**Spectral Equidistribution of Random Monomial Unitaries**

*A.I. Researcher\**

### Abstract

We analyze the spectral properties of random monomial unitary matrices Uₙ of dimension n. These matrices are constructed as the product Uₙ = DₙPₙ where Dₙ is a diagonal matrix with independent and identically distributed (i.i.d.) uniform entries on the unit circle 𝕋, and Pₙ is a permutation matrix chosen uniformly at random from the symmetric group Sₙ, independent of Dₙ. We prove that the empirical spectral measure of Uₙ converges weakly almost surely to the uniform distribution on 𝕋 as n→∞. Additionally, we establish a deterministic bound on the spectral discrepancy, showing it is bounded by the ratio of the number of cycles in the permutation to the dimension n. This implies a convergence rate of O((log n)/n) with high probability.

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### 1 Introduction

The study of limiting spectral distributions (LSD) of large random matrices is a central topic in modern probability theory. While classical ensembles often feature matrices with independent entries, structured random matrices provide insights into the interplay between algebraic structure and spectral behavior. In this paper, we investigate a specific model of structured random unitary matrices, termed random monomial unitaries, as proposed in [3].

**Definition 1.1 (Random Monomial Unitary).** Let n ∈ ℕ. Let Dₙ = diag(e^(iθ₁), ..., e^(iθₙ)) be a random diagonal unitary matrix, where {θⱼ}ⱼ₌₁ⁿ are i.i.d. uniform random variables on [0, 2π). Let Pₙ be the matrix corresponding to a permutation σₙ chosen uniformly at random from the symmetric group Sₙ, independent of Dₙ. We adopt the convention that the permutation matrix Pₙ associated with σₙ has entries (Pₙ)ᵢⱼ = δ\_{σₙ(i),j}. We define the random monomial unitary matrix as

Uₙ = DₙPₙ. (1)

Since Uₙ is unitary, its eigenvalues lie on the unit circle 𝕋 = {z ∈ ℂ : |z| = 1}. We study the distribution of these eigenvalues via the empirical spectral measure.

**Definition 1.2 (Empirical Spectral Measure (ESM)).** Let λ₁, ..., λₙ be the eigenvalues of Uₙ. The empirical spectral measure μₙ of Uₙ is the random probability measure on 𝕋 defined by

μₙ = (1/n) Σⱼ₌₁ⁿ δ\_{λⱼ} (2)

The spectrum of a random permutation matrix Pₙ itself is closely tied to the cycle structure of σₙ and is known not to be uniformly distributed [5]. The introduction of the random phases Dₙ significantly alters this behavior. Our main result confirms the conjecture in [3] that these phases lead to spectral equidistribution.

> "Based on the conjecture formulated in [3]."

**Theorem 1.3.** As n→∞, the empirical spectral measure μₙ converges weakly to the uniform distribution on the unit circle, Unif(𝕋), almost surely.

We also quantify the rate of this convergence by analyzing the discrepancy of the spectrum.

**Definition 1.4 (Discrepancy).** The discrepancy of a probability measure μ on 𝕋 with respect to the normalized Lebesgue measure m = dθ/(2π) is defined as

D(μ) = sup\_{A ⊂ 𝕋, arc} |μ(A) - m(A)| (3)

**Theorem 1.5.** For any realization of Uₙ corresponding to the permutation σₙ, the discrepancy of its empirical spectral measure μₙ is bounded by

D(μₙ) ≤ \#cycles(σₙ)/n (4)

Since the number of cycles in a uniform random permutation is concentrated around its mean, which is asymptotically log n (see [1]), Theorem 1.5 implies that the discrepancy is typically O((log n)/n).

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### 2 Spectral Structure of Monomial Unitaries

We first analyze the spectrum of U = DP for a fixed permutation σ and diagonal matrix D = diag(d₁, ..., dₙ). The spectrum is determined by the cycle structure of σ. Let σ = C₁ ⋅⋅⋅ Cₖ be the decomposition into disjoint cycles, with lengths l₁, ..., lₖ.

**Definition 2.1 (Cycle Phase Product).** The cycle phase product Φⱼ associated with cycle Cⱼ is defined as Φⱼ = Π\_{i∈Cⱼ} dᵢ.

**Proposition 2.2.** The characteristic polynomial of U = DP is given by

χᵤ(λ) = Πⱼ₌₁ᵏ (λˡʲ - Φⱼ). (5)

The spectrum of U is the union of the sets of lⱼ-th roots of Φⱼ for j = 1, ..., k.

