Quantum critical collapse in the large $D$ limit

Abstract: In analytically tractable models of Einstein-scalar critical collapse, it has been shown that the quantum self-energy of the collapsing matter in the dominant $s$-wave sector grows with Lyapunov exponent $D-2$, eventually competing with the classical instability that controls black hole formation. This competition shifts the naked singularity threshold and produces a finite mass gap within semiclassical control, yet that control may appear threatened as the spacetime dimension $D\to\infty$. We study this quantum growing mode in both continuously (CSS) and discretely (DSS) self-similar critical spacetimes at large $D$ and establish that the universal Lyapunov exponent survives both forms of self-similarity. In CSS, the classical approach to a naked singularity is interrupted by the appearance of trapped surfaces. At large $D$, the horizon tracing reduces to a Bessel problem, yielding a quantum wall whose radius obeys $r_h\propto(D-2)<sup>{-2}$. This radius can remain above the Planck scale at finite $D$, although maintaining this hierarchy becomes increasingly demanding as $D$ grows. In DSS, the same growth is modulated by echoing and can retain memory of previous echoes. The simplest anomaly-induced action exhibits accumulating memory but is singular at the spherical center. We construct a covariant one-loop action whose stress tensor remains smooth throughout the DSS domain and organize it into an echo-periodic structure without memory. This suggests that the smooth vacuum polarization of the genuine four-dimensional DSS Choptuik spacetime may similarly be governed by its exact echoing symmetry.