lean/OAI/Probability/SKGap/Stability/PlantedFieldArray.lean

import Mathlib
import OAI.Probability.SKGap.Entropy.ProdFst
import OAI.Probability.SKGap.Entropy.BoundGood
import OAI.Probability.SKGap.Terminal.GibbsEventProbabilityReal

namespace OAI

section
open scoped BigOperators
open scoped BigOperators
open scoped BigOperators
open scoped BigOperators
open scoped BigOperators
open scoped BigOperators NNReal
open MeasureTheory ProbabilityTheory
open MeasureTheory ProbabilityTheory Filter
open scoped BigOperators NNReal
open MeasureTheory ProbabilityTheory
open scoped BigOperators NNReal ENNReal
open MeasureTheory ProbabilityTheory Filter
open scoped BigOperators NNReal ENNReal
open MeasureTheory ProbabilityTheory
open scoped BigOperators Matrix Matrix.Norms.Elementwise
open scoped BigOperators
open MeasureTheory ProbabilityTheory
open scoped BigOperators Matrix Matrix.Norms.Elementwise
open scoped BigOperators
open scoped BigOperators NNReal ENNReal
open MeasureTheory Metric Set
open scoped BigOperators NNReal ENNReal
open MeasureTheory ProbabilityTheory Filter Set
open scoped BigOperators NNReal ENNReal Matrix.Norms.L2Operator
open MeasureTheory ProbabilityTheory Filter Set
open scoped BigOperators Matrix.Norms.L2Operator
open MeasureTheory ProbabilityTheory Filter Set
open scoped BigOperators Matrix Matrix.Norms.Elementwise
open MeasureTheory ProbabilityTheory Filter Set
open MeasureTheory ProbabilityTheory Filter
open scoped BigOperators ENNReal NNReal
open MeasureTheory ProbabilityTheory Filter
open scoped BigOperators NNReal ENNReal Matrix
open MeasureTheory ProbabilityTheory Filter
open scoped BigOperators ENNReal NNReal
open MeasureTheory ProbabilityTheory Filter
open scoped BigOperators NNReal ENNReal
open scoped BigOperators
open MeasureTheory ProbabilityTheory
open scoped BigOperators Matrix Matrix.Norms.Elementwise NNReal ENNReal
open scoped BigOperators
open Filter Topology
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal ENNReal BigOperators Topology
open MeasureTheory ProbabilityTheory Filter
open Matrix
open scoped NNReal ENNReal BigOperators Topology Matrix.Norms.Elementwise
open MeasureTheory ProbabilityTheory Filter
open scoped BigOperators NNReal ENNReal Topology
open MeasureTheory ProbabilityTheory Filter Matrix
open scoped NNReal ENNReal BigOperators Topology
open MeasureTheory ProbabilityTheory Filter
open scoped BigOperators NNReal ENNReal Topology
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal ENNReal BigOperators Topology
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal ENNReal BigOperators Topology
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal ENNReal BigOperators Topology
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal ENNReal BigOperators Topology
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal Topology
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal Topology BigOperators
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal Topology BigOperators
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal Topology BigOperators
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal Topology BigOperators
open MeasureTheory ProbabilityTheory Filter Matrix
open scoped NNReal ENNReal BigOperators Topology
open MeasureTheory ProbabilityTheory Filter Matrix
open scoped NNReal ENNReal BigOperators Topology
open MeasureTheory ProbabilityTheory Filter Matrix
open scoped NNReal ENNReal BigOperators Topology
open MeasureTheory ProbabilityTheory Filter Matrix
open scoped NNReal ENNReal BigOperators Topology
open MeasureTheory ProbabilityTheory Filter Matrix
open scoped NNReal ENNReal BigOperators Topology