*Proof.* We can conjugate U by a permutation matrix to obtain a block-diagonal matrix U' = ⊕ⱼ₌₁ᵏ Uⱼ, where Uⱼ corresponds to the action of U on the subspace spanned by the basis vectors indexed by the elements of Cⱼ. Let Cⱼ = (i₁, i₂, ..., i\_{lⱼ}). We order the basis for this subspace as (e\_{i₁}, ..., e\_{i\_{lⱼ}}). We analyze the action of U on the standard basis vectors eₛ. Recall the definition (P)ᵣₛ = δ\_{σ(r),s}. The entries of U = DP are (U)ᵣₛ = Σₜ DᵣₜPₜₛ = dᵣPᵣₛ = dᵣδ\_{σ(r),s}. The action of U on eₛ is:

Ueₛ = Σᵣ Uᵣₛeᵣ = Σᵣ dᵣδ\_{σ(r),s}eᵣ.

The term in the sum is non-zero if and only if s = σ(r), i.e., r = σ⁻¹(s). Thus,

Ueₛ = d\_{σ⁻¹(s)}e\_{σ⁻¹(s)}.

Applying this to the cycle Cⱼ. Let l = lⱼ:

  * For s = i₁, σ⁻¹(i₁) = iₗ. Ue\_{i₁} = d\_{iₗ}e\_{iₗ}.
  * For k \> 1, s = iₖ, σ⁻¹(iₖ) = iₖ₋₁. Ue\_{iₖ} = d\_{iₖ₋₁}e\_{iₖ₋₁}.

The matrix representation of the block Uⱼ in the ordered basis (e\_{i₁}, ..., e\_{iₗ}) is constructed by placing the coefficients of Ue\_{iₛ} in the s-th column.

\<pre\>
Uⱼ = | 0      d\_{i₁}   0      ...  0      |
| 0      0      d\_{i₂}   ...  0      |
| ⋮      ⋮      ...    ...  ⋮      |
| 0      0      0      ...  d\_{iₗ₋₁} |
| d\_{iₗ}   0      0      ...  0      |
\</pre\>

We compute the characteristic polynomial χ\_{Uⱼ}(λ) = det(λI - Uⱼ).

\<pre\>
λI - Uⱼ = | λ     -d\_{i₁}   0      ...  0      |
| 0      λ     -d\_{i₂}   ...  0      |
| ⋮      ⋮      ...    ...  ⋮      |
| 0      0      0      ... -d\_{iₗ₋₁} |
| -d\_{iₗ}  0      0      ...  λ      |
\</pre\>

We expand the determinant along the first column.

det(λI - Uⱼ) = λ ⋅ C₁₁ + (-d\_{iₗ}) ⋅ C\_{l,1} (6)

where Cᵣ,ₛ denotes the cofactor (-1)ʳ⁺ˢ Mᵣ,ₛ, and Mᵣ,ₛ is the determinant of the minor. The minor M₁₁ is an (l-1)×(l-1) upper triangular matrix with λ along the diagonal. M₁₁ = λˡ⁻¹. The first term in (6) is λ⋅λˡ⁻¹ = λˡ. The minor M\_{l,1} is an (l-1)×(l-1) lower triangular matrix:

\<pre\>
M\_{l,1} = det | -d\_{i₁}    0       ...  0       |
| λ       -d\_{i₂}    ...  0       |
| ⋮       ...     ...  ⋮       |
| 0       ...     λ    -d\_{iₗ₋₁} |
\</pre\>

The determinant is the product of the diagonal entries: M\_{l,1} = Π\_{k=1}^{l-1}(-d\_{iₖ}) = (-1)ˡ⁻¹ Π\_{k=1}^{l-1} d\_{iₖ}. The cofactor C\_{l,1} = (-1)ˡ⁺¹ M\_{l,1}. The second term in (6) is:

(-d\_{iₗ}) ⋅ C\_{l,1} = (-d\_{iₗ}) ⋅ (-1)ˡ⁺¹ ⋅ (-1)ˡ⁻¹ Π\_{k=1}^{l-1} d\_{iₖ}
\= (-d\_{iₗ}) ⋅ (-1)²ˡ Π\_{k=1}^{l-1} d\_{iₖ}
\= -Π\_{k=1}^{l} d\_{iₖ} = -Φⱼ.

Therefore, χ\_{Uⱼ}(λ) = λˡ - Φⱼ. The characteristic polynomial of U is the product of these factors over all cycles.

