Wormholes and Averaging over N

Abstract: The gravitational path integral produces an asymptotic expansion in $G_N$, a fact which is puzzling in the case of observables that are expected to fluctuate wildly. Wormholes appear to compute ensemble averages of functions of such observables, though in typical constructions of AdS/CFT, there are no parameters to average over except, in some examples, a single integer $N$. We introduce a procedure that we call ``Mellin averaging'' to define a sort of asymptotic average of a function of $N$. We argue that Mellin averaging over $N$ may suffice to reproduce the apparent randomness seen in wormhole physics, provided that the dual theory admits an analytic continuation in $N$ and the relevant observables fluctuate on superpolynomially small scales in $N$. As a test case, we consider the spectral form factor in the regime where the double cone is believed to dominate the gravitational path integral and compare to a random matrix theory in which $N$ behaves as a continuous variable. We also describe some toy models of analytic continuation in $N$: a qubit model that can be analytically continued in $N$, and an explicit construction of a deterministic function of $N$ that simulates a sequence of independent draws from a Gaussian ensemble.

• The paper introduces Mellin averaging to systematically extract the 1/N asymptotic expansion of highly oscillatory observables in gravitational path integrals.
• It demonstrates that Mellin averaging suppresses erratic wormhole-induced fluctuations, thereby recovering Gaussian statistics and semiclassical corrections.
• The study develops random matrix and analytic toy models that bridge discrete gauge parameters with continuum predictions in quantum gravity.

Mellin Averaging, Wormholes, and the Statistical Interpretation of the Gravitational Path Integral

Introduction: The Wormhole Averaging Dilemma in Quantum Gravity

The paper "Wormholes and Averaging over N" (2605.15180) addresses a key conundrum in the gravitational path integral’s application to holography and AdS/CFT. The gravitational path integral produces an asymptotic expansion in $G_N$, yet wormhole contributions yield statistical averages over quantities which—within string theory and AdS/CFT—are, at first sight, sharply defined rather than fluctuating observables. Historically, wormholes were thought to encode an ensemble average over couplings, but concrete AdS/CFT models often depend only on a discrete label, such as the rank $N$ of the gauge group, which calls into question the statistical underpinnings of semiclassical gravity’s predictions.

The core contribution of this work is the proposal and precise formalization of "Mellin averaging" over the discrete boundary theory parameter $N$ and an analysis of the circumstances under which such a procedure can reconcile the apparent ensemble averaging underpinning wormholes with holographic dualities possessing only discrete parameters.

Mellin Averaging and the Resolution of Statistical Fluctuations

The authors define Mellin averaging as a method of extracting the asymptotic $1/N$ expansion from observables $F(N)$ that are highly oscillatory as functions of $N$—typically exhibiting fluctuations on scales much smaller than any polynomial in $1/N$. This procedure is not a standard average over $N$, but a formal prescription using the analytic properties of the Mellin transform:

$M_F(s) = \int_1^\infty dN\, N^{s-1} F(N)$

The residues of $M_F(s)$ at positive integer $N$0 define the asymptotic expansion coefficients for $N$1, and under certain regularity conditions (analyticity except for isolated poles), the Mellin average recovers exactly all $N$2 corrections that are accessible in a semiclassical gravitational path integral.

This approach only provides an asymptotic expansion, not a definition of a smoothed observable as a function of $N$3. It acts trivially on observables with well-defined $N$4 expansions (remaining invariant under Mellin averaging), but suppresses the erratic, rapidly oscillating parts—those associated with wormhole-induced statistical fluctuations.

(Figure 1)

Figure 1: Schematic depiction of contours in the Mellin transform; the Mellin averaging contour $N$5 encloses the poles at integer $N$6, capturing the asymptotics.

Statistical Independence and Superpolynomial Fluctuations

A core technical result is that Mellin averaging correctly recovers the Gaussian statistics reflected in the late-time behavior of the spectral form factor—so long as the observable $N$7 fluctuates on superpolynomially small scales in $N$8. That is, if as a function of $N$9, $N$0 decorrelates on intervals much smaller than any inverse power of $N$1, then the averaging reproduces the suppression of non-Gaussian corrections observed in the gravitational path integral.

The authors formalize and exemplify this by constructing explicit deterministic models where $N$2 behaves as a sequence of statistically independent random variables in the Mellin sense. For instance, models such as $N$3 with a transcendental $N$4 demonstrate statistical independence without violating determinism due to the structure of the averaging.

Moreover, by constructing generalizations with variable changes and linear combinations (as in Eq. (6.37)), the Mellin average of powers of $N$5 can be tuned to match higher moments of the Gaussian ensemble, including necessary $N$6 corrections.