open MeasureTheory ProbabilityTheory Filter Matrix
open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise
open MeasureTheory ProbabilityTheory Filter Matrix
open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise
open MeasureTheory ProbabilityTheory Filter Matrix
open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise
open MeasureTheory ProbabilityTheory Filter Matrix
open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise
open MeasureTheory ProbabilityTheory Filter Matrix
open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise
open MeasureTheory ProbabilityTheory Filter Matrix
open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise
open MeasureTheory ProbabilityTheory Filter Matrix
open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise
open MeasureTheory ProbabilityTheory Filter Set Matrix
open scoped BigOperators NNReal ENNReal Matrix.Norms.L2Operator
open MeasureTheory ProbabilityTheory Filter Matrix
open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise
open MeasureTheory ProbabilityTheory Filter Matrix
open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise
open MeasureTheory ProbabilityTheory Filter Matrix
open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise
open MeasureTheory ProbabilityTheory Filter Matrix
open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise
open MeasureTheory ProbabilityTheory Filter Matrix
open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise
open Filter MeasureTheory ProbabilityTheory
open scoped Topology NNReal ENNReal
open Filter MeasureTheory ProbabilityTheory
open scoped Topology NNReal ENNReal
open MeasureTheory Filter
open scoped Topology NNReal ENNReal
open MeasureTheory Filter ProbabilityTheory
open scoped Topology NNReal ENNReal
open MeasureTheory Filter
open scoped Topology
open MeasureTheory Filter ProbabilityTheory
open scoped Topology NNReal ENNReal
open MeasureTheory Filter ProbabilityTheory
open scoped Topology NNReal ENNReal
open MeasureTheory Filter ProbabilityTheory
open scoped Topology NNReal ENNReal
open MeasureTheory Filter ProbabilityTheory
open scoped Topology NNReal ENNReal
open MeasureTheory Filter ProbabilityTheory ContinuousLinearMap
open scoped Topology NNReal ENNReal
open Filter MeasureTheory ProbabilityTheory
open scoped Topology NNReal ENNReal
open MeasureTheory Filter
open scoped BigOperators Topology
open MeasureTheory Filter
open scoped BigOperators Topology
open MeasureTheory Filter
open scoped BigOperators Topology
open MeasureTheory Filter
open scoped BigOperators Topology
open MeasureTheory Filter
open scoped BigOperators Topology
open Filter Set Metric
open scoped Topology RealInnerProductSpace
open scoped BigOperators
open ContinuousLinearMap
open scoped BigOperators
open ContinuousLinearMap
open scoped Topology Interval
open MeasureTheory
open MeasureTheory
open scoped BigOperators Topology Interval
open MeasureTheory
open scoped BigOperators Topology Interval
open scoped Topology
open MeasureTheory
open scoped BigOperators Topology Interval
open scoped BigOperators Topology
open MeasureTheory
open scoped BigOperators Topology Interval RealInnerProductSpace
open MeasureTheory Filter
open scoped BigOperators Topology Interval
open MeasureTheory Filter
open scoped Topology
open MeasureTheory Filter
open scoped Topology
open MeasureTheory Filter
open scoped Topology
open MeasureTheory ProbabilityTheory Filter
open scoped BigOperators NNReal ENNReal Matrix.Norms.L2Operator
open MeasureTheory ProbabilityTheory Filter Matrix
open scoped BigOperators ENNReal NNReal Topology Matrix.Norms.L2Operator
namespace SKGapCutoff.Regression

