Liouville Quantum Duality and Random Planar Maps II

Abstract: This is Part II of our project on block-weighted planar maps and Liouville quantum duality. Focusing on the scaling properties at the dual critical point, we derive the conditional distribution of the root block size given the total size, as well as, conversely, the distribution of the total size for a fixed root block size. We show that these laws are in perfect agreement with the results of Liouville quantum gravity (LQG), obtained by modifying the standard Liouville random measure with additional atomic contributions representing localized quantum areas. The ratio of dual and direct partition functions with punctures is shown to be universal, its explicit LQG expression exactly matching its combinatorial analogue. We also investigate the block distance profile for doubly rooted maps, which is here rigorously related to the distance profile of maps consisting of a single block. Finally, we analyze the multifractal properties of the usual and dual Liouville measures, predicting the associated spectra, from both quantum and Euclidean perpectives. We illustrate our results through specific realizations of block-weighted planar maps, i.e., quadrangulations decomposed into simple blocks, tree-like structures formed by attaching quartic maps, and bicubic maps decomposed into 3-connected blocks. For each model, we give the single non-universal constant which uniquely determines the strength of the corresponding atomic Liouville measure.

• The paper establishes that block-weighted planar maps at dual criticality precisely match the scaling behavior and partition function ratios of atomic Liouville quantum gravity.
• It employs explicit probabilistic laws, generating function substitutions, and multifractal analysis to bridge discrete combinatorial models with continuum quantum geometry.
• The study validates universal scaling laws and provides numerical checks that unify theoretical predictions with statistical properties of random maps.

Liouville Quantum Duality and Scaling in Block-Weighted Planar Maps

Overview and Motivation

"Liouville Quantum Duality and Random Planar Maps II" (2604.24180) addresses the scaling properties and universality of block-weighted planar maps at their dual critical point, establishing rigorous connections to Liouville quantum gravity (LQG) duality in both discrete and continuum frameworks. The paper provides explicit probabilistic laws and multifractal spectra for planar maps decomposed into blocks, and demonstrates the precise matching of their distributional and scaling behavior with dual LQG measures — particularly those involving atomic (localized) quantum area contributions.

This work delivers a combinatorial justification of atomic dual LQG, serving as a bridge between rigorous probabilistic combinatorics of random maps and measure-theoretic quantum geometry. It presents strong numerical checks and universal ratios, and develops multifractal analysis from both Euclidean and quantum perspectives.

Block-Weighted Planar Maps and Dual Criticality

Block-weighted planar maps are constructed by decomposing maps into maximal submaps (blocks) forming tree-like structures. The key analytic object is the substitution relation between generating functions, which directly encodes the block structure:

$M_u(g) - 1 = u [ B( g M_u(g)^2 ) - 1 ],$

where $M_u(g)$ enumerates maps with weight $u$ per block and $B(t)$ enumerates maps of a single block. For generic $B(t)$ with singular expansion characterized by exponent $\alpha$ ($1 < \alpha < 2$), the critical point $u = u^*$ demarcates a phase transition between macroscopic and microscopic block regimes.

At the dual critical point ($u = u^*$), the block size distribution and total size distribution display stretched exponential scaling governed by a universal parameter $\alpha$ related to LQG duality:

$M_u(g)$0

with string susceptibility exponents $M_u(g)$1 and $M_u(g)$2 encoding direct and dual phases.

Explicit Block Size and Map Size Distributions

For the root block size $M_u(g)$3 in maps of fixed total size $M_u(g)$4, the probability law transitions from discrete to scaling regime as $M_u(g)$5, $M_u(g)$6, and $M_u(g)$7 fixed:

• The probability that the root block is finite approaches $M_u(g)$8 as $M_u(g)$9.
• For doubly rooted maps (two marked edges in the root block), the scaling density $u$0 emerges, normalized so that root block size scales as $u$1.

(Figure 1)

Figure 1: Plot of $u$2 versus $u$3 for varying $u$4, illustrating the universal scaling law for large root block sizes.

