Factorization, Ensembles, and the Gravitational Path Integral

For two decoupled copies of one specified boundary theory, the partition function is exactly a product. A gravitational prescription with a nonzero two-boundary cumulant must therefore be computing a different object—such as a same-draw ensemble moment or an unconditioned baby-universe expectation—or it must be missing contributions that restore the fixed-theory identity. The safest diagnostic is simple: state what is held fixed, condition on it, and recompute the complete two-boundary quantity.

Required background. Euclidean Gravitational Path Integrals, Topology Sums, and Boundary Conditions defines the multi-boundary amplitude and its contour. Fixed-Theory, Ensemble, and Superselection Claims fixes the meanings of a fixed theory, an average, and a sector.

Helpful background. Why Continuum QFT Does Not Factorize Naively treats the different problem of assigning Hilbert-space factors to spatial subregions. Non-Unique JT Matrix-Integral Completion: Fixed-Theory, Disorder-Average, and Ensemble Distinctions supplies the main low-dimensional laboratory.

Exact factorization for two decoupled copies

Let $T$ be a fixed microscopic theory, and let $B_i$ collect the geometry, sources, spin structure, thermal circumference, and other boundary data for copy $i$. The factorization question has a definite answer when four assumptions hold:

1. Both copies use the same specified theory $T$, rather than a distribution over theories.
2. Their actions, measures, regulators, and gauge constraints contain no cross-coupling or shared integration variable.
3. The boundary condition and state preparation are products across the two copies.
4. No average, coarse-graining, or trace over an additional common sector is taken afterward.

The Euclidean functional integral then separates:

$$\begin{aligned} S_{T\otimes T}[\phi_1,\phi_2;B_1,B_2] &=S_T[\phi_1;B_1]+S_T[\phi_2;B_2],\\ Z_{T\otimes T}[B_1\sqcup B_2] &=Z_T[B_1]Z_T[B_2]. \end{aligned}$$

For thermal circles, the same statement follows directly from the tensor-product trace:

$$\begin{aligned} H_{12}&=H_T\otimes 1+1\otimes H_T,\\ Z_{T\otimes T}(\beta_1,\beta_2) &=\operatorname{Tr}_{\mathcal H_T\otimes\mathcal H_T} e^{-\beta_1H_T\otimes1-\beta_2 1\otimes H_T}\\ &=Z_T(\beta_1)Z_T(\beta_2). \end{aligned}$$

With vacuum bubbles treated consistently—for example by using normalized generating functionals—the logarithm is additive:

$$W_T[J_1,J_2]\equiv\log Z_T[J_1,J_2] =W_T[J_1]+W_T[J_2].$$

Consequently every mixed connected response between the two copies vanishes,

$$\frac{\delta^2W_T}{\delta J_1(x)\,\delta J_2(y)}=0.$$

“The same theory on both copies” does not mean that a random Hamiltonian is drawn twice. It means that one already-fixed Hamiltonian governs each decoupled copy. Conversely, an entangled preparation, an explicit double-trace interaction, or a shared constraint can correlate the copies, but then one of the assumptions above has been changed. This exact-copy statement is also independent of the type-III and edge-mode subtleties that obstruct a naive tensor factorization of adjacent spatial regions.

The gravitational cumulant is an algebraic object first

Write the gravitational prescription with boundary data $B$ as $\mathcal Z_{\mathrm{grav}}[B]$. If its disconnected sector inherits component-factorizing actions, measures, contours, and vacuum normalization, then

$$\mathcal Z_{\mathrm{grav}}[B_1\sqcup B_2] =\mathcal Z_{\mathrm{grav}}[B_1]\mathcal Z_{\mathrm{grav}}[B_2] +\mathcal C_{\mathrm{grav}}[B_1,B_2],$$

where

$$\mathcal C_{\mathrm{grav}}[B_1,B_2] \equiv \mathcal Z_{\mathrm{grav}}[B_1\sqcup B_2] -\mathcal Z_{\mathrm{grav}}[B_1]\mathcal Z_{\mathrm{grav}}[B_2]$$

is the two-boundary cumulant of that prescription. It becomes a probability covariance only after a normalized averaging measure has been supplied.