**Remark 2.3.** In the random model (Definition 1.1), dᵢ = e^(iθᵢ) with θᵢ i.i.d. uniform on [0, 2π). The cycle phase product Φⱼ = exp(i Σ\_{i∈Cⱼ} θᵢ) is also uniformly distributed on 𝕋. This follows because the sum of independent continuous random variables on the circle group, where at least one is uniform, is itself uniform. Since the cycles are disjoint, they involve disjoint sets of θᵢ, ensuring that the {Φⱼ} are independent random variables. The spectrum of Uₙ is thus a union of independent, randomly rotated regular polygons.

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### 3 Proof of the Discrepancy Bound

We now prove Theorem 1.5, utilizing the structure derived in Section 2 and a fundamental result from discrepancy theory regarding equally spaced points.

**Lemma 3.1.** Let Λₗ = {e^(i(φ+2πk/l)) : k=0,...,l-1} be a set of l equally spaced points on 𝕋. Let μₗ be its empirical measure. The discrepancy of μₗ is D(μₗ) = 1/l. Consequently, for any arc A ⊂ 𝕋 with normalized measure m(A),

| |Λₗ ∩ A| - l m(A) | ≤ 1. (7)

*Proof.* We identify 𝕋 with the interval [0,1) via the map x ↦ e^(i2πx). The points correspond to xₖ = (φ/(2π) + k/l) (mod 1). The discrepancy is invariant under rotation, so we may assume φ=0. The points are xₖ = k/l for k=0,...,l-1. Let Fₗ(x) be the empirical cumulative distribution function (CDF), Fₗ(x) = μₗ([0,x)). The uniform CDF is F(x) = x. The starred discrepancy D\*(μₗ) is defined as sup\_{x∈[0,1)} |Fₗ(x) - x|. For xₖ = k/l the empirical CDF is a step function. Fₗ(x) = (k+1)/l for x ∈ [k/l, (k+1)/l). We analyze the deviation Fₗ(x) - x = (k+1)/l - x. On this interval, this function is decreasing. It is maximized as x approaches k/l from the right, giving a supremum of (k+1)/l - k/l = 1/l. We also analyze the deviation x - Fₗ(x⁻), where Fₗ(x⁻) = lim\_{ε→0⁺} Fₗ(x-ε). On the interval (k/l, (k+1)/l], Fₗ(x⁻) = k/l. The deviation is x - k/l. This is maximized as x approaches (k+1)/l from the left, giving a supremum of 1/l. Thus, D\*(μₗ) = 1/l. For a monotonically increasing sequence, the discrepancy D(μₗ) (supremum over all intervals/arcs) equals the starred discrepancy D\*(μₗ) (see [4], Chapter 2, Theorem 1.4). Thus D(μₗ) = 1/l. The inequality follows immediately from the definition of discrepancy: |μₗ(A) - m(A)| ≤ D(μₗ) = 1/l. Multiplying by l yields | |Λₗ ∩ A| - l m(A) | ≤ 1.

*Proof of Theorem 1.5.* Let σₙ have k = \#cycles(σₙ) cycles C₁, ..., Cₖ of lengths l₁, ..., lₖ. Let Λ be the spectrum of Uₙ. By Proposition 2.2, Λ = ∪ⱼ₌₁ᵏ Λⱼ, where Λⱼ is the set of lⱼ-th roots of Φⱼ, forming a regular lⱼ-gon. Let A ⊂ 𝕋 be an arc, and let m(A) be its normalized Lebesgue measure. We estimate the deviation of the ESM μₙ(A) = (1/n)|Λ ∩ A| from m(A).

|nμₙ(A) - nm(A)| = | |Λ ∩ A| - nm(A) |
\= | Σⱼ₌₁ᵏ |Λⱼ ∩ A| - Σⱼ₌₁ᵏ lⱼm(A) | (since Σⱼ₌₁ᵏ lⱼ = n)
≤ Σⱼ₌₁ᵏ | |Λⱼ ∩ A| - lⱼm(A) |.

By Lemma 3.1, each term in the sum is bounded by 1.

|nμₙ(A) - nm(A)| ≤ Σⱼ₌₁ᵏ 1 = k.

Dividing by n, we obtain

|μₙ(A) - m(A)| ≤ k/n = \#cycles(σₙ)/n.

Since this bound is independent of the arc A, it bounds the discrepancy D(μₙ).

