feat(Expander): spectral gap of box products of permutation multigraphs

Complexity.lean

@@ -2472,6 +2472,7 @@ public import Complexity.Foundations.Combinatorics.Expander.AffineFamily
 public import Complexity.Foundations.Combinatorics.Expander.AffinePlane
 public import Complexity.Foundations.Combinatorics.Expander.AffineTower
 public import Complexity.Foundations.Combinatorics.Expander.Basic
+public import Complexity.Foundations.Combinatorics.Expander.BoxProduct
 public import Complexity.Foundations.Combinatorics.Expander.Cayley
 public import Complexity.Foundations.Combinatorics.Expander.Cheeger
 public import Complexity.Foundations.Combinatorics.Expander.Connected

Complexity/Foundations/Combinatorics/Expander/BoxProduct.lean

@@ -0,0 +1,184 @@
+/-
+Copyright (c) 2026 Christian Reitwiessner. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Christian Reitwiessner
+-/
+
+module
+-- upstream: candidate
+
+public import Complexity.Foundations.Combinatorics.Expander.Lazy
+public import Complexity.Foundations.Combinatorics.Expander.Perm
+import Complexity.Foundations.Data.Finset.Expect
+
+/-!
+# Box products of permutation multigraphs
+
+Two families of permutations `σ : K → Equiv.Perm V` and `τ : K' → Equiv.Perm W` give the family
+`boxProdPerms σ τ : K ⊕ K' → Equiv.Perm (V × W)` that moves one coordinate at a time: `σ k` on the
+first, `τ k'` on the second. Its multigraph `ofPerms (boxProdPerms σ τ)` is the box (Cartesian)
+product of the multigraphs of `σ` and `τ`. Its random walk picks a coordinate, with probability
+proportional to the number of its permutations, and takes a step there.
+
+A Poincaré inequality with constant `c` for both families gives one for the product with the same
+constant (`mul_expect_sq_le_boxProdPerms`). The proof splits a function of expectation zero on
+`V × W` orthogonally into its averages over the fibers `{v} × W` and a part of average zero on every
+fiber. The second family controls the fiber part, and the first controls the fiber averages, whose
+changes along `σ` are at most the averages of the changes of the function (Jensen). This is the
+two-factor case of the tensorization of the spectral gap of product chains: the gap of the product
+walk is at least the smaller of the gaps of the factors, weighted by their shares of the degree
+(`secondEigenvalue_ofPerms_boxProdPerms_le`). For `J` factors of equal degree and gap `γ`, iterating
+gives the gap `γ / J`. This is the walk that the OpenAI preprint on derandomizing logarithmic space
+applies to a uniformly selected component of its sampling environment, with gap `γ / J` for `J`
+components [OpenAI, proof of Lemma 6.2][OpenAI2026LogSpace]; the statements and proofs here are
+ours.
+
+## Main definitions
+
+* `Cslib.RotationMap.boxProdPerms`: the permutations of `V × W` that move one coordinate.
+
+## Main results
+
+* `Cslib.RotationMap.mul_expect_sq_le_boxProdPerms`: Poincaré inequalities for the two factors give
+  one for the box product, with the same constant.
+* `Cslib.RotationMap.secondEigenvalue_ofPerms_boxProdPerms_le`: the second eigenvalue of the box
+  product, `λ₂ ≤ 1 - min (|K| (1 - λ₂(σ))) (|K'| (1 - λ₂(τ))) / (|K| + |K'|)`.
+* `Cslib.RotationMap.spectralExpansion_lazy_ofPerms_boxProdPerms_le`: the same bound, halved, for
+  the spectral expansion of the lazy box product.
+
+## References
+
+* [OpenAI, *Exact Derandomization of Logarithmic Space: L = RL = BPL*][OpenAI2026LogSpace]
+-/
+
+public section
+
+namespace Cslib.RotationMap
+
+open Finset
+open scoped BigOperators
+
+variable {V W K K' : Type*}
+
+/-- The permutations of `V × W` that move one coordinate: `σ k` on the first, `τ k'` on the second.
