hbound] with p hp |
| rw [Real.norm_eq_abs, abs_of_nonneg (tsum_nonneg (fun _ => sq_nonneg _))] |
| exact hp.2.1 |
| refine ⟨hint, ?_⟩ |
| calc |
| _ ≤ ∫ _p, (1 + (d : ℝ) * Real.tanh beta ^ 2 / |
| (1 - ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2)) |
| ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t) := |
| integral_mono_ae hint (integrable_const _) (hbound.mono (fun p hp => hp.2.1)) |
| _ = _ := by simp |
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| end Problem367 |
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| end |
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| end OAI |
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