noncomputable def plantedFieldArray (β : ℝ) (n : ℕ) (g : Option (Fin n) → ℝ) : Fin n → ℝ :=
  sharedGaussianField (β^2*((n:ℝ)-1)/(n:ℝ))
    (β*Real.sqrt (((n:ℝ)-2)/(n:ℝ))) (β/Real.sqrt (n:ℝ)) g

lemma plantedFieldArray_measurable (β : ℝ) (n : ℕ) : Measurable (plantedFieldArray β n) := by
  unfold plantedFieldArray sharedGaussianField
  fun_prop

lemma planted_field_event_eq (β : ℝ) {n : ℕ} (hn : 2≤n) (x : Spin n)
    (E : Set (Fin n → ℝ)) (hE : MeasurableSet E) :
    singleSpinPlantedLaw β x ((gaugedField x) ⁻¹' E) =
      standardArrayLaw (Option (Fin n)) ((plantedFieldArray β n) ⁻¹' E) := by
  have hh := congrArg (fun μ : Measure (Fin n → ℝ) => μ E) (planted_field_representation β hn x)
  change (singleSpinPlantedLaw β x).map (gaugedField x) E =
    (standardArrayLaw (Option (Fin n))).map (plantedFieldArray β n) E at hh
  rw [Measure.map_apply (continuous_gaugedField x).measurable hE,
    Measure.map_apply (plantedFieldArray_measurable β n) hE] at hh
  exact hh

lemma plantedField_common_convergence (β : ℝ) (hβ : 0<β) :
    ExponentialConvergence (fun n => standardArrayLaw (Option (Fin n)))
      (fun n g => (β/Real.sqrt (n:ℝ))*g none) 0 := by
  intro ε hε
  refine ⟨2,ε^2/(2*β^2),by norm_num,by positivity,?_⟩
  filter_upwards [eventually_gt_atTop 0] with n hn
  have hnR : (0:ℝ)<n := Nat.cast_pos.mpr hn
  have ht := subgaussian_abs_tail
    (gaussian_hasSubgaussianMGF (coordinate_hasLaw (none : Option (Fin n))))
      (ε*Real.sqrt (n:ℝ)/β) (by positivity)
  simp only [Real.dist_eq,sub_zero]
  apply (measure_mono (show {g : Option (Fin n) → ℝ | ε ≤ |β/Real.sqrt (n:ℝ)*g none|} ⊆
    {g | ε*Real.sqrt (n:ℝ)/β ≤ |g none|} from ?_)).trans
  · apply ht.trans_eq
    rw [ENNReal.ofReal_mul (by norm_num : (0:ℝ)≤2)]
    norm_num only [ENNReal.ofReal_ofNat,NNReal.coe_one,mul_one]
    congr 2
    rw [div_pow,mul_pow,Real.sq_sqrt hnR.le]
    ring_nf
  · intro g hg
    change ε ≤ |β/Real.sqrt (n:ℝ)*g none| at hg
    rw [abs_mul,abs_of_pos (by positivity : 0<β/Real.sqrt (n:ℝ))] at hg
    apply (div_le_iff₀ hβ).mpr
    have hh := (le_div_iff₀ (Real.sqrt_pos.mpr hnR)).mp
      (show ε ≤ β*|g none|/Real.sqrt (n:ℝ) by simpa only [div_mul_eq_mul_div] using hg)
    nlinarith

lemma plantedField_mean_bound (β : ℝ) (n : ℕ) :
    |β^2*((n:ℝ)-1)/(n:ℝ)| ≤ β^2 := by
  by_cases hn : n=0
  · simp [hn, sq_nonneg]
  have hn1 : (1:ℝ)≤n := by exact_mod_cast (Nat.one_le_iff_ne_zero.mpr hn)
  rw [abs_of_nonneg (by positivity)]
  apply (div_le_iff₀ (by positivity : (0:ℝ)<n)).mpr
  nlinarith [sq_nonneg β]

lemma plantedField_scale_bound (β : ℝ) (n : ℕ) :
    |β*Real.sqrt (((n:ℝ)-2)/(n:ℝ))| ≤ |β| := by
  have hs : Real.sqrt (((n:ℝ)-2)/(n:ℝ)) ≤ 1 := by
    apply Real.sqrt_le_one.mpr
    by_cases hn : n=0
    · simp [hn]
    · apply (div_le_iff₀ (Nat.cast_pos.mpr (Nat.pos_of_ne_zero hn))).mpr
      linarith
  rw [abs_mul,abs_of_nonneg (Real.sqrt_nonneg _)]
  nlinarith [abs_nonneg β]