Spectral Form Factor, Wormholes, and Correlations Across $N$7

A focal observable is the spectral form factor:

$N$8

where $N$9 is a smooth microcanonical window and $1/N$0 is the CFT Hamiltonian at rank $1/N$1. The gravitational path integral, dominated at late times $1/N$2 by the double cone wormhole, predicts Gaussian statistics and factorization properties echoing those of random matrix theory (RMT).

(Figure 2)

Figure 2: Behavior of the microcanonical spectral form factor, illustrating the smooth decay ("slope"), the emergence of the ramp (erratic, linearly growing behavior), and eventual plateau.

Applying Mellin averaging relies crucially on the correlation length between observables at different $1/N$3. The analysis demonstrates, via gravitational and RMT analogues, that for sufficiently late times $1/N$4, the spectral form factors at $1/N$5 and $1/N$6 rapidly decay in correlation, with an effective correlation length in $1/N$7 of $1/N$8. For $1/N$9 superpolynomial in $F(N)$0, the independence necessary for the statistical interpretation becomes exact.

The presence of highly atypical states ("BPS floors") can significantly delay the dip time $F(N)$1, potentially requiring exceedingly large times for the random matrix statistics—and therefore the wormhole saddle—to dominate. The authors carefully analyze the impact of $F(N)$2-BPS spectra in $F(N)$3 SYM on this process, showing that for highly irrational times, the floor is superpolynomial—meaning the wormhole effect, and hence Mellin averaging, is only justified at superpolynomial times.

(Figure 3)

Figure 3: Left: The BPS spectral form factor for $F(N)$4. Right: The value at irrational time $F(N)$5 (golden ratio), numerically verifying the superpolynomial floor scaling.

Random Matrix Theory as an Analogue

The authors develop a precise random matrix model that interpolates smoothly in $F(N)$6, analogizing the CFT Hilbert space to the size of a random matrix and "removing" fractional qubits via projection. Loop equations and free probability theory techniques are employed to extract the correlation structure between matrix observables at different effective $F(N)$7, demonstrating that except for polynomial windows, the connected correlator at different $F(N)$8 decays exponentially in the entropy difference, matching the AdS/CFT double cone predictions.

The match with random matrix universality is further confirmed with explicit expansions in the Chebyshev polynomial basis, showing the robustness of the Mellin averaging prescription.

Toy Models of Analytic Continuation in $F(N)$9

A prominent challenge is the analytic continuation from discrete $N$0 to the continuum thermodynamic parameter required for Mellin averaging. The paper constructs toy models allowing for continuous $N$1 by embedding the finite $N$2 "black hole" degrees of freedom as a subchain of an infinite system, while the rest is interpreted as environment or "outside the black hole." Hamiltonians are deformed smoothly so that increasing $N$3 adds degrees of freedom to the system, with the rest remaining fixed—a concrete realization of analytic continuation in the context of quantum many-body theory.

Implications and Outlook

The formalism and models advanced in this work have several significant implications:

• Emergent Probability and Factorization: The connection between superpolynomial oscillations in $N$4 and the emergence of statistical independence via Mellin averaging demonstrates how deterministic holographic duals can nonetheless reproduce ensemble averages consistent with wormhole physics.
• Constraints on Nonperturbative Effects: Mellin averaging yields only the asymptotic $N$5 expansion; it cannot capture nonperturbatively small effects (such as multi-boundary wormhole corrections) which would require genuine ensemble interpretation or an extension of the analytic framework (as discussed with Laplace-Mellin generalizations).
• Nontrivial Impact of Atypical States: The analysis reveals that rare spectral features (such as BPS sectors) can obstruct the regime of validity for semiclassical wormhole dominance in a nontrivial manner, with potential implications for the universality of the statistical interpretation in holography.
• Blueprint for Analytic Continuation in Quantum Systems: The analytic toy models underscore how, at least in principle, discrete quantum systems can be embedded in continuous parameter families, opening directions for further formalization of non-integer $N$6 descriptions in gauge/gravity duality.

Conclusion

"Wormholes and Averaging over N" formalizes Mellin averaging as a candidate procedure for recovering the statistical output of the gravitational path integral in holographic duals depending on discrete labels such as $N$7. Under analytic continuation and in the presence of observables fluctuating on nonperturbatively small scales in $N$8, Mellin averaging can reproduce all power-law corrections expected from semiclassical wormholes, provided suitable limits are taken.

The framework delineated places constraints on the circumstances under which semiclassical ensemble statistics genuinely emerge in holography and proposes concrete analytic approaches to situations previously opaque due to the discreteness of microscopic dual labels. Further investigation is warranted into the extension to nonperturbative saddle contributions and the physical realization of analytic continuation in quantum gravity.