Gaussian Ensembles Integration

This section details the integration of polynomial functions over Gaussian Random Matrix Ensembles: the Gaussian Unitary Ensemble (GUE), the Gaussian Orthogonal Ensemble (GOE), and the Gaussian Symplectic Ensemble (GSE).

Theory: Wick's Theorem

Unlike Haar measure integration which requires Weingarten calculus, Gaussian integration relies on Wick's Theorem (or Isserlis' Theorem). The integral of a product of centered Gaussian random variables is given by the sum over all possible pair contractions (matchings).

GUE (Gaussian Unitary Ensemble)

\[H\]

is a complex Hermitian matrix ($H = H^\dagger$). The entries are independent complex Gaussian variables (subject to Hermiticity). The contraction rule is:

\[\langle H_{ij} \bar{H}_{kl} \rangle_{GUE} = \delta_{il} \delta_{jk}\]

This effectively "connects" the indices in a specific way corresponding to the unitary symmetry.

GOE (Gaussian Orthogonal Ensemble)

\[H\]

is a real symmetric matrix ($H = H^T$). The entries are real Gaussian variables. The contraction rule includes an extra term due to symmetry ($H_{kl} = H_{lk}$):

\[\langle H_{ij} H_{kl} \rangle_{GOE} = \delta_{ik} \delta_{jl} + \delta_{il} \delta_{jk}\]

GSE (Gaussian Symplectic Ensemble)

\[H\]

is a Hermitian quaternionic self-dual matrix ($d=2n$). The integrals relate to GOE via specific duality relations (often mapping $d \to -d$ or $d \to 2d$). The contraction rule involves the symplectic form $J$:

\[\langle H_{ij} H_{kl} \rangle_{GSE} = \delta_{il} \delta_{jk} + (J)_{ik} (J)_{jl}\]

IntegrateUnitary.jl implements GSE integration by mapping it to contractions involving the definition of the symplectic metric. For concrete integer dimensions, dGSE(n) requires even n; odd n throws an ArgumentError.

Ginibre Ensembles

Ginibre ensembles consist of non-Hermitian matrices where each entry is an independent Gaussian random variable.

GinUE (Complex Ginibre Ensemble)

Matrices $G$ with i.i.d. complex Gaussian entries. The contraction rule is:

\[\langle G_{ij} \bar{G}_{kl} \rangle_{GinUE} = \delta_{ik} \delta_{jl}\]

Note that only contractions between $G$ and its complex conjugate $\bar{G}$ are non-zero.

GinOE (Real Ginibre Ensemble)

Matrices $G$ with i.i.d. real Gaussian entries. The contraction rule is:

\[\langle G_{ij} G_{kl} \rangle_{GinOE} = \delta_{ik} \delta_{jl}\]

GinSE (Symplectic Ginibre Ensemble)

Matrices $G$ with i.i.d. quaternionic Gaussian entries. Integrals are computed using duality relations. For concrete integer dimensions, dGinSE(n) requires even n; odd n throws an ArgumentError.

Usage

You can define the Gaussian measures using dGUE, dGOE, dGSE, dGinUE, dGinOE, and dGinSE. For concrete integer dimensions, dGSE(n) and dGinSE(n) require even n.

GUE Example

using IntegrateUnitary, Symbolics

# GUE Measure with symbolic dimension
# Average Trace of H^2
# < tr(H^2) > = d^2
res = @integrate tr(H^2) dGUE(d)
println(res)
# Output: d^2

# Average Trace of H^4
# < tr(H^4) > = 2d^3 + d
res4 = @integrate tr(H^4) dGUE(d)

# Average Trace of H^6
# < tr(H^6) > = 5d^4 + 10d^2
res6 = @integrate tr(H^6) dGUE(d)

GinUE Example

# GinUE Measure with symbolic dimension
# Average Trace of G G'
# < tr(G G') > = d^2
res_ginue = @integrate tr(G * G') dGinUE(d)
println(res_ginue)
# Output: d^2

# Average Trace of (G G')^2
# < tr(G G' G G') > = 2d^3
res_ginue_sq = @integrate tr(G * G' * G * G') dGinUE(d)

# Wishart-style moments
# < tr(G G')^2 > = d^4 + d^2
x2 = @integrate tr(G * G')^2 dGinUE(d)

# < tr((G G')^2) > = 2d^3
y2 = @integrate tr((G * G')^2) dGinUE(d)

GOE Example

# GOE Measure
# Average Trace of H^2
# < tr(H^2) > = d^2 + d
res_goe = @integrate tr(H^2) dGOE(d)
println(res_goe)
# Output: d^2 + d

GSE Example

# GSE Measure
# Average Trace of H^2
# < tr(H^2) > = d^2 - d
res_gse = @integrate tr(H^2) dGSE(d)
println(res_gse)
# Output: d^2 - d

Scaling Conventions

IntegrateUnitary.jl computes the raw Gaussian moments (combinatorial counts). This corresponds to the normalization where the variance of off-diagonal entries is 1.

  • Standard Physics Normalization (Wigner's semicircle law radius 2): Requires scaling $H \to H/\sqrt{d}$. $\langle \text{Tr}(H^2) \rangle \to d$.
  • IntegrateUnitary.jl Normalization: $\langle \text{Tr}(H^2) \rangle_{GUE} = d^2$.

Implementation Details

IntegrateUnitary.jl automates the following steps:

  1. Index Collection: Parses the expression to find all occurrences of $H$ (or $G$).
  2. Pair Partitioning: Generates all ways to pair up the matrix factors.
  3. Contraction: For each pair, applies the specific ensemble contraction rule.
  4. Summation: Sums the contributions.

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.

References

  1. Mehta, M. L. (2004). Random Matrices. Elsevier.
  2. Livan, G., Novaes, M., & Vivo, P. (2018). Introduction to Random Matrices: Theory and Practice. Springer.
  3. Wick, G. C. (1950). The evaluation of the collision matrix. Physical Review, 80(2), 268.
  4. Ginibre, J. (1965). Statistical ensembles of complex, real, and quaternionic matrices. Journal of Mathematical Physics, 6(3), 440-449.

Pre-computed Moments

For common moments like $\langle \text{Tr}(H^2) \rangle$, $\langle \text{Tr}(H^4) \rangle$, and $\langle \text{Tr}(H^6) \rangle$, IntegrateUnitary.jl uses a Pre-computed Integral Library to provide results instantly.

[!TIP] Cached paths include:

  • low-order trace moments for GUE/GOE/GSE,
  • Gaussian element-wise second moments,
  • GinUE low-order trace moments (tr(GG^\dagger), tr((GG^\dagger)^2), tr(GG^\dagger)^2),
  • second trace moments for GinOE/GinSE.

These are returned in $\mathcal{O}(1)$ by the Integral Library.

See Also