A connected topology and a nonzero cumulant are not synonymous. A connected saddle must lie on the declared integration cycle, carry its fluctuation and zero-mode factors, and survive cancellation against every same-order contribution. A shared normalization or constraint can also correlate disconnected components. The detailed topology-versus-cumulant analysis establishes these distinctions; here $\mathcal C_{\mathrm{grav}}$ always denotes the complete answer at the stated level of approximation, not one attractive geometry.

If the intended dual is one fixed theory $T$ under the four assumptions above, the exact target is

$$\mathcal C_{\mathrm{grav}}[B_1,B_2]=0.$$

A nonzero semiclassical value is therefore a factorization problem. It does not by itself select an ensemble interpretation.

Same-draw ensembles reproduce connected terms

Now let $d\mu(T)$ be a normalized probability measure over theories or Hamiltonians. An overbar will mean that both boundary observables use the same draw $T$:

$$\overline{Z_1Z_2} \equiv \int d\mu(T)\,Z_T[B_1]Z_T[B_2], \qquad \overline{Z_i} \equiv \int d\mu(T)\,Z_T[B_i].$$

The connected ensemble moment is

$$\kappa_2^{\mu}(B_1,B_2) =\overline{Z_1Z_2}-\overline{Z_1}\,\overline{Z_2}.$$

For real thermal partition functions this is the usual covariance and may be nonzero. For complex sources or complex inverse temperatures it is a connected moment without complex conjugation and need not be positive.

The origin of the connection is transparent in a two-member ensemble. If $T=a,b$ with equal probabilities, then

$$\kappa_2^{\mu}(B_1,B_2) =\frac14 \bigl(Z_a[B_1]-Z_b[B_1]\bigr) \bigl(Z_a[B_2]-Z_b[B_2]\bigr).$$

No member violates factorization: for either fixed label $r=a,b$, the two-copy answer is $Z_r[B_1]Z_r[B_2]$. The covariance appears because the same hidden label is used on both boundaries and is forgotten afterward.

Drawing a theory independently for each boundary gives a different observable:

$$\begin{aligned} &\int d\mu(T_1)d\mu(T_2)\, Z_{T_1}[B_1]Z_{T_2}[B_2]\\ &\qquad =\overline{Z_1}\,\overline{Z_2}. \end{aligned}$$

Thus a proposed dictionary must say same draw or independent draws. The word “ensemble” alone is insufficient.

Object being computed	Two-boundary quantity	Connected part	What factorizes?
Two decoupled copies of fixed $T$	$Z_T[B_1]Z_T[B_2]$	$0$	The exact fixed-theory observable.
Same draw from $d\mu(T)$	$\overline{Z_1Z_2}$	$\kappa_2^\mu$	Every member factorizes; the averaged moment need not.
Independent draws from $d\mu(T)$	$\overline{Z_1}\,\overline{Z_2}$	$0$	The averaging operation factorizes too.
Unconditioned sector mixture	$\int d\alpha\,p(\alpha)G_\alpha[B_1,B_2]$	Between-sector covariance plus any within-sector connection	Undetermined until the sector is resolved.
Finite sector window $A$	Conditional average over $\alpha\in A$	Residual connection plus covariance within $A$	Approximate unless the window has zero variance and no residual connection.
One sharp sector $\alpha$	$G_\alpha[B_1,B_2]$	$C_\alpha[B_1,B_2]$	Exact only if the residual $C_\alpha$ vanishes for the tested algebra.

Matching $\mathcal C_{\mathrm{grav}}$ to $\kappa_2^\mu$ is a valid interpretation only when an independent construction identifies the measure, normalization, observables, and shared draw. One matching two-point function does not uniquely reconstruct a probability measure or its higher moments.

The JT double trumpet is a controlled benchmark

Orientable pure JT gravity supplies the cleanest calculation. Let $C>0$ be the Schwarzian boundary coupling and take $\operatorname{Re}\beta_i>0$. With the trumpet normalization used on the connected-boundary page,

$$Z_{\mathrm T}(\beta,b) =\sqrt{\frac{C}{2\pi\beta}} \exp\!\left(-\frac{Cb^2}{2\beta}\right).$$

The genus-zero connected two-boundary coefficient is therefore

$$\widehat Z_{0,2}(\beta_1,\beta_2) =\int_0^\infty b\,db\, Z_{\mathrm T}(\beta_1,b)Z_{\mathrm T}(\beta_2,b) =\frac{\sqrt{\beta_1\beta_2}} {2\pi(\beta_1+\beta_2)}.$$