+Their multigraph is the box product of the multigraphs of `σ` and `τ`. -/
+@[expose]
+def boxProdPerms (σ : K → Equiv.Perm V) (τ : K' → Equiv.Perm W) : K ⊕ K' → Equiv.Perm (V × W) :=
+  Sum.elim (fun k ↦ (σ k).prodCongr (Equiv.refl W)) fun k ↦ (Equiv.refl V).prodCongr (τ k)
+
+variable (σ : K → Equiv.Perm V) (τ : K' → Equiv.Perm W)
+
+@[simp]
+theorem boxProdPerms_inl (k : K) (p : V × W) : boxProdPerms σ τ (.inl k) p = (σ k p.1, p.2) := rfl
+
+@[simp]
+theorem boxProdPerms_inr (k : K') (p : V × W) : boxProdPerms σ τ (.inr k) p = (p.1, τ k p.2) := rfl
+
+variable [Fintype V] [Fintype W] [Fintype K] [Fintype K']
+
+variable {σ τ} in
+/-- **Poincaré inequalities tensorize**: if both families satisfy the Poincaré
+inequality `c 𝔼 f² ≤ ∑ k, 𝔼 v, (f (σ k v) - f v) ^ 2` for every function `f` of expectation zero,
+then so does their box product, with the same constant. -/
+theorem mul_expect_sq_le_boxProdPerms {c : ℝ}
+    (hσ : ∀ f : V → ℝ, 𝔼 v, f v = 0 → c * 𝔼 v, f v ^ 2 ≤ ∑ k, 𝔼 v, (f (σ k v) - f v) ^ 2)
+    (hτ : ∀ g : W → ℝ, 𝔼 w, g w = 0 → c * 𝔼 w, g w ^ 2 ≤ ∑ k, 𝔼 w, (g (τ k w) - g w) ^ 2)
+    (f : V × W → ℝ) (hf : 𝔼 p, f p = 0) :
+    c * 𝔼 p, f p ^ 2 ≤ ∑ k, 𝔼 p, (f (boxProdPerms σ τ k p) - f p) ^ 2 := by
+  rcases isEmpty_or_nonempty W with hW | hW
+  · simp
+  -- The fiber averages `g` and the decomposition `𝔼 f² = 𝔼 g² + 𝔼 (f - g)²`.
+  set g : V → ℝ := fun v ↦ 𝔼 w, f (v, w) with hg
+  have hsplit : 𝔼 p, f p ^ 2 = 𝔼 v, g v ^ 2 + 𝔼 v, 𝔼 w, (f (v, w) - g v) ^ 2 := by
+    have h := Fintype.expect_add_sq_of_expect_fiber_eq_zero g
+      (g := fun p ↦ f p - g p.1) (Fintype.expect_sub_expect_fiber f)
+    simpa only [add_sub_cancel, Fintype.expect_prod_type, Fintype.expect_const] using h
+  -- The second family controls the parts of average zero on the fibers.
+  have hfib (v : V) :
+      c * 𝔼 w, (f (v, w) - g v) ^ 2 ≤ ∑ k, 𝔼 w, (f (v, τ k w) - f (v, w)) ^ 2 := by
+    have := hτ (fun w ↦ f (v, w) - g v) (Fintype.expect_sub_expect_fiber f v)
+    simpa only [sub_sub_sub_cancel_right] using this
+  -- The first family controls the fiber averages, by Jensen's inequality.
+  have hg0 : 𝔼 v, g v = 0 := by rw [← Fintype.expect_prod_type]; exact hf
+  have hjensen (k : K) (v : V) : (g (σ k v) - g v) ^ 2 ≤ 𝔼 w, (f (σ k v, w) - f (v, w)) ^ 2 := by
+    simpa only [hg, expect_sub_distrib] using
+      sq_expect_le_expect_sq univ fun w ↦ f (σ k v, w) - f (v, w)
+  calc c * 𝔼 p, f p ^ 2
+      = c * 𝔼 v, g v ^ 2 + 𝔼 v, c * 𝔼 w, (f (v, w) - g v) ^ 2 := by
+        rw [hsplit, mul_add, mul_expect _ fun v ↦ 𝔼 w, (f (v, w) - g v) ^ 2]
+    _ ≤ ∑ k, 𝔼 v, (g (σ k v) - g v) ^ 2 + 𝔼 v, ∑ k, 𝔼 w, (f (v, τ k w) - f (v, w)) ^ 2 :=
+        add_le_add (hσ g hg0) (expect_le_expect fun v _ ↦ hfib v)
+    _ ≤ ∑ k, 𝔼 v, 𝔼 w, (f (σ k v, w) - f (v, w)) ^ 2 +
+          𝔼 v, ∑ k, 𝔼 w, (f (v, τ k w) - f (v, w)) ^ 2 := by
+        gcongr with k _ v
+        exact hjensen k v
+    _ = ∑ k, 𝔼 p, (f (boxProdPerms σ τ k p) - f p) ^ 2 := by
+        rw [Fintype.sum_sum_type, expect_sum_comm]
+        simp only [boxProdPerms_inl, boxProdPerms_inr, Fintype.expect_prod_type]
+
+/-- A bound on the second eigenvalue of a permutation multigraph gives a Poincaré inequality for
+its permutations, with constant `2 |K| (1 - λ₂)`. -/
+private theorem mul_expect_sq_le_of_secondEigenvalue [Nonempty K] (σ : K → Equiv.Perm V)
+    (f : V → ℝ) (hf : 𝔼 v, f v = 0) :
+    2 * Fintype.card K * (1 - (ofPerms σ).secondEigenvalue) * 𝔼 v, f v ^ 2 ≤
+      ∑ k, 𝔼 v, (f (σ k v) - f v) ^ 2 := by
