Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Area measures and branched polymers in supercritical Liouville quantum gravity

Manan Bhatia, Ewain Gwynne, Jinwoo Sung

TL;DR

The paper rigorously analyzes supercritical Liouville quantum gravity in the regime cL in (1,25) by constructing a discrete, loop-decorated planar-map model that converges to LQG in the scaling limit. Conditioning these maps to be finite reveals a degeneration to the continuum random tree, reconciling the infinite-spike and branched-polymer viewpoints via a CLE4-coupled multiplicative cascade, while also proving the nonexistence of a locally finite bulk LQG volume measure satisfying locality and coordinate covariance. The authors develop a two-pronged approach, combining a continuous framework of nested CLE4 with multiplicative cascades and a discrete random-map analysis using Boltzmann maps and perimeter cascades, to establish CRT convergence and quantify the rarity of finiteness through exponential decay. Their results have foundational implications for the geometry of supercritical LQG, showing that while infinite spikes dominate generically, finitary conditioning yields a tree-like continuum geometry, and they rule out a natural bulk area form in this regime. The work introduces powerful cascade techniques that may generalize to other models related to outer-loop decompositions and may impact the study of universality classes for random geometries.

Abstract

We study Liouville quantum gravity (LQG) in the supercritical (a.k.a. strongly coupled) phase, which has background charge $Q \in (0,2)$ and central charge $\mathbf{c}_{\mathrm{L}} = 1+6Q^2 \in (1,25)$. Recent works have shown how to define LQG in this phase as a planar random geometry associated with a variant of the Gaussian free field, which exhibits "infinite spikes." In contrast, a number of results from physics, dating back to the 1980s, suggest that supercritical LQG surfaces should behave like "branched polymers": i.e., they should look like the continuum random tree. We prove a result which reconciles these two descriptions of supercritical LQG. More precisely, we show that for a family of random planar maps with boundary in the universality class of supercritical LQG, if we condition on the (small probability) event that the planar map is finite, then the scaling limit is the continuum random tree. We also show that there does not exist any locally finite measure associated with supercritical LQG which is locally determined by the field and satisfies the LQG coordinate change formula. Our proofs are based on a branching process description of supercritical LQG which comes from its coupling with CLE$_4$ (Ang and Gwynne, arXiv:2308.11832).

Area measures and branched polymers in supercritical Liouville quantum gravity

TL;DR

The paper rigorously analyzes supercritical Liouville quantum gravity in the regime cL in (1,25) by constructing a discrete, loop-decorated planar-map model that converges to LQG in the scaling limit. Conditioning these maps to be finite reveals a degeneration to the continuum random tree, reconciling the infinite-spike and branched-polymer viewpoints via a CLE4-coupled multiplicative cascade, while also proving the nonexistence of a locally finite bulk LQG volume measure satisfying locality and coordinate covariance. The authors develop a two-pronged approach, combining a continuous framework of nested CLE4 with multiplicative cascades and a discrete random-map analysis using Boltzmann maps and perimeter cascades, to establish CRT convergence and quantify the rarity of finiteness through exponential decay. Their results have foundational implications for the geometry of supercritical LQG, showing that while infinite spikes dominate generically, finitary conditioning yields a tree-like continuum geometry, and they rule out a natural bulk area form in this regime. The work introduces powerful cascade techniques that may generalize to other models related to outer-loop decompositions and may impact the study of universality classes for random geometries.

Abstract

We study Liouville quantum gravity (LQG) in the supercritical (a.k.a. strongly coupled) phase, which has background charge and central charge . Recent works have shown how to define LQG in this phase as a planar random geometry associated with a variant of the Gaussian free field, which exhibits "infinite spikes." In contrast, a number of results from physics, dating back to the 1980s, suggest that supercritical LQG surfaces should behave like "branched polymers": i.e., they should look like the continuum random tree. We prove a result which reconciles these two descriptions of supercritical LQG. More precisely, we show that for a family of random planar maps with boundary in the universality class of supercritical LQG, if we condition on the (small probability) event that the planar map is finite, then the scaling limit is the continuum random tree. We also show that there does not exist any locally finite measure associated with supercritical LQG which is locally determined by the field and satisfies the LQG coordinate change formula. Our proofs are based on a branching process description of supercritical LQG which comes from its coupling with CLE (Ang and Gwynne, arXiv:2308.11832).

