Update documentation for Ginibre ensembles

Author: lpawela

README.md

@@ -104,6 +104,21 @@ integrate(abs(S_mat[1,1])^2, dSp(S_mat, d))
 # Output: 1 / d
 ```
 
+### Ginibre Ensembles
+Ginibre ensembles consist of non-Hermitian matrices with i.i.d. Gaussian entries. 
+- **GinUE (Complex Ginibre Ensemble)**: i.i.d. complex Gaussian entries. Use `dGinUE`.
+- **GinOE (Real Ginibre Ensemble)**: i.i.d. real Gaussian entries. Use `dGinOE`.
+- **GinSE (Symplectic Ginibre Ensemble)**: i.i.d. quaternionic Gaussian entries. Use `dGinSE`.
+
+```julia
+@variables d
+# Use T=Complex{Num} for GinUE to ensure conj(G) != G
+G = [Symbolics.variable(:G, i, j, T=Complex{Num}) for i = 1:2, j = 1:2]
+# E[Tr(G G')] = d^2 = 4 (for 2x2 matrix)
+integrate(tr(G * G'), dGinUE(G, d))
+# Output: 4
+```
+
 ### Circular Ensembles
 IntU also supports Circular Ensembles (CUE, COE, CSE) which are commonly used in random matrix theory.
 - **CUE (Circular Unitary Ensemble)**: Equivalent to the Haar measure on $U(d)$. Use `dCUE`.

docs/src/api.md

@@ -29,6 +29,9 @@ dSp
 dGUE
 dGOE
 dGSE
+dGinUE
+dGinOE
+dGinSE
 ```
 
 ### Pure States
@@ -76,6 +79,8 @@ LazyTrace
 LazySum
 IntU.integrate_indices_gue
 IntU.integrate_indices_goe
+IntU.integrate_indices_ginue
+IntU.integrate_indices_gin_oe
 IntU.integrate_indices_orthogonal
 IntU.integrate_indices_symplectic
 IntU.integrate_indices_coe

docs/src/asymptotic.md

@@ -11,7 +11,7 @@ asymptotic(expr, measure, order=1)
 ```
 
 - **expr**: The symbolic expression to integrate.
-- **measure**: The integration measure (Haar, PureState, etc.).
+- **measure**: The integration measure (Haar, PureState, GinUE, etc.).
 - **order**: The maximum power of $1/d$ to retain (default 1).
 
 ## Example

docs/src/gaussian_integration.md

@@ -37,9 +37,32 @@ The contraction rule involves the symplectic form $J$:
 ```
 IntU.jl implements GSE integration by mapping it to contractions involving the definition of the symplectic metric.
 
+## 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:
+```math
+\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:
+```math
+\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.
+
 ## Usage
 
-You can define the Gaussian measures using `dGUE`, `dGOE`, and `dGSE`.
+You can define the Gaussian measures using `dGUE`, `dGOE`, `dGSE`, `dGinUE`, `dGinOE`, and `dGinSE`.
 
 ### GUE Example
 
@@ -59,6 +82,20 @@ println(res)
 # Output: d^2
 ```
 
+### GinUE Example
+
+```julia
+# GinUE Measure with symbolic dimension
+G = SymbolicMatrix(:G)
+measure_GinUE = dGinUE(G, d)
+
+# Average Trace of G G'
+# < Tr(G G') > = d^2
+res_ginue = integrate(IntU.tr(G * G'), measure_GinUE)
+println(res_ginue)
+# Output: d^2
+```
+
 ### GOE Example
 
 ```julia
@@ -98,16 +135,17 @@ This corresponds to the normalization where the variance of off-diagonal entries
 ## Implementation Details
 
 IntU.jl automates the following steps:
-1.  **Index Collection**: Parses the expression to find all occurrences of $H$.
-2.  **Pair Partitioning**: Generates all ways to pair up the $H$ factors ($\sim (2k-1)!!$ terms).
-3.  **Contraction**: For each pair, applies the specific ensemble contraction rule (GUE, GOE, or GSE).
+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.
 
 ## 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