Existence of a finite-temperature critical subsystem fraction p_crit in random quantum circuits

Establish the existence, in random quantum circuits that model finite-temperature dynamics (e.g., with continuous symmetries or conserved quantities), of a sharp transition at a less-than-half critical subsystem fraction p_crit such that for p < p_crit the subsystem complexity saturates essentially instantaneously, whereas for p_crit < p < 1/2 it grows for a finite time before saturating, as predicted by holography.

Background

Holographic analysis predicts a second spatial transition at finite temperature: a critical fraction p_crit < 1/2 at which the subsystem complexity changes its early-time behavior. Below p_crit the complexity saturates nearly immediately; above p_crit (but still below 1/2) it shows a rise-and-fall over a finite time.

In random quantum circuits at infinite temperature, p_crit = 0. The authors propose that circuits with continuous symmetries (or other finite-temperature features) could exhibit a nonzero p_crit, paralleling the holographic prediction, and they explicitly leave this as a conjecture for future work.