Paper Structure

This paper contains 33 sections, 48 theorems, 170 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Physics predictions on the behavior of supercritical LQG and the main contributions of this paper
  3. A discrete model of supercritical LQG
  4. Main result 1: supercritical LQG disk conditioned to be finite is a branched polymer
  5. Main result 2: non-existence of supercritical LQG volume measure
  6. Proof outline
  7. Notational comments
  8. Acknowledgements
  9. Background on the supercritical LQG disk
  10. Supercritical LQG disk
  11. Coupling with CLE(4)
  12. Coupling with nested CLE(4) and the perimeter cascade
  13. Non-existence of the supercritical volume measure
  14. The perimeter process contains infinitely many macroscopic loops
  15. Proof of non-existence
  16. ...and 18 more sections

Key Result

Theorem 1.2

Consider the loop-decorated planar map sampled from the unconditioned law $\mathbb{P}_\infty^{(p)}$ defined in Definition def:model. For each $n\in \mathbb{N}$, let $(\chi_i^n)_{i\in \mathbb{N}}$ be the inner half-perimeters of the $n$th generation rings $\{\mathrm{per}_\mathrm{in}(f): f\in \mathfra jointly with respect to the product topology on $\mathbb{R}^{\mathbb{N}\times \mathbb{N}}$.

Figures (5)

  • Figure 1: An example of a ring sampled from $\mathbb P_\mathrm{ring}^{(4)}$ with the outer boundary colored in green and the inner boundary in purple. For this ring, $\mathrm{per}(f_\mathrm{out})=4$ and $\mathrm{per}(f_\mathrm{in})=10$.
  • Figure 2: An illustration of the iterative construction in Definition \ref{['def:model']} with $p=13$. (a) The outermost gasket $M_0$ is sampled from $\mathbb P_\mathbf{q}^{(p)}$. The collection $\mathfrak{F}_0$ comprises the inner faces of $M_0$. (b) Given $M_0$, a ring $R(f)$ is sampled conditionally independently for each face $f\in \mathfrak{F}_0$. The outer/inner boundaries of the rings are colored in green/purple, respectively. The ring in the top-right of the figure has inner perimeter zero. (c) The rings are attached to the corresponding faces, with the possible rotations chosen uniformly. The rings are identified with red loops that separate inner and outer boundaries. These loops are the discrete analogs of CLE$_4$ loops in the coupling of supercritical LQG disk with CLE$_4$. (d) Given the previous figure, for each ring $R(f)$, sample conditionally independent Boltzmann maps from $\mathbb{P}_\mathbf{q}^{\mathrm{per}_\mathrm{in}(f)}$ where $\mathrm{per}_\mathrm{in}(f)$ is the inner half-perimeter of the ring $R(f)$. (e) The Boltzmann maps are glued to the inner boundaries of the rings, with the possible rotations chosen uniformly. These comprise the map $M_1$ colored purple and their inner faces $\mathfrak{F}_1$ colored yellow. (f) The map after another iteration, with $M_0$ colored in green, $M_1$ in purple, and $M_2$ in blue. In this case, $\mathfrak{F}_2=\varnothing$ as the Boltzmann maps comprising $M_2$ do not have non-root (inner) faces. The construction terminates at this stage, giving us a finite map.
  • Figure 3: The unique value of $\theta^*$ satisfying $\mu_Q=(1/\theta^*)\log \phi_Q(\theta^*)$, obtained by solving \ref{['eq:194']} numerically. We observe $\theta^*\approx 1.9647$ as $Q\to 0$.
  • Figure 4: From left to right: a tree $\mathcal{T}$, the corresponding looptree $\mathrm{Loop}(\mathcal{T})$, and the contracted looptree $\overline{\mathrm{Loop}}(\mathcal{T})$. The root vertex $\emptyset \in V_{\mathcal{T}}$ and the corresponding vertices on $\mathrm{Loop}(\mathcal{T})$ and $\pi(\emptyset)\in \overline{\mathrm{Loop}}(\mathcal{T})$ are marked with arrows.
  • Figure 5: A planar map (left) and the corresponding scooped-out map (right).

Theorems & Definitions (103)

  • Definition 1.1
  • Theorem 1.2: See Proposition \ref{['prop:26']} for a precise statement
  • Proposition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Definition 2.1
  • Definition 2.2: Critical LQG disk
  • Definition 2.3: Supercritical LQG disk
  • Proposition 2.4
  • ...and 93 more