Asymptotic Expansions

For large Hilbert space dimension $d$, exact Weingarten results can be complicated rational functions. IntU.jl provides utilities to expand these results as a Taylor series in $1/d$.

Usage

asymptotic(expr, measure, order=1)
  • expr: The symbolic expression to integrate.
  • measure: The integration measure (Haar, PureState, GinUE, etc.).
  • order: The maximum power of $1/d$ to retain (default 1).

Example

Evaluating the fourth moment of a matrix entry $|U_{11}|^4$:

@variables d
U = SymbolicMatrix(:U, :U, d)
res = asymptotic(abs(U[1,1])^4, dU(d), 4)
# Output: 2/d^2 - 2/d^3 + 2/d^4

[!TIP] When substituting numeric values into asymptotic results using evaluate, IntU.jl automatically handles removable singularities (e.g., $0/0$ forms) by simplifying the expression if a denominator evaluates to zero.

This approximation is useful for checking convergence properties or leading-order behavior in high-dimensional quantum systems.

References

  • Collins, B. (2003). Moments and Cumulants of Polynomial random variables on unitary groups, the Itzykson-Zuber integral and free probability. International Mathematics Research Notices.
  • Puchała, Z., & Miszczak, J. A. (2017). Symbolic integration with respect to the Haar measure on the unitary group. Bulletin of the Polish Academy of Sciences: Technical Sciences.