| import OAI.Probability.InvariantIsing.Core.Ward |
|
|
| namespace OAI |
|
|
| /-! Endpoint Gibbs averages bound a finite log-partition difference. These |
| are the convex secant inequalities used to change spectral multiplicities. -/ |
|
|
| noncomputable section |
| open IsingPerceptron |
| open scoped BigOperators |
|
|
| namespace InvariantIsing |
|
|
| lemma log_finitePartition_sub_ge_gibbsAverage |
| {S : Type*} [Fintype S] {w : S → ℝ} (hw : GibbsReference w) |
| (H K : S → ℝ) : |
| gibbsAverage w H (fun s => K s - H s) ≤ |
| Real.log (finitePartition w K) - Real.log (finitePartition w H) := by |
| let Y := fun s => K s - H s |
| have hd := hasDerivAt_log_finitePartition hw (t := (0 : ℝ)) |
| (fun s => (hasDerivAt_const 0 (H s)).add ((hasDerivAt_id 0).mul_const (Y s))) |
| have hc := convexOn_log_finitePartition hw H Y |
| have hs := hc.le_slope_of_hasDerivAt (Set.mem_univ (0 : ℝ)) |
| (Set.mem_univ (1 : ℝ)) (by norm_num) hd |
| simpa only [slope_def_field, sub_zero, div_one, zero_add, zero_mul, one_mul, |
| add_sub_cancel, Y, gibbsAverage, Pi.add_apply, id_eq, add_zero] using hs |
|
|
| lemma log_finitePartition_sub_le_gibbsAverage |
| {S : Type*} [Fintype S] {w : S → ℝ} (hw : GibbsReference w) |
| (H K : S → ℝ) : |
| Real.log (finitePartition w K) - Real.log (finitePartition w H) ≤ |
| gibbsAverage w K (fun s => K s - H s) := by |
| have h := log_finitePartition_sub_ge_gibbsAverage hw K H |
| have he : gibbsAverage w K (fun s => H s - K s) = |
| -gibbsAverage w K (fun s => K s - H s) := by |
| simp only [gibbsAverage, mul_sub, Finset.sum_sub_distrib] |
| ring |
| rw [he] at h |
| linarith |
|
|
| lemma log_finitePartition_secant_bounds |
| {S : Type*} [Fintype S] {w : S → ℝ} (hw : GibbsReference w) |
| (H K : S → ℝ) : |
| gibbsAverage w H (fun s => K s - H s) ≤ |
| Real.log (finitePartition w K) - Real.log (finitePartition w H) ∧ |
| Real.log (finitePartition w K) - Real.log (finitePartition w H) ≤ |
| gibbsAverage w K (fun s => K s - H s) := |
| ⟨log_finitePartition_sub_ge_gibbsAverage hw H K, |
| log_finitePartition_sub_le_gibbsAverage hw H K⟩ |
|
|
| end InvariantIsing |
|
|
| end |
|
|
| end OAI |
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|