The block size scaling law is:

$u$5

where $u$6 is a universal function involving Wright functions.

Analogously, for maps conditioned on root block size $u$7, the total size $u$8 scales as $u$9, and the Laplace transform of properly scaled total size matches in the continuum dual LQG setting:

$B(t)$0

with $B(t)$1 a model-dependent constant.

(Figure 2)

Figure 2: Rescaled probability densities for root block size and total map size in block-weighted quadrangulations, compared to theoretical scaling predictions.

Partition Function Ratios and Universality

Universal ratios between partition functions of $B(t)$2-rooted maps (punctured spheres) in direct and dual phases are computed combinatorially and shown to match exactly with corresponding continuum LQG partition function ratios. This yields explicit formulas such as:

$B(t)$3

aligning discrete and quantum geometric predictions.

Multifractal Spectra in Liouville Quantum Gravity and Duality

Rigorous multifractal analysis is carried out for both standard ($B(t)$4) and dual ($B(t)$5) LQG measures. Moment scaling exponents and $B(t)$6-spectra are derived:

• For $B(t)$7-LQG, expected and almost sure $B(t)$8-spectra are quadratic in $B(t)$9 up to $B(t)$0, with explicitly computed Hausdorff dimension spectra of singularities.
• For dual $B(t)$1-LQG atomic measures, spectra are exactly dual under the map $B(t)$2, reproducing the entire multifractal formalism via combinatorial scaling and rigorous field-theoretic constructions.

(Figure 3)

Figure 3: Left: a.s.\ LQG $B(t)$3-spectrum $B(t)$4. Right: a.s.\ LQG multifractal dimension spectrum $B(t)$5 for $B(t)$6.

(Figure 4)

Figure 4: Red: a.s.\ dimension spectrum $B(t)$7. Orange: dual a.s.\ dimension spectrum $B(t)$8; maximal values correspond to quantum surface dimension without atomic localization.

(Figure 5)

Figure 5: Red/blue: a.s.\ $B(t)$9-spectrum for quantum balls $\alpha$0. Orange/purple: dual a.s.\ $\alpha$1-spectrum $\alpha$2; linear extensions align at transition points.

Block Distance Profiles and Duality

The block distance profile (distribution of geodesic distances between marked edges in the same block) at the dual critical point is shown to be directly related, via convolution, to that of maps formed of a single block. Scaling exponents for Hausdorff dimensions obey:

$\alpha$3

and explicit distance profile formulas for specific classes (e.g., quadrangulations) confirm these relations.

(Figure 6)

Figure 6: Block distance profile $\alpha$4 for block-weighted quadrangulations at dual criticality and its comparison with the single-block profile $\alpha$5.

Implications, Applications, and Future Directions

The rigorous combinatorial realization of Liouville quantum duality via random planar maps at dual criticality establishes the universal scaling nature of atomic LQG measures. This synthesis enables:

• Transfer of explicit probabilistic laws between discrete combinatorial models and continuum quantum geometry.
• Precision matching of partition functions and multifractal spectra across duality transformations, underlining the universality class structure in two-dimensional quantum gravity.
• Numerical validation and explicit formulas for large-scale statistical properties of random maps, relevant for statistical physics, quantum geometry, and complex networks.

Future developments may include:

• Extending the framework to decorated maps with additional statistical mechanics models (e.g., Ising, O(N)), exploring universality classes for $\alpha$6.
• Rigorous derivations of dual multifractal spectra using advanced probabilistic tools, further strengthening measure-theoretic foundations.
• Investigation of embeddings of random planar maps (e.g., Tutte or Smith) for direct comparison with quantum geometric constructions.

Conclusion

This paper rigorously connects block-weighted random planar maps at their dual critical point to the dual regime of Liouville quantum gravity, justifying atomic LQG constructions and multifractal spectra via analytic combinatorics. Universality of scaling laws, partition function ratios, and multifractal properties are established across discrete and continuum theories. The results unify paradigms in random geometry, deepening both theoretical understanding and practical methodologies for quantum gravity and statistical mechanics.