+  have hK : (0 : ℝ) < 2 * Fintype.card K := by
+    have := Fintype.card_pos (α := K)
+    positivity
+  have h := (ofPerms σ).expect_mul_walk_le_secondEigenvalue_mul hf
+  rw [expect_mul_walk_ofPerms] at h
+  have := mul_le_mul_of_nonneg_left h hK.le
+  rw [mul_sub, mul_div_cancel₀ _ hK.ne'] at this
+  linarith
+
+/-- **The spectral gap of a box product**: the second eigenvalue of the box product of two
+permutation multigraphs is at most
+`1 - min (|K| (1 - λ₂(σ))) (|K'| (1 - λ₂(τ))) / (|K| + |K'|)`. For factors with the same number of
+permutations, the gap is at least half the smaller gap of the factors. -/
+theorem secondEigenvalue_ofPerms_boxProdPerms_le [Nonempty K] [Nonempty K'] :
+    (ofPerms (boxProdPerms σ τ)).secondEigenvalue ≤
+      1 - min (Fintype.card K * (1 - (ofPerms σ).secondEigenvalue))
+        (Fintype.card K' * (1 - (ofPerms τ).secondEigenvalue)) /
+        (Fintype.card K + Fintype.card K') := by
+  set a : ℝ := Fintype.card K * (1 - (ofPerms σ).secondEigenvalue)
+  set b : ℝ := Fintype.card K' * (1 - (ofPerms τ).secondEigenvalue)
+  have hKK : (0 : ℝ) < Fintype.card K + Fintype.card K' := by
+    have := Fintype.card_pos (α := K)
+    positivity
+  rcases subsingleton_or_nontrivial (V × W) with hV | hV
+  · -- On at most one vertex `λ₂ = 0`, and the bound is nonnegative as `λ₂ ≥ -1` for the factors.
+    rw [secondEigenvalue_eq_zero_of_subsingleton, sub_nonneg, div_le_one hKK]
+    have ha2 : a ≤ 2 * Fintype.card K := by
+      have := (ofPerms σ).neg_one_le_secondEigenvalue
+      have : (0 : ℝ) ≤ Fintype.card K := Nat.cast_nonneg _
+      simp only [a]
+      nlinarith
+    have hb2 : b ≤ 2 * Fintype.card K' := by
+      have := (ofPerms τ).neg_one_le_secondEigenvalue
+      have : (0 : ℝ) ≤ Fintype.card K' := Nat.cast_nonneg _
+      simp only [b]
+      nlinarith
+    linarith [min_le_left a b, min_le_right a b]
+  have h := secondEigenvalue_ofPerms_le (boxProdPerms σ τ) (c := 2 * min a b) fun f hf ↦
+    mul_expect_sq_le_boxProdPerms
+      (fun g hg ↦ (mul_le_mul_of_nonneg_right (by linarith [min_le_left a b])
+        (expect_nonneg fun _ _ ↦ sq_nonneg _)).trans (mul_expect_sq_le_of_secondEigenvalue σ g hg))
+      (fun g hg ↦ (mul_le_mul_of_nonneg_right (by linarith [min_le_right a b])
+        (expect_nonneg fun _ _ ↦ sq_nonneg _)).trans (mul_expect_sq_le_of_secondEigenvalue τ g hg))
+      f hf
+  refine h.trans_eq ?_
+  rw [Fintype.card_sum, Nat.cast_add, mul_div_mul_left _ _ two_ne_zero]
+
+/-- **The lazy box product is an expander** when both factors are: its spectral expansion is at most
+`1 - min (|K| (1 - λ₂(σ))) (|K'| (1 - λ₂(τ))) / (2 (|K| + |K'|))`. -/
+theorem spectralExpansion_lazy_ofPerms_boxProdPerms_le [Nonempty K] [Nonempty K'] :
+    (ofPerms (boxProdPerms σ τ)).lazy.spectralExpansion ≤
+      1 - min (Fintype.card K * (1 - (ofPerms σ).secondEigenvalue))
+        (Fintype.card K' * (1 - (ofPerms τ).secondEigenvalue)) /
+        (2 * (Fintype.card K + Fintype.card K')) := by
+  have h₁ := (ofPerms (boxProdPerms σ τ)).spectralExpansion_lazy_le_one_add_secondEigenvalue_div_two
+  have h₂ := secondEigenvalue_ofPerms_boxProdPerms_le σ τ
+  rw [div_mul_eq_div_div_swap]
+  linarith
+
+end Cslib.RotationMap

ComplexityTests/BoxProductExpander.lean

@@ -0,0 +1,72 @@
+/-
+Copyright (c) 2026 Christian Reitwiessner. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Christian Reitwiessner
+-/
+
+import Complexity.Foundations.Combinatorics.Expander.BoxProduct
+import Complexity.Foundations.Combinatorics.Expander.ShearTranslation
+
+/-!
