Quantum Loewner evolution in quantum natural time: phases and Markov properties
Quantum Loewner evolution (QLE)$(γ^2, η)$ is a family of growth processes in random environments, introduced by Miller and Sheffield (arXiv:1312.5745) as candidate scaling limits of growth processes (such as diffusion-limited aggregation) on random planar maps. The random environments are Liouville quantum gravity (LQG) surfaces with parameter $γ$, and the parameter $η$ plays a role analogous to that in dielectric breakdown models. Their construction applies to pairs $(γ^2, η)$ lying on a curve in parameter space, and the associated time parametrization is independent of the underlying LQG surface. In later work (arXiv:1507.00719), they defined a quantum natural time variant of QLE$(8/3, 0)$ whose time parametrization encodes a notion of distance in the LQG geometry, leading to the identification of $\sqrt{8/3}$-LQG with the Brownian map. In this paper we construct quantum natural time QLE$(γ^2, η)$ for a different but overlapping subset of the same parameter curve. Its time parametrization conjecturally corresponds to the scaling limit of time parametrizations of discrete growth processes on random planar maps. We prove that it exhibits three phases, mirroring those of Schramm-Loewner evolution (SLE); this answers a question of Miller and Sheffield for quantum natural time QLE. Moreover, we establish stationarity of the unexplored surface and, in the relevant phases, identify the random surfaces cut out or swallowed by the process as quantum disks. Our construction builds on recent radial LQG-SLE coupling results of Ang and Yu (arXiv:2309.05176, arXiv:2411.19810).