This is the double trumpet. It is a fixed-topology moduli integral, not a full stationary classical JT solution with respect to the gluing modulus Saad, Shenker, and Stanford 2019, §3.4.2. More generally, the formal perturbative dictionary is

$$\left\langle\prod_{a=1}^{n}Z(\beta_a)\right\rangle_{c,\mathrm{pert}} \sim \sum_{g\geq0}e^{(2-2g-n)S_0} \widehat Z_{g,n}(\beta_1,\ldots,\beta_n).$$

For two boundaries this becomes

$$\big\langle Z(\beta_1)Z(\beta_2)\big\rangle_{c,\mathrm{pert}} \sim \sum_{g\geq0}e^{-2gS_0} \widehat Z_{g,2}(\beta_1,\beta_2).$$

The two disconnected disks scale as $e^{2S_0}$, whereas the double trumpet scales as $e^0$. Its ratio to the leading product is therefore of order $e^{-2S_0}$ at fixed $\beta_i$ in the semiclassical regime. Small relative size is not exact factorization:

$$\frac{\mathcal C_{\mathrm{grav}}} {\mathcal Z_{\mathrm{grav}}[B_1]\mathcal Z_{\mathrm{grav}}[B_2]} \longrightarrow0 \quad\not\Rightarrow\quad \mathcal C_{\mathrm{grav}}=0.$$

Saad, Shenker, and Stanford showed that the JT amplitudes with arbitrary numbers of boundaries reproduce the connected correlators of a particular double-scaled matrix integral order by order in the genus expansion. In that independently constructed dictionary, the double trumpet is precisely a same-matrix ensemble connected moment Saad, Shenker, and Stanford 2019, §§1 and 3.4.1, Eqs. (133)–(138). The same paper stresses that for a non-disordered theory $Z(\beta)$ is a fixed function, so the corresponding ensemble connected correlator has no immediate fixed-theory meaning Saad, Shenker, and Stanford 2019, §6.2, printed p. 59, PDF.

This example establishes an ensemble interpretation for the specified JT matrix-integral completion. It does not establish that every gravitational wormhole computes an ensemble average, and the perturbative genus series does not select a unique nonperturbative matrix completion.

Conditioning on an α sector

A baby-universe state can produce the same shared-label algebra. Let $p_\Psi(\alpha)$ be the normalized distribution over labels seen by the relevant commuting boundary operators. To avoid assuming the conclusion, allow a residual conditioned connection:

$$G_\alpha[B_1,B_2] =Z_\alpha[B_1]Z_\alpha[B_2] +C_\alpha[B_1,B_2].$$

The unconditioned connected part is then

$$\begin{aligned} C_\Psi[B_1,B_2] &=\int d\alpha\,p_\Psi(\alpha) C_\alpha[B_1,B_2]\\ &\quad+\operatorname{Cov}_{p_\Psi} \bigl(Z_\alpha[B_1],Z_\alpha[B_2]\bigr). \end{aligned}$$

This is the law of total covariance in the present notation. Conditioning on a sharp $\alpha$ removes the second, between-sector term. It removes the full connection only if

$$C_\alpha[B_1,B_2]=0$$

for the observable algebra being tested.

In the Marolf–Maxfield construction, the boundary-creating operators $\widehat{Z[J]}$ commute and admit simultaneous eigenstates $|\alpha\rangle$. Their expectation values factorize for that commuting algebra, while the Hartle–Hawking state gives a classical distribution over the eigenvalues Marolf and Maxfield 2020, §2.3, Eqs. (17)–(22). That is a concrete realization of $C_\alpha=0$ in its stated setting. Extending the conclusion requires showing that the proposed sector exists in the theory of interest, that the relevant asymptotic observables preserve it, and that no additional operator or preparation detects a residual connection. The adjacent baby-universe and α-parameter page develops those requirements.

Crucially, fixed-sector factorization constrains the sum of contributions. It does not require every connected geometry to disappear. In the Marolf–Maxfield and Saad–Shenker–Yao models, connected configurations can remain in intermediate expansions and cancel or reorganize against other many-universe contributions Saad, Shenker, and Yao 2024, §§1.1–1.2 and 3.4.