+# Box products of permutation multigraphs
+
+Concrete moves of a box product, the Poincaré inequality for the box product of two copies of the
+two-point walk, and a uniform spectral gap for the box product of two shear-translation graphs of
+unrelated sizes, with every hypothesis discharged.
+-/
+
+namespace ComplexityTests.BoxProductExpander
+
+open Cslib Cslib.RotationMap Finset
+open scoped BigOperators
+
+-- A move of the first factor leaves the second coordinate alone, and vice versa.
+example : boxProdPerms (shearTranslationMove 5) (shearTranslationMove 3) (.inl 0)
+    ((1, 2), (2, 2)) = ((3, 2), (2, 2)) := by decide
+example : boxProdPerms (shearTranslationMove 5) (shearTranslationMove 3) (.inr 1)
+    ((1, 2), (2, 2)) = ((1, 2), (2, 1)) := by decide
+
+/-- The flip of `ZMod 2`. -/
+abbrev flip : Unit → Equiv.Perm (ZMod 2) := fun _ ↦ Equiv.addRight 1
+
+/-- The two-point walk satisfies the Poincaré inequality with constant `4`, with equality. -/
+theorem poincare_flip (f : ZMod 2 → ℝ) (hf : 𝔼 v, f v = 0) :
+    4 * 𝔼 v, f v ^ 2 ≤ ∑ k, 𝔼 v, (f (flip k v) - f v) ^ 2 := by
+  have hsum (g : ZMod 2 → ℝ) : ∑ v, g v = g 0 + g 1 := Fin.sum_univ_two g
+  simp only [Fintype.expect_eq_sum_div_card, ZMod.card, hsum, Fintype.univ_unit, sum_singleton,
+    flip, Equiv.coe_addRight] at hf ⊢
+  have h1 : f 1 = -f 0 := by linarith [(div_eq_zero_iff.1 hf).resolve_right (by norm_num)]
+  have e1 : (1 : ZMod 2) + 1 = 0 := rfl
+  rw [zero_add, e1, h1]
+  nlinarith [sq_nonneg (f 0)]
+
+/-- The square `ZMod 2 × ZMod 2`, the box product of two edges, inherits the constant `4`. -/
+example (f : ZMod 2 × ZMod 2 → ℝ) (hf : 𝔼 p, f p = 0) :
+    4 * 𝔼 p, f p ^ 2 ≤ ∑ k, 𝔼 p, (f (boxProdPerms flip flip k p) - f p) ^ 2 :=
+  mul_expect_sq_le_boxProdPerms poincare_flip poincare_flip f hf
+
+/-- **A uniform gap for a product of two components**: the lazy box product of the
+shear-translation graphs on `ZMod n × ZMod n` and `ZMod m × ZMod m` has spectral expansion at most
+`1 - 1 / 16384`, for all `n` and `m`. -/
+example (n m : ℕ) [NeZero n] [NeZero m] :
+    spectralExpansion
+        (ofPerms (boxProdPerms (shearTranslationMove n) (shearTranslationMove m))).lazy ≤
+      1 - 1 / 16384 := by
+  have h := spectralExpansion_lazy_ofPerms_boxProdPerms_le (shearTranslationMove n)
+    (shearTranslationMove m)
+  have hn := secondEigenvalue_shearTranslation_le n
+  have hm := secondEigenvalue_shearTranslation_le m
+  rw [shearTranslation] at hn hm
+  simp only [Fintype.card_fin, Nat.cast_ofNat] at h
+  have hmin : 4 * (1 / 4096 : ℝ) ≤
+      min (4 * (1 - (ofPerms (shearTranslationMove n)).secondEigenvalue))
+        (4 * (1 - (ofPerms (shearTranslationMove m)).secondEigenvalue)) :=
+    le_min (by linarith) (by linarith)
+  have : 4 * (1 / 4096 : ℝ) / (2 * (4 + 4)) ≤
+      min (4 * (1 - (ofPerms (shearTranslationMove n)).secondEigenvalue))
+        (4 * (1 - (ofPerms (shearTranslationMove m)).secondEigenvalue)) / (2 * (4 + 4)) := by
+    gcongr
+  norm_num at this ⊢
+  linarith
+
+end ComplexityTests.BoxProductExpander