Circular Ensembles

IntegrateUnitary.jl provides support for the three classical Circular Ensembles of Random Matrix Theory:

  • CUE (Circular Unitary Ensemble): Corresponds to the Haar measure on the Unitary group $U(d)$.
  • COE (Circular Orthogonal Ensemble): Ensemble of symmetric unitary matrices ($S^T = S$).
  • CSE (Circular Symplectic Ensemble): Ensemble of self-dual unitary matrices ($S^R \equiv J S^T J^T = S$, where $J$ is the standard symplectic form) of even dimension $d=2N$.

These ensembles are defined on the unitary group, unlike the Gaussian ensembles which are defined on the space of Hermitian matrices.

Measures

IntegrateUnitary.dCUE — Function
dCUE(dim)

Defines the Circular Unitary Ensemble measure for U(d). This is mathematically equivalent to the Haar measure on U(d).

IntegrateUnitary.dCOE — Function
dCOE(dim)

Defines the Circular Orthogonal Ensemble (COE) measure on U(N). Integration engine identifies variables via metadata tag :COE.

IntegrateUnitary.dCSE — Function
dCSE(dim)

Defines the Circular Symplectic Ensemble (CSE) measure on U(2N). Integration engine identifies variables via metadata tag :CSE.

Examples

COE (Circular Orthogonal Ensemble)

For the COE, the matrix $S$ is symmetric unitary. The diagonal entries have different statistical properties than off-diagonal entries.

  1. Basic Integration using @integrate

The @integrate macro automatically identifies S as the random matrix when dCOE is used.

using IntegrateUnitary, Symbolics
@variables d
# COE diagonal 2nd moment E[|S_{1,1}|^2]
@integrate abs(S[1, 1])^2 dCOE(d)
# Output: 2 / (d + 1)

# COE off-diagonal 2nd moment E[|S_{1,2}|^2]
@integrate abs(S[1, 2])^2 dCOE(d)
# Output: 1 / (d + 1)

# COE diagonal 4th moment E[|S_{1,1}|^4]
@integrate abs(S[1, 1])^4 dCOE(d)
# Output: 8 / ((d + 1) * (d + 3))

# COE off-diagonal 4th moment E[|S_{1,2}|^4]
@integrate abs(S[1, 2])^4 dCOE(d)
# Output: 2 / (d * (d + 3))

# COE correlation moment E[|S_{1,1}|^2 |S_{1,2}|^2]
@integrate abs(S[1, 1])^2 * abs(S[1, 2])^2 dCOE(d)
# Output: 2 / ((d + 1) * (d + 3))
  1. Manual Integration
using IntegrateUnitary, Symbolics
@variables d
S = SymbolicMatrix(:S, :COE)
# COE moment E[|S_{1,1}|^2]
integrate(abs(S[1, 1])^2, dCOE(d))
# Output: 2 / (d + 1)

CSE (Circular Symplectic Ensemble)

For the CSE, the matrix $S$ is defined on a space of dimension $2N$ and satisfies $S = J S^T J^T$.

  1. Basic Integration using @integrate
using IntegrateUnitary, Symbolics
@variables d
# CSE diagonal 2nd moment E[|S_{1,1}|^2]
@integrate abs(S[1, 1])^2 dCSE(d)
# Output: 1 / (d - 1)

# CSE diagonal 4th moment E[|S_{1,1}|^4]
@integrate abs(S[1, 1])^4 dCSE(d)
# Output: 2 / (d * (d - 1))
  1. Manual Integration
using IntegrateUnitary, Symbolics
@variables d
S = SymbolicMatrix(:S, :CSE)
# CSE moment E[|S_{1,1}|^2]
integrate(abs(S[1, 1])^2, dCSE(d))
# Output: 1 / (d - 1)

CUE (Circular Unitary Ensemble)

The CUE is statistically identical to the standard Unitary Haar measure.

  1. Basic Integration using @integrate
using IntegrateUnitary, Symbolics
@variables d
# CUE moment E[|U_{1,1}|^2]
@integrate abs(U[1, 1])^2 dCUE(d)
# Output: 1 / d
  1. Manual Integration
using IntegrateUnitary, Symbolics
@variables d
U = SymbolicMatrix(:U, :U)
# CUE moment E[|U_{1,1}|^2]
res = integrate(abs(U[1, 1])^2, dCUE(d))
# Output: 1 / d

Potential Pitfalls

[!IMPORTANT]

Symbolic (d) Pitfalls

  • Small Dimensions: For Haar-related measures (Unitary, Orthogonal, Circular), element-wise results are rational functions with poles at small $d$ (typically $d < n$ for degree $n$ moments). Pure trace moments $|\mathrm{tr}(U)|^{2k}$ are an exception: they depend on $d$ as a step function and require a concrete integer dimension.
  • Removable Singularities: Substituting numeric values can yield $0/0$ forms (e.g., at $d=1$ or $d=2$).
  • Automatic Handling: IntegrateUnitary.jl's evaluate function automatically simplifies expressions to resolve removable singularities when a denominator evaluates to zero.

See Also