Four interpretations and their burdens of proof

A nonzero semiclassical $\mathcal C_{\mathrm{grav}}$ leaves several logically distinct possibilities. More than one description can coexist after a full dictionary is constructed—for example, an unconditioned state can define an ensemble over sectors—but one connected saddle cannot establish any of them by itself.

Interpretation	What the path integral computes	Evidence needed	What would defeat it?
Ensemble moment	A same-draw moment under a specified $d\mu(T)$	A normalized measure and a dictionary reproducing the relevant hierarchy of moments, positivity conditions, and observables	Conditioning on one member leaves the same unexplained connection, or higher moments fail.
Unconditioned superselection mixture	An expectation over sectors in a state $\Psi$	A sector Hilbert space, commuting observable algebra, preparation rule, and proof that allowed operations preserve $\alpha$	A sector-changing observable or nonzero $C_\alpha$ for the claimed fixed-sector quantity.
Exact effective object	A reduced or nonlocal effective description after other degrees of freedom were integrated out	A derivation from a complete theory identifying the traced sector and induced multi-copy terms	The claimed observable is asserted to be the product partition function of the complete decoupled theory.
Incomplete fixed-theory expansion	Only part of the exact fixed-theory answer	Explicit additional contributions with the right magnitude, phase, and source dependence to make the complete cumulant vanish	The proposed completion leaves a nonzero exact cumulant or fails another fixed-theory observable.

The third possibility requires genuinely shared or reduced data. Integrating out local fields independently on disconnected components still preserves the product. By contrast, tracing a joint state or integrating over a common global sector can induce nonlocal multi-integral terms and a reduced object that need not factorize Hernández-Cuenca 2025, §§2.3 and 3.2. This changes the object being computed; it does not repeal fixed-theory factorization.

The fourth possibility is realized in controlled low-dimensional constructions. Correlated branes, approximate $\alpha$-state descriptions, and half-wormhole-like terms can cancel or reorganize ordinary wormhole contributions and restore a fixed-spectrum answer Blommaert, Iliesiu, and Kruthoff 2022, §§1–2; Saad, Shenker, and Yao 2024, §§1–2. These models demonstrate mechanisms, not a universal cancellation theorem for higher-dimensional gravity.

The adversarial conditioning test

To test a claimed fixed-theory interpretation, keep the boundary data and approximation scheme fixed while removing only the proposed source of averaging.

1. Name and normalize the observable. Distinguish the unnormalized gravitational amplitude, its algebraic cumulant, a normalized state expectation, and an ensemble moment.
2. Condition on the hidden label. Replace $d\mu(T)$ by a delta measure at one $T$, or $p_\Psi(\alpha)$ by a sharp sector. Same-draw covariance must disappear.
3. Recompute the complete answer. Include every connected and disconnected term, shared constraint, contour coefficient, and nonperturbative contribution at the relevant order. Do not delete a wormhole by notation.
4. Test the claimed algebra. Verify that the operations used to prepare and measure the two copies preserve the sector and introduce no coupling or common trace.
5. Demand the exact target. If the claim is two decoupled copies of a fixed theory, the final cumulant must be zero, not merely small compared with the leading product.

This test also fixes three common order-of-limits errors.

Condition before truncating topology. Exact sharp-$\alpha$ states are nonperturbative, and geometric calculations in them can be poorly controlled. Conditioning a full answer and conditioning a finite topology expansion need not commute Marolf and Maxfield 2020, §5.2; Saad, Shenker, and Yao 2024, §§1.1–1.2 and 5.3.

A finite spectral window is not yet a sharp sector: it retains the covariance of eigenvalues inside the window. The zero-width limit must be controlled, and generalized continuous-spectrum eigenstates need not be normalizable before smearing.

Keep exact identities separate from large-parameter limits. Sending $S_0$ or $N$ to infinity can suppress a connected-to-disconnected ratio. It cannot prove equality at finite $S_0$ or finite $N$. If exponentially small terms are invoked to restore the identity, their coefficient and phase must be calculated.

Do not exchange late time with the genus expansion without control. In the symmetric continuation, the double trumpet grows as

$$\widehat Z_{0,2}(\beta+it,\beta-it) \sim\frac{t}{4\pi\beta} \qquad(t\gg\beta),$$

which is the perturbative ramp Saad, Shenker, and Stanford 2019, §3.4.1, Eq. (136). At times comparable with the inverse level spacing, spectral discreteness and nonperturbative completion become essential. The genus series itself is asymptotic, so its large-$S_0$ and late-time limits are not interchangeable. A fixed-$H$ partition function remains a fixed function before any time window, smoothing, or ensemble average is applied.

