How do spectral three-point correlators explain holographic complexity saturation?

A generating function replaces the complexity operator

The paper does not begin by defining a complexity operator directly. Instead it studies the generating function of complexity, an average of the exponential e^{-αC} in the thermofield double state, and recovers the expectation value of complexity by taking the limit α→0. This matters because the generating function can be expanded in the energy eigenbasis, where each term separates into a spectral correlation function of the density of states and a matrix element of the complexity operator between energy eigenstates.

For codimension-one complexity, that spectral factor is a two-point correlation of the density of states. The paper's central structural claim is that moving to codimension-zero regions changes this: the generating function is instead governed by three-point correlation functions of the density of states. In gravitational language, those correlations are interpreted as nonperturbative Euclidean spacetime wormholes, so the spectral expansion links a bulk geometric observable to wormhole contributions.

Codimension-zero complexity reduces to two boundary terms

A codimension-zero observable is attached to a bulk spacetime region bounded by a future hypersurface Σ+ and a past hypersurface Σ−, both anchored on the same boundary time slice. Using Stokes' theorem, the paper represents such an observable as C_Any = C+ + C−, the sum of two codimension-one contributions associated with those two boundaries. Classically the two terms produce the same kind of linear growth as ordinary codimension-one complexity, with growth rates M+ and M− proportional to the ADM mass for ordinary AdS black holes.

This decomposition is what makes the three-point structure unavoidable. A single codimension-one quantity involves one pair of energy labels; a codimension-zero quantity involves two such quantities, and the overlap between the two fixed-complexity states brings in a third energy variable. The paper shows that the resulting probability distribution is a Kirkwood–Dirac quasiprobability rather than a squared overlap, because it contains the overlap between two fixed-complexity states. In the semiclassical limit that overlap collapses onto the diagonal, but wormhole corrections make it nontrivial.

Linear growth comes from a universal pole structure

The paper traces the early-time linear growth to a specific feature of the matrix elements of the complexity operator in the energy basis. For fixed-complexity states, the wavefunction is oscillatory, and its overlap between nearby energies produces a pair of simple poles at E_ij = ±iα̃. Fourier transforming this pole structure against the time-evolution phase yields the linear growth, with the growth rate set by the parameter M that controls the semiclassical growth of the observable.

The paper presents this as a general mechanism rather than a feature of one model. It states that the construction applies to the broad class of holographic observables with this pole structure, including the geodesic length in JT gravity and the chord-number operator in the double-scaled SYK model, the latter also realizing the Krylov spread-complexity operator. In the appendix, the geodesic length in JT gravity is worked out explicitly, and its matrix element is shown to have an infinite sequence of simple poles in the complex energy plane.

Spectral level repulsion cancels the near-diagonal growth

The linear regime cannot persist forever in a finite-dimensional quantum system. The paper identifies the Heisenberg time T_H = 2πℏ divided by the average energy spacing as the scale at which the discreteness of the spectrum becomes important. The disconnected part of the three-point spectral correlator dominates the semiclassical linear growth, while the connected parts encode the nonperturbative corrections that produce the late-time plateau.

The cancellation mechanism is level repulsion. As two energies approach each other, the two-point and three-point correlation functions vanish, R2(Ei,Ej)→0 and R3(Ei,Ej,Ek)→0. This suppresses precisely the near-diagonal contribution responsible for the linear growth, so the growth rate asymptotically vanishes and a plateau forms. The paper emphasizes that level repulsion is a generic feature of quantum chaotic systems, which is why it presents the saturation as a universal mechanism rather than a model-specific accident. The appendix extends the check across the three Dyson classes, where the same short-distance suppression holds even though the detailed time dependence need not match the GUE polynomial.

Wormholes make the fixed-complexity basis overcomplete

The paper offers a second, operator-level route to the same conclusion. At the semiclassical level, fixed-complexity states form a continuously labeled family whose oscillatory wavefunctions reproduce the required pole structure. Once nonperturbative Euclidean wormhole corrections are included, distinct fixed-complexity states acquire nontrivial overlaps, so the continuous family becomes overcomplete and the probability distribution fails to be normalizable.

The remedy is to extract a complete orthonormal basis from that family using the Gram–Schmidt procedure. With respect to this basis, well-defined complexity operators can be constructed for the codimension-one case and, by summing two independently orthogonalized operators, for the codimension-zero case. The resulting expectation value reproduces the same qualitative time evolution: linear growth with rate M+ + M− for times much shorter than T_H, and a plateau for times much longer than T_H. The paper notes that this plateau is the finite-dimensional statement that a finite number of black hole microstates cannot support indefinite growth.

The first moment hides the three-point information

One result deserves separate attention because it is easy to misread. Although the codimension-zero generating function is governed by three-point spectral correlations, the expectation value of the codimension-zero complexity operator depends only on two-point correlations. The paper explains why: the operator is the sum C_Any,O = C+,O + C−,O, so its first moment is the sum of the two individual first moments, and linearity removes the mixed terms.

The three-point information is not lost, however. Higher moments contain mixed products of the two operators, and the mixed second moment involves the overlap between the two orthonormal bases, which couples them and brings back the three-point correlations. The same information reappears at quadratic order in the generating function through the ordered mixed contribution. Readers comparing spectral formulations should therefore distinguish the first moment from higher moments rather than treating the three-point structure as a property of every observable built from the codimension-zero construction.

Universality holds for the growth-to-plateau transition

The paper's universality claim is specific. What is claimed to be universal is the qualitative time evolution, an exponentially long linear regime followed by saturation at the Heisenberg time, together with the mechanism producing it. The detailed time dependence is not claimed to be identical across Dyson classes; the appendix states that the explicit Fourier transforms and plateau coefficients depend on the symmetry class, while the short-distance suppression underlying the cancellation is common to all three.

The paper also marks where its construction stops. The explicit microscopic wavefunction description is generally unavailable in full gravity except for a few low-dimensional examples discussed in the appendix. The switchback effect, another expected diagnostic of holographic complexity, is not computed here; the paper sketches that a single perturbation would be governed by a four-point function and that the crossed amplitude exhibits a similar pole structure, with a detailed derivation deferred to separate work. The plateau is also defined as the absence of persistent linear growth and an asymptotically vanishing growth rate, so subleading time-dependent contributions such as a logarithmic term may remain.

The spectral picture is the paper's real contribution

The most useful way to read this paper is as a reorganization of holographic complexity around spectral statistics. The linear growth is traced to a pole structure in complexity matrix elements; the saturation is traced to level repulsion in the density of states; and the bridge between them is the three-point spectral correlator, interpreted as Euclidean spacetime wormholes. That chain, rather than any single formula, is the paper's distinctive claim.

For a reader deciding what to carry forward, the priority is the distinction between the generating function and its first moment. The generating function is where three-point correlations and wormhole corrections enter, and where the codimension-zero construction differs structurally from the codimension-one case. The first moment reduces to two codimension-one pieces and hides that information. The paper's unified connection among black hole interiors, chaotic spectral statistics, and Euclidean wormholes therefore lives most clearly in the generating function and in higher moments, not in the expectation value of complexity alone.

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