Asymptotic Expansions

For large Hilbert space dimension $d$, exact Weingarten results are often complicated rational functions of $d$. IntegrateUnitary.jl provides utilities to expand these into a Laurent series in $1/d$, extracting the physically relevant scaling behaviour.

API

asymptotic(expr, measure, order=1)   # integrate then expand
asymptotic(expr, d, order=1)         # expand a rational function directly

Arguments

The first form computes the exact symbolic integral first, then performs the Laurent expansion. The second form expands any rational expression directly, skipping integration.

[!NOTE] order counts the highest negative power of $d$ retained. For example, order=3 keeps terms up to and including $1/d^3$. For most physics applications order=2 or order=3 is sufficient.

Unitary group examples

Leading-order trace-polynomial moment

For trace-polynomial moments that are rational in $d$, asymptotic(expr, measure, order) returns the large-$d$ scaling directly:

using IntegrateUnitary, Symbolics
@variables d
U = symbolic_unitary(:U, d)
A = SymbolicMatrix(:A)
B = SymbolicMatrix(:B)

asymptotic(tr(U * A * U' * B), dU(d), 2)
# Output: tr(A)*tr(B)/d

Pure-trace moments $|\mathrm{tr}(U)|^{2k}$ are a special case: their exact value is step-function in $d$, so they should be evaluated at concrete dimension with integrate.

Entry-wise high-degree moment

The 10th moment of a single entry has exact form $120 / (d(d+1)(d+2)(d+3)(d+4))$. The asymptotic expansion reveals the leading scaling immediately:

using IntegrateUnitary, Symbolics
@variables d
Ud = SymbolicMatrix(:Ud, :U, d)

asymptotic(abs(Ud[1,1])^10, dU(d), 6)
# Output: 120/d^5 - 1200/d^6 + ...

The leading $1/d^5$ matches the expectation from the stable-rank estimate.

Gaussian ensemble examples

For GUE the second trace moment $\langle \mathrm{tr}(H^2) \rangle = d^2$ grows quadratically, while $\langle \mathrm{tr}(H)^4 \rangle = 3d^2$ matches the fourth moment of a Gaussian (three Wick pairings):

using IntegrateUnitary, Symbolics
@variables d
H = SymbolicMatrix(:H, :GUE, d)

# Leading large-d behaviour of the GUE 4th trace moment
asymptotic(abs(tr(H))^4, dGUE(d), 2)
# Output: 3d^2

Orthogonal group example

using IntegrateUnitary, Symbolics
@variables d
O = SymbolicMatrix(:O, :O, d)

# E[O_{11}^4] = 3/(d*(d+2)); leading large-d behaviour:
asymptotic(O[1,1]^4, dO(d), 3)
# Output: 3/d^2 - 6/d^3 + ...

Expanding a rational function directly

The second form of asymptotic accepts any rational expression and expands it without performing any integration. This is useful when you already have an exact result (e.g., from the integral library or a manual calculation):

using Symbolics
@variables d

# Page's purity formula: 2n/(n^2+1) for equal subsystems of size n
page = 2d / (d^2 + 1)
asymptotic(page, d, 5)
# Output: 2/d - 2/d^3 + 2/d^5
# Leading 2/d: purity is O(1/d); maximally mixed would be exactly 1/d

# Exact Weingarten result 2/(d*(d+1)) expanded to four terms:
asymptotic(2 / (d * (d + 1)), d, 4)
# Output: 2/d^2 - 2/d^3 + 2/d^4 - 2/d^5

[!IMPORTANT]

Symbolic (d) Pitfalls

• Poles at small dimensions: For Haar-related measures (Unitary, Orthogonal, Circular), element-wise results are rational functions with poles at small integer $d$ (typically $d < n$ for degree-$n$ moments).
• Non-rational results: Pure trace moments $|\mathrm{tr}(U)|^{2k}$ depend on $d$ as a step function and require a concrete integer dimension. Prefer integrate(abs(tr(U))^(2k), dU(n)) over asymptotic(expr, measure) for these cases.
• Removable singularities: substituting numeric values can yield $0/0$ forms (e.g. at $d = 1$ or $d = 2$). Use evaluate to resolve these automatically.

See Also

• Integral Library — O(1) retrieval for common results
• Unitary Integration — exact Weingarten engine
• Gaussian Ensembles — GUE/GOE/GSE moments
• QI Helpers — combining asymptotic with partial_trace

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.
• Page, D. N. (1993). Average entropy of a subsystem. Physical Review Letters, 71(9), 1291–1294.

See the API Reference for the full asymptotic signature.