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### 4 Proof of Spectral Equidistribution

We prove Theorem 1.3 using the method of moments (Weyl's criterion). We must show that for any fixed integer m ≥ 1, the m-th moment of the ESM converges almost surely to the m-th moment of Unif(𝕋), which is 0. Let X\_{n,m} be the normalized m-th moment:

X\_{n,m} := ∫\_{𝕋} zᵐ dμₙ(z) = (1/n) Tr(Uₙᵐ). (8)

We need to show X\_{n,m} → 0 almost surely as n → ∞. We analyze the expectation and variance of X\_{n,m}. Let E denote the total expectation, Eᴅ the expectation over the random phases Dₙ (conditioned on Pₙ), and Eᴘ the expectation over the random permutation Pₙ.

#### 4.1 Trace Formula

**Lemma 4.1.** Let Fₘ(σ) = {i : σᵐ(i) = i} be the set of fixed points of σᵐ. Then

Tr(Uₙᵐ) = Σ\_{i∈Fₘ(σ)} Π\_{k=0}^{m-1} d\_{σᵏ(i)}. (9)

*Proof.* We analyze the powers of Uₙ = DₙPₙ. Let Dₙ = diag(d₁, ..., dₙ). We prove by induction that ((DP)ᵐ)ᵢⱼ = (Π\_{k=0}^{m-1} d\_{σᵏ(i)}) δ\_{σᵐ(i),j}.

  * **Base case m=1:** (DP)ᵢⱼ = Σₖ DᵢₖPₖⱼ = dᵢPᵢⱼ = dᵢδ\_{σ(i),j}. The formula gives d\_{σ⁰(i)}δ\_{σ(i),j} = dᵢδ\_{σ(i),j}. This holds.
  * **Inductive step:** Assume the formula holds for m.
    ((DP)ᵐ⁺¹)ᵢⱼ = Σₗ ((DP)ᵐ)ᵢₗ(DP)ₗⱼ
    \= Σₗ (Π\_{k=0}^{m-1} d\_{σᵏ(i)}) δ\_{σᵐ(i),l} ⋅ dₗδ\_{σ(l),j}.
    The sum is non-zero only when l = σᵐ(i).
    ((DP)ᵐ⁺¹)ᵢⱼ = (Π\_{k=0}^{m-1} d\_{σᵏ(i)}) d\_{σᵐ(i)} δ\_{σ(σᵐ(i)),j}
    \= (Π\_{k=0}^{m} d\_{σᵏ(i)}) δ\_{σᵐ⁺¹(i),j}.
    The induction holds. The trace is the sum of the diagonal elements:

Tr(Uₙᵐ) = Σ\_{i=1}^{n} ((DP)ᵐ)ᵢᵢ = Σ\_{i=1}^{n} (Π\_{k=0}^{m-1} d\_{σᵏ(i)}) δ\_{σᵐ(i),i}.

The Kronecker delta restricts the sum to the fixed points Fₘ(σ).

#### 4.2 Expectation and Variance

**Lemma 4.2.** For any n ≥ 1 and m ≥ 1, E[X\_{n,m}] = 0.

*Proof.* We compute the expectation over the phases Dₙ for a fixed σ.

Eᴅ[Tr(Uₙᵐ)] = Σ\_{i∈Fₘ(σ)} Eᴅ[Π\_{k=0}^{m-1} d\_{σᵏ(i)}].

Consider a term corresponding to i ∈ Fₘ(σ). Let C be the cycle containing i, and l its length. Since σᵐ(i) = i, l must divide m. Let m = ql. The product of phases along the trajectory of i repeats the cycle product Φᴄ exactly q times:

Π\_{k=0}^{m-1} d\_{σᵏ(i)} = (Π\_{k=0}^{l-1} d\_{σᵏ(i)})^q = (Φᴄ)^q.

As noted in Remark 2.3, Φᴄ is uniformly distributed on 𝕋. Since q ≥ 1,

Eᴅ[(Φᴄ)^q] = ∫\_{𝕋} z^q dUnif(z) = 0.

Thus, Eᴅ[Tr(Uₙᵐ)] = 0. Consequently, E[X\_{n,m}] = Eᴘ[Eᴅ[X\_{n,m}]] = 0.

To estimate the variance, we rely on the statistics of the cycle structure of uniform random permutations.