lemma plantedFieldArray_squareTails (β : ℝ) (hβ : 0<β) :
    ExponentialSquareTails (fun n => standardArrayLaw (Option (Fin n)))
      (plantedFieldArray β) := by
  let ρ := fun n => standardArrayLaw (Option (Fin n))
  have hi : ExponentialSquareTails ρ (fun _ g i => g (some i)) := by
    exact iid_exponential_squareTails ρ _ (fun _ _ => measurable_pi_apply _)
      (fun _ => coordinates_independent.precomp (Option.some_injective _))
      (gaussianReal 0 1) (fun _ _ => coordinate_hasLaw _) gaussian_integrable_exp_quarter_sq
  have ha := hi.bounded_mul
    (fun n _ _ => β*Real.sqrt (((n:ℝ)-2)/(n:ℝ))) |β| (abs_nonneg β)
    (fun n _ _ => plantedField_scale_bound β n)
  have hm := exponentialSquareTails_of_bounded ρ
    (fun n _ _ => β^2*((n:ℝ)-1)/(n:ℝ)) (β^2) (fun n _ _ => plantedField_mean_bound β n)
  have hb := hm.add ha
  apply hb.l2_stability
  have hz := (plantedField_common_convergence β hβ).continuous_map
    (f := fun x : ℝ => x^2) (by fun_prop)
  simp only [zero_pow (by norm_num : (2:ℕ)≠0)] at hz
  intro ε hε
  apply (hz ε hε).mono
  filter_upwards [eventually_gt_atTop 0] with n hn
  intro g hg
  have he : (∑ i : Fin n,
      (β^2*((n:ℝ)-1)/(n:ℝ)+β*Real.sqrt (((n:ℝ)-2)/(n:ℝ))*g (some i) -
        plantedFieldArray β n g i)^2)/(n:ℝ) = ((β/Real.sqrt (n:ℝ))*g none)^2 := by
    simp only [plantedFieldArray,sharedGaussianField,sub_add_cancel_left,
      neg_sq,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]
    field_simp
  change ε ≤ dist _ 0 at hg ⊢
  rwa [he] at hg

noncomputable def plantedFieldLimitLaw (β : ℝ) : Measure ℝ :=
  (gaussianReal 0 1).map (fun z => β^2+β*z)

instance plantedFieldLimitLaw_probability (β : ℝ) : IsProbabilityMeasure (plantedFieldLimitLaw β) := by
  unfold plantedFieldLimitLaw
  infer_instance

lemma plantedFieldArray_empirical (β : ℝ) (hβ : 0<β) :
    ExponentialEmpiricalConcentration (fun n => standardArrayLaw (Option (Fin n)))
      (plantedFieldArray β) (plantedFieldLimitLaw β) := by
  intro f K hf B hB ε hε
  have hf' : LipschitzWith (K+1) f := hf.weaken (by simp)
  obtain ⟨c,hc,ht⟩ := planted_field_average_exponential β hβ f (by positivity) hf'
    (|B|+1) (by positivity) (fun z => (hB z).trans (by linarith [le_abs_self B])) ε hε
  refine ⟨1,c,by norm_num,hc,?_⟩
  filter_upwards [ht,eventually_ge_atTop 2] with n hn hn2
  have hm : (∫ z, f z ∂plantedFieldLimitLaw β) = ∫ z, f (β^2+β*z) ∂gaussianReal 0 1 := by
    exact integral_map (by fun_prop) hf.continuous.aestronglyMeasurable
  rw [hm]
  have hh := hn (fun _ => false)
  have hE : MeasurableSet {v : Fin n → ℝ | ε ≤ |(∑ i, f (v i))/(n:ℝ) -
      ∫ z, f (β^2+β*z) ∂gaussianReal 0 1|} := by
    apply measurableSet_le measurable_const
    exact (((Finset.measurable_sum _ (fun i _ => hf.continuous.measurable.comp
      (measurable_pi_apply i))).div_const _).sub_const _).abs
  rw [show singleSpinPlantedLaw β (fun _ : Fin n => false) {g | ε ≤
      |(∑ i, f (gaugedField (fun _ => false) g i))/(n:ℝ) -
      ∫ z, f (β^2+β*z) ∂gaussianReal 0 1|} = _ from
        planted_field_event_eq β hn2 (fun _ => false) _ hE] at hh
  simpa only [one_mul,Set.preimage_ofPred_eq] using hh

