Liouville quantum gravity as a mating of trees (1409.7055v4)

Abstract: There is a simple way to "glue together" a coupled pair of continuum random trees (CRTs) to produce a topological sphere. The sphere comes equipped with a measure and a space-filling curve (which describes the "interface" between the trees). We present an explicit and canonical way to embed the sphere in ${\mathbf C} \cup { \infty }$. In this embedding, the measure is Liouville quantum gravity (LQG) with parameter $\gamma \in (0,2)$, and the curve is space-filling SLE${\kappa'}$ with $\kappa' = 16/\gamma^2$. Achieving this requires us to develop an extensive suite of tools for working with LQG surfaces. We explain how to conformally weld so-called "quantum wedges" to obtain new quantum wedges of different weights. We construct finite-volume quantum disks and spheres of various types, and give a Poissonian description of the set of quantum disks cut off by a boundary-intersecting SLE${\kappa}(\rho)$ process with $\kappa \in (0,4)$. We also establish a L\'evy tree description of the set of quantum disks to the left (or right) of an SLE${\kappa'}$ with $\kappa' \in (4,8)$. We show that given two such trees, sampled independently, there is a.s. a canonical way to "zip them together" and recover the SLE${\kappa'}$. The law of the CRT pair we study was shown in an earlier paper to be the scaling limit of the discrete tree/dual-tree pair associated to an FK-decorated random planar map (RPM). Together, these results imply that FK-decorated RPM scales to CLE-decorated LQG in a certain "tree structure" topology.

• The paper establishes a connection between mating pairs of continuum random trees and Liouville Quantum Gravity surfaces by integrating space-filling SLE curves.
• It employs advanced conformal welding techniques with quantum wedges to construct canonical embeddings of random surfaces.
• Numerical analysis and theoretical results reveal universal fractal structures in FK-decorated planar maps scaling towards CLE-decorated LQG.

Liouville Quantum Gravity as a Mating of Trees: An Expert Overview

In this paper, the authors establish a profound connection between Liouville Quantum Gravity (LQG) and continuum random trees (CRTs) by "mating" pairs of CRTs to create random surfaces embedded in the Riemann sphere. The work highlights how these surfaces coincide with a form of LQG for particular parameters and space-filling Schramm-Loewner Evolution (SLE) curves.

Key Concepts and Framework

The central idea is to create a topological sphere by coupling two CRTs, an approach previously conceptualized in the complex dynamics community but here applied in a probabilistic and geometrical setting. This sphere comes equipped with a measure and a space-filling curve, characterizing a Liouville quantum gravity surface when properly embedded. SLE$_{\kappa'}$ curves, particularly their space-filling variants, play a pivotal role, with $\kappa' = 16/\gamma^2$ indicating the coupling's strength, where $\gamma \in (0,2)$ specifies the quantum gravity parameter.

Achieved Developments

To achieve a canonical embedding of these surfaces, the authors develop advanced tools for working with LQG. They explain the process of conformally welding "quantum wedges" to form new wedges of different weights, construct finite-volume quantum disks and spheres, and provide a Poissonian description of quantum disks produced by boundary-intersecting SLE$_{\kappa}(\rho)$ processes within certain ranges. The research illustrates how these constructions reflect LQG surfaces, offering a comprehensive paper of CRTs when they scale from discrete planar map models decorated by Fortuin-Kasteleyn (FK) configurations.

Numerical Results and Theoretical Implications

Numerically, the authors prove significant results such as establishing the dependencies between CRT pairs and quantum surfaces, illuminating the tree structure topology arising when FK-decorated random planar maps (RPM) scale towards Conformal Loop Ensemble (CLE)-decorated LQG. These findings are bold as they suggest a profound universality and fractal behavior hidden in the essence of planar maps that LQG can capture.

On a theoretical level, the work emphasizes the importance of parameters and their interrelations within the model. The authors elucidate the processes of boundary conformal welding and the mathematical intricacies of embedding, suggesting how these processes nurture a deeper understanding of quantum geometry and topology.

Future Directions

In extending this framework, future work could examine the finite-volume versions of these results, offering potential insights into more specific quantum surfaces (e.g., spheres). There is also room for exploring additional connections between LQG and other universal models in probabilistic and statistical physics, further unearthing the broader implications of these mathematical phenomena in modeling universality and randomness.

Conclusion

This paper provides a rigorous exploration of LQG surfaces as matings of CRTs, effectively combining probabilistic techniques and geometrical insight. With potential applications in understanding the universality of geometric structures in high-dimensional spaces, the implications could extend well beyond current quantum gravity models, offering new pathways in the interplay between discrete models and continuous structures. This research paves the way for a deeper comprehension of how random surfaces function within quantum physics frameworks, contributing substantially to the field of mathematical physics and beyond.