**Lemma 4.3.** Let Nₗ(σ) be the number of cycles of length l in σ ∈ Sₙ. If σ is chosen uniformly at random, then for 1 ≤ l ≤ n,

Eᴘ[Nₗ(σ)] = 1/l. (10)

*Proof.* Let Cₗ be the set of all possible cycles of length l formed by elements of {1, ..., n}. The number of ways to choose l elements is C(n, l). The number of distinct cycles on these l elements is (l-1)\!. Thus, |Cₗ| = C(n, l)(l-1)\!. Let Iᴄ be the indicator variable that the cycle C ∈ Cₗ appears in the decomposition of σ. Then Nₗ(σ) = Σ\_{C∈Cₗ} Iᴄ. By linearity of expectation, Eᴘ[Nₗ(σ)] = Σ\_{C∈Cₗ} Eᴘ[Iᴄ]. For a fixed cycle C, Eᴘ[Iᴄ] is the probability that σ contains C. This occurs if σ permutes the l elements of C according to C (1 way), and permutes the remaining n-l elements among themselves ((n-l)\! ways). The total number of permutations is n\!. Therefore,

Eᴘ[Iᴄ] = (n-l)\! / n\!.
Eᴘ[Nₗ(σ)] = |Cₗ| ⋅ Eᴘ[Iᴄ] = C(n, l)(l-1)\! (n-l)\! / n\!
\= (n\! / (l\!(n-l)\!)) (l-1)\! ((n-l)\! / n\!) = (l-1)\! / l\! = 1/l.

**Lemma 4.4.** For a fixed m ≥ 1, let τ₁(m) = Σ\_{l|m} l be the sum of the divisors of m. The variance of the normalized trace satisfies

Var(X\_{n,m}) = E[|X\_{n,m}|²] ≤ τ₁(m)/n². (11)

*Proof.* We compute E[|X\_{n,m}|²] = (1/n²)E[|Tr(Uₙᵐ)|²]. We first compute the expectation over phases for a fixed σ. Since Eᴅ[X\_{n,m}] = 0, we compute Eᴅ[|X\_{n,m}|²].

|Tr(Uₙᵐ)|² = Σ\_{i,j∈Fₘ(σ)} (Π\_{k=0}^{m-1} d\_{σᵏ(i)}) (Π\_{l=0}^{m-1} d\_{σˡ(j)})̅.

We take the expectation Eᴅ. The variables {dᵣ} are independent and uniform on 𝕋. The expectation of a product of d's and d̅'s is non-zero if and only if every variable dᵣ appears the same number of times as its conjugate dᵣ̅.
Let Oᵢ and Oⱼ be the multisets of indices visited by the orbits of i and j under σ iterated m times. Eᴅ[⋅] is 1 if Oᵢ = Oⱼ as multisets, and 0 otherwise.
Let Cᵢ and Cⱼ be the cycles containing i and j. If Cᵢ ≠ Cⱼ, the sets of indices are disjoint. Since m ≥ 1, Oᵢ and Oⱼ are non-empty, so Oᵢ ≠ Oⱼ. The expectation is 0.
If Cᵢ = Cⱼ = C. Let l be the length of C. Since i, j ∈ Fₘ(σ), l|m. Let m = ql. The multiset Oᵢ consists of the elements of C, each repeated exactly q times. The multiset Oⱼ is identical. Thus, the expectation is 1. Therefore,

Eᴅ[|Tr(Uₙᵐ)|²] = Σ\_{i,j∈Fₘ(σ)} I(Cᵢ = Cⱼ).

We regroup the sum by cycles {Cᵣ} of σ.

Eᴅ[|Tr(Uₙᵐ)|²] = Σᵣ Σ\_{i∈Cᵣ∩Fₘ(σ), j∈Cᵣ∩Fₘ(σ)} 1.

A cycle Cᵣ of length lᵣ contributes to the sum if and only if lᵣ|m. In this case, Cᵣ ∩ Fₘ(σ) = Cᵣ.

Eᴅ[|Tr(Uₙᵐ)|²] = Σ\_{r: lᵣ|m} |Cᵣ|² = Σ\_{r: lᵣ|m} lᵣ².

Using the notation Nₗ(σ), we have

Eᴅ[|Tr(Uₙᵐ)|²] = Σ\_{l|m} l² Nₗ(σ).

Now we take the expectation over the permutations Eᴘ.

E[|Tr(Uₙᵐ)|²] = Eᴘ[Σ\_{l|m} l² Nₗ(σ)] = Σ\_{l|m, l≤n} l² Eᴘ[Nₗ(σ)].

We apply Lemma 4.3.

E[|Tr(Uₙᵐ)|²] = Σ\_{l|m, l≤n} l² ⋅ (1/l) = Σ\_{l|m, l≤n} l.

This sum is bounded by the sum over all divisors, τ₁(m).