lemma plantedFieldLimitLaw_secondMoment (β : ℝ) :
    Integrable (fun z : ℝ => z^2) (plantedFieldLimitLaw β) ∧
      (∫ z, z^2 ∂plantedFieldLimitLaw β)=β^2+β^4 := by
  have hl : HasLaw (fun z : ℝ => β^2+β*z) (plantedFieldLimitLaw β) (gaussianReal 0 1) :=
    ⟨by fun_prop,rfl⟩
  have hi2 : MemLp (id : ℝ → ℝ) 2 (plantedFieldLimitLaw β) := by
    apply (memLp_map_measure_iff (by fun_prop) (by fun_prop)).mpr
    exact (memLp_const (β^2)).add
      (((IsGaussian.hasGaussianLaw_id (μ := gaussianReal 0 1)).memLp_two).const_mul β)
  have hi := hi2.integrable_sq
  refine ⟨hi,?_⟩
  have hmean : (∫ z, z ∂plantedFieldLimitLaw β)=β^2 := by
    rw [← hl.integral_eq]
    have hm : Integrable (fun z : ℝ => β*z) (gaussianReal 0 1) :=
      (IsGaussian.hasGaussianLaw_id (μ := gaussianReal 0 1)).integrable.const_mul β
    rw [integral_add (integrable_const _) hm,integral_const,integral_const_mul]
    simp [integral_id_gaussianReal]
  have hv : Var[id; plantedFieldLimitLaw β]=β^2 := by
    rw [← hl.variance_eq,variance_const_add (by fun_prop),variance_const_mul]
    simp
  have hh := variance_eq_sub (X := id) hi2
  simp only [Pi.pow_apply,id_eq,hmean] at hh
  rw [hv] at hh
  nlinarith

lemma plantedFieldArray_secondMoment (β : ℝ) (hβ : 0<β) :
    ExponentialConvergence (fun n => standardArrayLaw (Option (Fin n)))
      (fun n g => (∑ i, plantedFieldArray β n g i^2)/(n:ℝ)) (β^2+β^4) := by
  have hh := (plantedFieldArray_empirical β hβ).secondMoment
    (plantedFieldArray_squareTails β hβ) (plantedFieldLimitLaw_secondMoment β).1
  rwa [(plantedFieldLimitLaw_secondMoment β).2] at hh

lemma ExponentiallyRare.without_prefactor {H : ℕ → Type*} [∀ n, MeasurableSpace (H n)]
    {ρ : ∀ n, Measure (H n)} {A : ∀ n, Set (H n)} (hA : ExponentiallyRare ρ A) :
    ∃ c : ℝ, 0<c ∧ ∀ᶠ n in atTop, ρ n (A n) ≤ ENNReal.ofReal (Real.exp (-c*n)) := by
  obtain ⟨C,c,hC,hc,he⟩ := hA
  refine ⟨c/2,by positivity,?_⟩
  have hnc := (tendsto_natCast_atTop_atTop : Tendsto (fun n : ℕ => (n:ℝ)) atTop atTop)
  filter_upwards [he,hnc.eventually (eventually_ge_atTop (2*Real.log C/c))] with n hn hsize
  apply hn.trans
  apply ENNReal.ofReal_le_ofReal
  rw [show C*Real.exp (-c*n)=Real.exp (Real.log C-c*n) by
    rw [Real.exp_sub,Real.exp_log hC,neg_mul,Real.exp_neg,div_eq_mul_inv] ]
  apply Real.exp_le_exp.mpr
  have hh := (div_le_iff₀ hc).mp hsize
  nlinarith

noncomputable def spinFieldSquareAverage {n : ℕ} (x : Spin n)
    (g : GaussianCoordinates n) : ℝ := (∑ i, gaugedField x g i^2)/(n:ℝ)

