  It has been known that the large-$q$ complex SYK model falls under the same
universality class as that of van der Waals (mean-field) and saturates the
Maldacena-Shenker-Stanford bound, both features shared by various black holes.
This makes the SYK model a useful tool in probing the fundamental nature of
quantum chaos and holographic duality. This work establishes the robustness of
this shared universality class and chaotic properties for SYK-like models by
extending to a system of coupled large-$q$ complex SYK models of different
orders. We provide a detailed derivation of thermodynamic properties,
specifically the critical exponents for an observed phase transition, as well
as dynamical properties, in particular the Lyapunov exponent, via the
out-of-time correlator calculations. Our analysis reveals that, despite the
introduction of an additional scaling parameter through interaction strength
ratios, the system undergoes a continuous phase transition at low temperatures,
similar to that of the single SYK model. The critical exponents align with the
Landau-Ginzburg (mean-field) universality class, shared with van der Waals
gases and various AdS black holes. Furthermore, we demonstrate that the coupled
SYK system remains maximally chaotic in the large-$q$ limit at low
temperatures, adhering to the Maldacena-Shenker-Stanford bound, a feature
consistent with the single SYK model. These findings establish robustness and
open avenues for broader inquiries into the universality and chaos in complex
quantum systems. We provide a detailed outlook for future work by considering
the ``very" low-temperature regime, where we discuss relations with the
Hawking-Page phase transition observed in the holographic dual black holes. We
present preliminary calculations and discuss the possible follow-ups that might
be be taken to make the connection robust.