E[|X\_{n,m}|²] = (1/n²) E[|Tr(Uₙᵐ)|²] ≤ τ₁(m)/n². (12)

#### 4.3 Almost Sure Convergence

*Proof of Theorem 1.3.* We have shown that for any fixed m ≥ 1, E[X\_{n,m}] = 0 (Lemma 4.2) and Var(X\_{n,m}) ≤ Cₘ/n² (Lemma 4.4), where Cₘ = τ₁(m). By Chebyshev's inequality, for any ε \> 0,

P(|X\_{n,m}| \> ε) ≤ Var(X\_{n,m})/ε² ≤ Cₘ / (ε²n²).

Since the series Σ\_{n=1}^{∞} 1/n² converges, we have

Σ\_{n=1}^{∞} P(|X\_{n,m}| \> ε) \< ∞.

By the Borel-Cantelli lemma, the probability that |X\_{n,m}| \> ε occurs infinitely often is zero. This implies X\_{n,m} → 0 almost surely as n → ∞. Since this holds for all m ≥ 1 (and similarly for m \< 0 by complex conjugation), by the Weyl criterion for equidistribution on the circle, the empirical spectral measure μₙ converges weakly almost surely to Unif(𝕋).

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### 5 Example

We illustrate the spectral structure and the discrepancy bound with a concrete example from [3].

**Example 5.1.** Let n=7. Let the permutation be σ = (1 3 4 7)(2 6)(5) ∈ S₇. The cycles are C₁=(1 3 4 7) (length l₁=4), C₂=(2 6) (length l₂=2), and C₃=(5) (length l₃=1). Let D₇ = diag(e^(iθ₁), ..., e^(iθ₇)). The cycle phase products are:

  * Φ₁ = e^(i(θ₁+θ₃+θ₄+θ₇))
  * Φ₂ = e^(i(θ₂+θ₆))
  * Φ₃ = e^(iθ₅)

By Proposition 2.2, the characteristic polynomial is

χ\_{U₇}(λ) = (λ⁴ - Φ₁)(λ² - Φ₂)(λ - Φ₃).

The spectrum Λ = Λ₁ ∪ Λ₂ ∪ Λ₃ consists of:

1.  Λ₁: 4th roots of Φ₁. A regular 4-gon.
2.  Λ₂: Square roots of Φ₂. An antipodal pair.
3.  Λ₃: {Φ₃}. A singleton.

The spectrum is a superposition of these three sets, independently and randomly rotated.

**Discrepancy Bound.** The number of cycles is k=3. By Theorem 1.5, D(μ₇) ≤ 3/7. We verify this using the derivation in Section 3. For any arc A, the deviation is bounded by the sum of the deviations of the individual lattices:

| |Λ ∩ A| - 7m(A) | ≤ | |Λ₁ ∩ A| - 4m(A) | + | |Λ₂ ∩ A| - 2m(A) | + | |Λ₃ ∩ A| - 1m(A) |
≤ 1 + 1 + 1 = 3.

**Concrete Realization.** Consider the specific phases provided in [3]: θ₁=0, θ₃=π/2, θ₄=0, θ₇=0 ⇒ Φ₁ = e^(iπ/2). θ₂=π/3, θ₆=π/6 ⇒ Φ₂ = e^(i(π/3+π/6)) = e^(iπ/2). θ₅=2π/5 ⇒ Φ₃ = e^(i2π/5).
The eigenvalues are:

  * Λ₁: Roots of λ⁴ = e^(iπ/2) → {e^(i(π/8 + kπ/2))} for k=0..3.
  * Λ₂: Roots of λ² = e^(iπ/2) → {e^(i(π/4 + kπ))} for k=0..1.
  * Λ₃: {e^(i2π/5)}.

The eigenangles (in degrees) are {22.5°, 112.5°, 202.5°, 292.5°} ∪ {45°, 225°} ∪ {72°}.

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### 6 Conclusion

We have rigorously established the spectral equidistribution of random monomial unitary matrices Uₙ = DₙPₙ as n→∞. The key mechanism driving this result is the randomization introduced by the independent uniform phases in Dₙ, which effectively decouples the spectral contributions of the cycles of the random permutation Pₙ. The analysis of the moments demonstrates that the expected trace powers vanish, and their variances decay sufficiently fast (O(1/n²)) to ensure almost sure convergence to zero. Furthermore, the deterministic discrepancy bound highlights the role of the permutation's cycle structure in controlling the uniformity of the spectrum for finite n, yielding a typical convergence rate of O((log n)/n).

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