lemma continuous_spinFieldSquareAverage {n : ℕ} (x : Spin n) :
    Continuous (spinFieldSquareAverage x) := by
  change Continuous (fun g => (∑ i, gaugedField x g i^2)/(n:ℝ))
  exact (continuous_finsetSum _ (fun i _ =>
    ((continuous_apply i).comp (continuous_gaugedField x)).pow 2)).div_const _

lemma spinFieldSquareAverage_planted_exponential (β : ℝ) (hβ : 0<β)
    (ε : ℝ) (hε : 0<ε) : ∃ c : ℝ, 0<c ∧ ∀ᶠ n : ℕ in atTop,
      ∀ x : Spin n, singleSpinPlantedLaw β x {g | ε ≤ |spinFieldSquareAverage x g-(β^2+β^4)|} ≤
        ENNReal.ofReal (Real.exp (-c*n)) := by
  obtain ⟨c,hc,ht⟩ := ((plantedFieldArray_secondMoment β hβ) ε hε).without_prefactor
  refine ⟨c,hc,?_⟩
  filter_upwards [ht,eventually_ge_atTop 2] with n hn hn2
  intro x
  have hE : MeasurableSet {v : Fin n → ℝ | ε ≤ |(∑ i, v i^2)/(n:ℝ)-(β^2+β^4)|} := by
    apply measurableSet_le measurable_const
    exact (((Finset.measurable_sum _ (fun i _ => (measurable_pi_apply i).pow_const 2)).div_const _).sub_const _).abs
  have he := planted_field_event_eq β hn2 x _ hE
  change singleSpinPlantedLaw β x {g | ε ≤ |spinFieldSquareAverage x g-(β^2+β^4)|} = _ at he
  rw [he]
  simpa only [Real.dist_eq,Set.preimage_ofPred_eq] using hn

theorem quenched_spinFieldSquareAverage_exponential (β : ℝ) (hβ : 0<β) (hβ1 : β<1)
    (ε : ℝ) (hε : 0<ε) : ∃ c : ℝ, 0<c ∧ Tendsto (fun n => disorderLaw β n {g |
      ENNReal.ofReal (Real.exp (-c*n)) ≤ gibbsEventProbability
        (fun x : Spin n => {h | ε ≤ |spinFieldSquareAverage x h-(β^2+β^4)|}) g})
      atTop (nhds 0) := by
  obtain ⟨c,hc,hs⟩ := spinFieldSquareAverage_planted_exponential β hβ ε hε
  refine ⟨c/2,by positivity,?_⟩
  apply quenched_gibbs_events_exponential β (by nlinarith) c hc _ _ hs
  intro n x
  exact measurableSet_le measurable_const ((continuous_spinFieldSquareAverage x).measurable.sub_const _).abs

lemma spinFieldSquareAverage_abs_le_opNorm_sq {n : ℕ} (hn : 0<n) (x : Spin n)
    (g : GaussianCoordinates n) : |spinFieldSquareAverage x g| ≤
      ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (sampledInteraction g)‖^2 := by
  have hh := matrix_mulVec_sq_bound (sampledInteraction g) (spin x) le_rfl
  have he : (∑ i, gaugedField x g i^2) = ∑ i, (sampledInteraction g *ᵥ spin x) i^2 := by
    simp only [gaugedField,mul_pow,spin_sq,one_mul,field,Matrix.mulVec,dotProduct]
  have hsum : (∑ i : Fin n, spin x i^2)=(n:ℝ) := by simp [spin_sq]
  rw [hsum] at hh
  rw [spinFieldSquareAverage,abs_of_nonneg (by positivity),he]
  exact (div_le_iff₀ (Nat.cast_pos.mpr hn)).mpr hh

noncomputable def gibbsSpinFieldSquareAverage {n : ℕ} (g : GaussianCoordinates n) : ℝ :=
  gibbsExpectation (sampledInteraction g) (fun x => spinFieldSquareAverage x g)

