# Meaning of “Exactly solvable in the large $N$ limit” for the SYK model

Every presentation on the SYK Model (check any youtube lecture by Douglas Stanford, Juan Maldacena, Subir Sachdev, Alexei Kitaev, etc.) claims that it is exactly solvable in the large $$N$$ limit, thus providing key insight into the realm of strongly correlated materials. However, once I got into the gory details of the papers, the only solutions seem to be either

1. Numerical solutions of the Schwinger-Dyson self-consistent equations for the green's function $$G(\tau_1,\tau_2)$$ and the self energy $$\Sigma(\tau_1,\tau_2)$$.
2. Low temperature/Strong Coupling limit $$\beta J >>1$$ of the Schwinger-Dyson equations or the path integral formulation.
3. First order approximation of the conformal symmetry breaking term in the action which gives the Schwartzian contribution to the action. This still needs exact calculation of the Schwinger-Dyson equations to determine a constant sitting outside the integral.
4. Exact calculations can be made in the limit $$q\rightarrow \infty$$ where $$q$$ is the number of terms in the SYK interaction term.

None of this alternatives seem "Exact" to me. Numerical calculations can be done, in principle, for any system. Furthermore, every analytical calculation seems to heavily rely on some very specific approximations, regarding either temperature, coupling or time-scales.

So my question is: What are the exact results that can be obtained using the SYK model that are so enlightening for strongly correlated physics? In what sense do they mean "Exact"?

The thing that is 'exact' is the conversion from a complicated class of quantum mechanics problems to a single classical problem. In the large-N limit, you can take a saddle-point action, thus obviating the need for a path integral.

You are correct that the saddle-point equations- the equations of motion for our classical system- can't be solved in closed form by any known method. This is analogous to saying that we can 'exactly' write down the two-point function for some complicated quadratic QFT (higher-derivative terms, a curved spacetime, whatever keeps you up at night). We would most likely end up with a differential equation we can't exactly solve. But we got this-exactly- from a far uglier-looking path integral. The path integral is something that would give supercomputers nightmares. The differential equation- not so much.