Pre-computed Integral Library
IntU.jl includes a library of pre-computed integrals for common random matrix measures. These results are retrieved instantly, avoiding the computational overhead of the Weingarten engine.
Unitary Group $U(d)$
The library provides the following general result for the trace of a unitary conjugation:
\[\int tr(U A U^\dagger B) dU = \frac{tr(A) tr(B)}{d}\]
This is particularly useful when working with Symbolic Trace Logic.
Example
using IntU
@variables d
U = SymbolicMatrix(:U, false, :U)
A = SymbolicMatrix(:A)
B = SymbolicMatrix(:B)
# Retrieved instantly from library
res = integrate(tr(U * A * U' * B), dU(d))
# Result: (tr(A)*tr(B))/dGaussian Ensembles
Low-order moments of Gaussian ensembles are available for both matrix and trace-level expressions.
| Moment | GUE ($d^2$) | GOE ($d^2+d$) | GSE ($d^2-d$) |
|---|---|---|---|
| $\langle Tr(H^2) \rangle$ | $d^2$ | $d^2 + d$ | $d^2 - d$ |
| $\langle Tr(H^4) \rangle$ | $2d^3 + d$ | $2d^3 + 5d^2 + 5d$ | $2d^3 - 5d^2 + 5d$ |
| $\langle Tr(H^6) \rangle$ | $5d^4 + 10d^2$ | - | - |
Example
using IntU
@variables d
H = SymbolicMatrix(:H, :GUE)
# <tr(H^4)>_GUE
res = integrate(tr(H^4), dGUE(d))
# Result: 2d^3 + dFallback Mechanism
If an expression is not found in the library, IntU.jl automatically falls back to full symbolic integration using Weingarten calculus (for Haar) or Wick's theorem (for Gaussian), ensuring that all valid integrals can still be calculated.