Usatyuk and Zhao provide a useful current stress test in a related setting: assuming factorization for one holographic theory, including a stronger finite-boundary factorization assumption where stated, they obtain a unique closed-universe state; ensemble averaging instead produces multiple smooth semiclassical wavefunctions Usatyuk and Zhao 2025, §§2–4. The result sharpens the conditioning question but does not prove that an arbitrary gravitational path-integral prescription satisfies their hypotheses.

Evidence and claim ceiling

The literature-dependent claims in this section were checked through 30 August 2026.

Result	Domain and control	Strongest licensed conclusion	Remaining uncertainty
Fixed-copy product	Specified theory, product preparation, decoupled actions and measures	The two-copy partition function factorizes exactly.	Whether a proposed gravity prescription computes this object.
JT double trumpet	Exact fixed-topology integral in orientable pure JT at positive $\beta_i$	A finite connected coefficient with the displayed normalization.	Its interpretation without an additional dictionary.
JT matrix-integral match	Perturbative genus expansion of a specified double-scaled matrix ensemble	JT connected amplitudes equal same-draw ensemble cumulants order by order.	Nonunique nonperturbative completion and applicability beyond the model.
Marolf–Maxfield α states	Constructed baby-universe Hilbert space and commuting boundary algebra	Sharp eigenstates factorize the stated boundary operators.	Existence, accessibility, and completeness of analogous sectors in a given holographic theory.
Fixed-spectrum model mechanisms	Controlled two-dimensional, matrix, and SYK-like laboratories	Additional correlated contributions can restore factorization in those models.	No general derivation for arbitrary quantum gravity.
Fixed-theory closed-universe test	Factorization hypotheses plus explicit JT fixed-H laboratory	Exact fixed-theory factorization strongly constrains closed-universe states.	Finite-boundary assumptions and extension to other theories.

The publication-safe conclusion is conditional. A connected gravitational term is compatible with an ensemble moment when a same-draw ensemble dictionary is independently established. An $\alpha$ interpretation resolves the same connection only after conditioning is defined and the residual $C_\alpha$ vanishes for the claimed algebra. If one fixed boundary theory is intended, a nonzero complete cumulant remains a problem until explicit additional physics restores the exact product.

The broader Hilbert-space, positivity, spectral, and uniqueness requirements continue on Fixed-Theory Factorization and Nonperturbative Completion Tests.

Common pitfalls

Confusing “same theory” with “same random draw.” Two copies of a fixed $T$ use the same known Hamiltonian and factorize. A same-draw ensemble uses one unknown random label twice and develops covariance after that label is averaged out.

Calling every connected geometry a covariance. A geometry is part of a gravitational integration domain. A covariance requires a normalized probability measure, while the full cumulant can also receive cancellations or non-topological shared contributions.

Assuming a sharp α solves everything. Conditioning removes between-sector variance. It does not remove a residual $C_\alpha$, prove that the sector is preserved by all relevant observables, or define a complete boundary theory.

Replacing equality by leading-order agreement. A term of relative order $e^{-2S_0}$ may be negligible for a coarse observable and still violate exact factorization. State clearly whether the claim is approximate or exact.

Importing spatial-subregion caveats. Edge modes and type-III algebras complicate spatial entanglement in one QFT. They do not generate covariance between two independently prepared, decoupled copies.

Exercises

1. Compute a two-member covariance

Let $T=a,b$ with probabilities $p$ and $1-p$. Show that

$$\kappa_2^\mu(B_1,B_2) =p(1-p) \bigl(Z_a[B_1]-Z_b[B_1]\bigr) \bigl(Z_a[B_2]-Z_b[B_2]\bigr).$$

Solution

Write $A_i=Z_a[B_i]$ and $B_i'=Z_b[B_i]$. Then

$$\overline{Z_1Z_2}=pA_1A_2+(1-p)B_1'B_2'$$

and

$$\overline{Z_1}\,\overline{Z_2} =\bigl[pA_1+(1-p)B_1'\bigr] \bigl[pA_2+(1-p)B_2'\bigr].$$

Subtracting and collecting terms gives the stated result. At $p=0$ or $1$ the label is fixed and the covariance vanishes. Each member factorizes for every $p$; only ignorance of the shared label produces the connection.

