Quantum Loewner evolution in quantum natural time: phases and Markov properties

TL;DR

This paper constructs a quantum natural time variant of Quantum Loewner Evolution (QLE) on Liouville quantum gravity surfaces, demonstrating phase behavior similar to SLE and identifying the nature of cut surfaces.

Contribution

It extends the construction of quantum natural time QLE to a new parameter subset, proving phase properties and surface identifications, advancing understanding of LQG growth processes.

Findings

QLE exhibits three phases similar to SLE.

Stationarity of the unexplored surface is established.

Identifies cut surfaces as quantum disks in relevant phases.

Abstract

Quantum Loewner evolution (QLE)$(\gamma^2, \eta)$ 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 $\gamma$, and the parameter $\eta$ plays a role analogous to that in dielectric breakdown models. Their construction applies to pairs $(\gamma^2, \eta)$ 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$(\gamma^2, \eta)$ 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).