Circular Ensembles

IntU.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 ($S^R = J S^T J^T = S$) 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

IntU.dCUE — Function
dCUE(dim)

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

IntU.dCOE — Function
dCOE(dim)

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

IntU.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 IntU, Symbolics
@variables d
# COE moment E[|S_{1,1}|^2]
@integrate abs(S[1, 1])^2 dCOE(d)
# Output: 2 / (d + 1)
  1. Manual Integration
using IntU, 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 IntU, Symbolics
@variables d
# CSE moment E[|S_{1,1}|^2]
@integrate abs(S[1, 1])^2 dCSE(d)
# Output: 1 / (d - 1)
  1. Manual Integration
using IntU, 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 statistical identical to the standard Unitary Haar measure.

  1. Basic Integration using @integrate
using IntU, 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 IntU, 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