2. Compare one draw with two draws

Using the same two-member ensemble, compute the expectation of $Z_{T_1}[B_1]Z_{T_2}[B_2]$ when $T_1$ and $T_2$ are drawn independently. Explain why it differs from the same-draw moment.

Solution

Independence gives

$$\mathbb E_{\mu\times\mu} \bigl[Z_{T_1}[B_1]Z_{T_2}[B_2]\bigr] =\mathbb E_\mu[Z_T[B_1]] \mathbb E_\mu[Z_T[B_2]].$$

All four label pairs $(a,a),(a,b),(b,a),(b,b)$ occur with product probabilities. In the same-draw observable only $(a,a)$ and $(b,b)$ occur. The difference is exactly the covariance found in Exercise 1.

3. Derive the conditioned decomposition

Assume $G_\alpha=Z_{\alpha,1}Z_{\alpha,2}+C_\alpha$ and normalized $p(\alpha)$. Subtract the product of the averaged one-boundary quantities from $\int d\alpha\,p(\alpha)G_\alpha$.

Solution

The averaged one-boundary quantities are $\bar Z_i=\int d\alpha\,p(\alpha)Z_{\alpha,i}$. Therefore

$$\begin{aligned} \int pG_\alpha-\bar Z_1\bar Z_2 &=\int pC_\alpha\\ &\quad+\left(\int pZ_{\alpha,1}Z_{\alpha,2} -\bar Z_1\bar Z_2\right). \end{aligned}$$

The parenthesis is the between-sector covariance. Replacing $p$ by a delta distribution removes that parenthesis but leaves $C_\alpha$. Hence sharp conditioning is sufficient only when the residual conditioned connection vanishes.

4. Check the JT scaling and normalization

Evaluate $\widehat Z_{0,2}(\beta,\beta)$ and compare the Euler weights of the cylinder and two disks. What does the comparison establish?

Solution

The displayed double-trumpet formula gives

$$\widehat Z_{0,2}(\beta,\beta) =\frac{\beta}{2\pi(2\beta)} =\frac{1}{4\pi}.$$

The connected cylinder has Euler characteristic $0$ and weight $e^{0S_0}$; two disks have total Euler characteristic $2$ and weight $e^{2S_0}$. The connected-to-disconnected ratio is therefore suppressed by $e^{-2S_0}$ up to the boundary-dependent disk coefficients. This establishes semiclassical suppression, not exact factorization.

5. Test a proposed resolution

A calculation finds $\mathcal C_{\mathrm{grav}}\sim e^{-2S_0}$ relative to the product and says, “factorization is restored at large $S_0$.” Identify the missing step for an ensemble claim, an $\alpha$-sector claim, and a fixed-theory claim.

Solution

For an ensemble claim, one must specify a normalized measure, prove that both boundaries use the same draw, and match the relevant moments. For an $\alpha$ claim, one must construct the sector and algebra, condition on $\alpha$, and show $C_\alpha=0$. For a fixed-theory claim, one must calculate the additional contributions that cancel the complete cumulant exactly. Suppression alone proves none of these statements.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Blommaert, A., L. V. Iliesiu, and J. Kruthoff. “Gravity Factorized.” Journal of High Energy Physics 2022, 9 (2022): 080. DOI. Open PDF.
• Hernández-Cuenca, S. “Wormholes and Factorization in Exact Effective Theory.” Journal of High Energy Physics 2025, 5 (2025): 024. DOI. Open PDF.
• Marolf, D., and H. Maxfield. “Transcending the Ensemble: Baby Universes, Spacetime Wormholes, and the Order and Disorder of Black Hole Information.” Journal of High Energy Physics 2020, 8 (2020): 044. DOI. Open PDF.
• Saad, P., S. H. Shenker, and D. Stanford. “JT Gravity as a Matrix Integral.” arXiv:1903.11115 [hep-th] (2019). arXiv.
• Saad, P., S. Shenker, and S. Yao. “Comments on Wormholes and Factorization.” Journal of High Energy Physics 2024, 10 (2024): 076. DOI. Open PDF.
• Usatyuk, M., and Y. Zhao. “Closed Universes, Factorization, and Ensemble Averaging.” Journal of High Energy Physics 2025, 2 (2025): 052. DOI. Open PDF.