theorem gibbsSpinFieldSquareAverage_limit (β : ℝ) (hβ : 0<β) (hβ1 : β<1)
    (ε : ℝ) (hε : 0<ε) :
    Tendsto (fun n => disorderLaw β n {g | ε ≤ |gibbsSpinFieldSquareAverage g-(β^2+β^4)|})
      atTop (nhds 0) := by
  obtain ⟨c,hc,hevent⟩ := quenched_spinFieldSquareAverage_exponential β hβ hβ1 (ε/2) (by positivity)
  let C := 2*(β^2+10)
  let K := C^2+(β^2+β^4)
  have hK : 0 ≤ K := by dsimp [K,C]; positivity
  have hnorm := RandomMatrix.sampled_opNorm_tendsto β
  have ht : Tendsto (fun n : ℕ => K*Real.exp (-c*n)) atTop (nhds 0) := by
    have hh := Real.tendsto_exp_neg_atTop_nhds_zero.comp
      ((tendsto_natCast_atTop_atTop : Tendsto (fun n : ℕ => (n:ℝ)) atTop atTop).const_mul_atTop hc)
    simpa only [Function.comp_apply,neg_mul,mul_zero] using hh.const_mul K
  have hsum := hnorm.add hevent
  simp only [add_zero] at hsum
  apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hsum
    (Eventually.of_forall (fun _ => bot_le))
  filter_upwards [eventually_gt_atTop 0,ht.eventually (gt_mem_nhds (by positivity : (0:ℝ)<ε/2))]
    with n hn hsmall
  apply (measure_mono (show {g | ε ≤ |gibbsSpinFieldSquareAverage g-(β^2+β^4)|} ⊆
      {g | C < ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (sampledInteraction g)‖} ∪
      {g | ENNReal.ofReal (Real.exp (-c*n)) ≤
        gibbsEventProbability (fun x : Spin n => {h | ε/2 ≤ |spinFieldSquareAverage x h-(β^2+β^4)|}) g} from ?_)).trans
      (measure_union_le _ _)
  intro g hg
  by_cases ha : C < ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (sampledInteraction g)‖
  · exact Or.inl ha
  by_cases hb : ENNReal.ofReal (Real.exp (-c*n)) ≤
      gibbsEventProbability (fun x : Spin n => {h | ε/2 ≤ |spinFieldSquareAverage x h-(β^2+β^4)|}) g
  · exact Or.inr hb
  exfalso
  have hp : (gibbsEventProbability (fun x : Spin n => {h | ε/2 ≤ |spinFieldSquareAverage x h-(β^2+β^4)|}) g).toReal
      ≤ Real.exp (-c*n) := by
    have hh := ENNReal.toReal_mono (show ENNReal.ofReal (Real.exp (-c*n))≠⊤ from ENNReal.ofReal_ne_top)
      (lt_of_not_ge hb).le
    simpa only [ENNReal.toReal_ofReal (Real.exp_pos _).le] using hh
  have hdev := gibbsExpectation_deviation_bound g
    (fun x h => spinFieldSquareAverage x h) ((β^2+β^4)) (ε/2) K (by positivity) (fun x => by
      calc
        _ ≤ |spinFieldSquareAverage x g|+|(β^2+β^4)| := by simpa only [sub_zero,zero_sub,abs_neg] using abs_sub_le (spinFieldSquareAverage x g) (0:ℝ) ((β^2+β^4))
        _ ≤ C^2+(β^2+β^4) := by
          rw [abs_of_nonneg (by positivity : 0≤β^2+β^4)]
          exact add_le_add_left ((spinFieldSquareAverage_abs_le_opNorm_sq hn x g).trans
            (pow_le_pow_left₀ (norm_nonneg _) (le_of_not_gt ha) 2)) _)
  change ε ≤ |gibbsExpectation (sampledInteraction g) (fun x => spinFieldSquareAverage x g)-(β^2+β^4)| at hg
  have he := mul_le_mul_of_nonneg_left hp hK
  linarith

end SKGapCutoff.Regression

open MeasureTheory ProbabilityTheory Filter Matrix
open scoped BigOperators ENNReal Matrix.Norms.L2Operator

end

end OAI