JT Correlators, Bilocals, and Chaos

Matter correlators in JT gravity are Schwarzian averages of reparametrized conformal bilocals. Expanding around the thermal saddle gives a connected four-point function from soft-mode exchange; after a specified out-of-time-order continuation, it grows as $e^{2\pi t/\beta}$ only between dissipation and scrambling. The regulator, operator dimension, coupling $C$, and time window are part of the result.

Required background. JT Gravity and the Schwarzian Boundary Mode supplies the path integral. Out-of-Time-Order Correlators and Contour Regularization fixes the ordering.

Helpful background. Lyapunov Growth and Chaos Bounds supplies the bound’s hypotheses. Holographic Hydrodynamization, Attractors, and Gradient Asymptotics provides a comparison with another asymptotic effective regime.

Evidence cutoff: 25 July 2026.

Bilocals under reparametrization

For an operator of infrared dimension $h$, the thermal bilocal on a reparametrized boundary is

$$\mathcal B_h(u_1,u_2;f) =\left[ \frac{f'(u_1)f'(u_2)} {\left(\frac{\beta}{\pi}\right)^2 \sin^2\!\left(\frac{\pi[f(u_1)-f(u_2)]}{\beta}\right)} \right]^h .$$

The JT correlator is

$$\langle\mathcal B_h\rangle =\frac{1}{Z_{\mathrm{Sch}}} \int_{\mathrm{Diff}(S^1)/SL(2,\mathbb R)} \mathcal Df\,e^{-I_{\mathrm{Sch}}[f]}\mathcal B_h[f].$$

This expression assumes the matter probe does not appreciably change the dilaton saddle. Heavy operators or many insertions require their backreaction.

Set $f(u)=u+\varepsilon(u)$. To quadratic order,

$$I^{(2)} =\frac{C}{2}\int_0^\beta du\, \left[ (\varepsilon'')^2-\left(\frac{2\pi}{\beta}\right)^2(\varepsilon')^2 \right].$$

The Fourier modes $n=0,\pm1$ are the $SL(2,\mathbb R)$ zero modes and must be divided out. Contracting the remaining modes between two bilocals produces the leading connected four-point function, of order $1/C$.

The out-of-time-order application

Place the four operators on a regulated thermal contour and analytically continue the Euclidean insertion times to the ordering $V(0)W(t)V(0)W(t)$. Soft-mode exchange gives schematically

$$\frac{\langle V(0)W(t)V(0)W(t)\rangle} {\langle VV\rangle\langle WW\rangle} =1-\frac{\alpha_{VW}}{C} e^{2\pi t/\beta} +O(C^{-2}),$$

where $\alpha_{VW}$ depends on dimensions, normalizations, and contour separations. Thus

$$\lambda_L=\frac{2\pi}{\beta}, \qquad t_*=\frac{\beta}{2\pi}\log C+O(\beta)$$

when $C$ is the large parameter in the chosen normalization. The result follows directly from the Schwarzian soft mode Maldacena, Stanford, and Yang 2016. Exact Schwarzian bilocal amplitudes extend the calculation beyond the Gaussian approximation Mertens, Turiaci, and Verlinde 2017.

Contour and window adversaries

Move the insertions onto a time-ordered contour. The exponentially enhanced term is absent or appears in a different analytic sheet; fitting the same ansatz no longer measures an OTO Lyapunov exponent. Next include data at $t\gtrsim t_*$. There the connected correction is order one, so the one-soft-mode expansion has failed and cannot establish continued exponential growth.

At $t\lesssim\beta$, microscopic matter response contaminates the fit. At frequencies near the throat gap, non-Schwarzian operators change the kernel. At finite $C$, loop and nonperturbative corrections become relevant before an exact late-time conclusion can be drawn.

The licensed claim is maximal leading OTOC growth for specified bilocals and regulator in the Schwarzian window. It does not identify a unique microscopic Hamiltonian, prove random-matrix statistics, or determine exact finite-spectrum recurrences.

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

• Maldacena, Juan, Douglas Stanford, and Zhenbin Yang. “Conformal Symmetry and Its Breaking in Two-Dimensional Nearly Anti-de Sitter Space.” Progress of Theoretical and Experimental Physics 2016, 12C104 (2016). DOI.
• Mertens, Thomas G., Gustavo J. Turiaci, and Herman L. Verlinde. “Solving the Schwarzian via the Conformal Bootstrap.” Journal of High Energy Physics 2017, 136 (2017). DOI.