Block-Weighted Random Planar Maps

• Block-weighted random planar maps are combinatorial structures where each positive-size block is assigned a weight u, affecting connectivity and scaling behavior.
• They exhibit sharp phase transitions: subcritical regimes have a giant core, critical regimes show n^(2/3)-scale blocks, and supercritical regimes yield logarithmic block sizes.
• The model employs Galton–Watson trees and analytic combinatorics to reveal universal geometric limits and duality with Liouville quantum gravity.

Block-weighted random planar maps are random rooted planar maps in which each positive-size block receives a multiplicative weight $u>0$. In the basic model, a block is a maximal $2$-connected submap; in the quadrangulation counterpart, the analogous objects are simple components obtained by collapsing maximal $2$-cycles. The parameter $u$ tunes the density of separating elements, and the resulting ensembles exhibit a sharp phase transition. For planar maps weighted by $u$ per $2$-connected block, and for quadrangulations weighted by $u$ per simple component, the critical value is $u_C=9/5$: below $u_C$ a macroscopic block appears, at $u_C$ the largest blocks are of order $2$0, and above $2$1 all large blocks are only logarithmic. In the quadrangulation formulation, these three regimes correspond respectively to the Brownian sphere, the stable tree of parameter $2$2, and the Brownian tree as scaling limits (Fleurat et al., 2023). Later work showed that the same decomposition-to-tree mechanism extends across a broad range of map families and interfaces naturally with Liouville quantum gravity duality (Salvy, 2024, Duplantier et al., 16 Jul 2025, Duplantier et al., 27 Apr 2026).

1. Canonical model and block decomposition

A rooted planar map is a proper embedding of a connected finite multigraph into the sphere, considered up to orientation-preserving homeomorphisms, together with a distinguished oriented half-edge. Its size is the number of edges. A map is separable when it has a cut-vertex, otherwise it is $2$3-connected. Its blocks are the maximal $2$4-connected submaps of positive size. Tutte’s block-cut decomposition cuts the map at all cut-vertices and encodes the map by its blocks and the pendant submaps attached along the edges of the root block (Fleurat et al., 2023).

If $2$5 is the generating function of $2$6-connected maps, with $2$7 for the vertex map, then the classical decomposition gives

$2$8

In the block-weighted model, the corresponding bivariate generating function satisfies

$2$9

This equation makes the role of $2$0 explicit: increasing $2$1 biases the law toward maps with many blocks, hence toward many separating elements (Fleurat et al., 2023).

The decomposition is naturally encoded by a block tree $2$2. Its internal vertices correspond to blocks, its edges correspond to half-edges of the original map, and an internal node with $2$3 children encodes a block of size $2$4. Recursively decomposing pendant submaps yields a complete tree representation of the map. This block tree is the combinatorial backbone of the model and is the source of its probabilistic structure (Fleurat et al., 2023).

Quadrangulations fit the same framework through Tutte’s angular bijection. Under this bijection, maps of size $2$5 correspond to quadrangulations of size $2$6, and $2$7-connected maps of size at least $2$8 correspond to simple quadrangulations of the same size. The bijection preserves block trees, so the weighted map model and the weighted quadrangulation model are analytically parallel. In particular, weighting $2$9-connected blocks in maps corresponds to weighting simple components in quadrangulations (Fleurat et al., 2023).

2. Galton–Watson encoding and the origin of the critical point

The decisive structural fact is that the block tree becomes a conditioned Galton–Watson tree. Under the singular Boltzmann law, the block tree $u$0 has offspring law $u$1; under the fixed-size law on maps with $u$2 edges, $u$3 has the law of a Galton–Watson tree with offspring distribution $u$4, conditioned to have $u$5 edges. Conditionally on the tree, the blocks decorating internal nodes are independent and uniform among blocks with the prescribed sizes (Fleurat et al., 2023).

The mean offspring is

$u$6

Criticality occurs when this mean equals $u$7, equivalently

$u$8

Because $u$9 increases on $u$0 with $u$1, the critical value is

$u$2

This is the point at which the singularity coming from the implicit functional equation meets the intrinsic singularity of $u$3 (Fleurat et al., 2023).

The tail of the offspring distribution determines the phase. For $u$4, one has

$u$5

and $u$6, so the conditioned tree is subcritical with a heavy tail. At $u$7, the law is critical and still has $u$8 tail. For $u$9, one has $2$0 but now the tail is exponentially damped, so $2$1 has finite exponential moments (Fleurat et al., 2023).

This framework places the model in the simply-generated-tree paradigm. The heavy-tailed $2$2 laws at and below criticality yield stable behavior and condensation phenomena, while the exponentially decaying supercritical law yields finite-variance CRT behavior. The same mechanism already underlies the unweighted case $2$3, where a probabilistic approach identifies a largest $2$4-connected block of asymptotic size $2$5 in uniform random maps, together with $2$6-scale fluctuations and Fréchet-type extremes for the next largest blocks (Addario-Berry, 2015).

3. Phase transition and largest-block asymptotics

The three regimes are distinguished by the size of the largest blocks. In the subcritical regime $2$7, a condensation phenomenon occurs: precisely one vertex in the conditioned Galton–Watson tree has degree $2$8, while all others are $2$9. Translated back to maps or quadrangulations,

$u$0

Moreover, the joint scaling limit of the deficit of the giant block and the ordered mesoscopic blocks is described by the jumps of a $u$1-stable Lévy process (Fleurat et al., 2023).

At the critical point $u$2, no block is macroscopic. The largest degrees in the conditioned Galton–Watson tree are of order $u$3, and the ordered block sizes satisfy

$u$4

with the normalized sequence converging to the ordered atoms of a point process governed by the jumps of a $u$5-stable excursion (Fleurat et al., 2023).

In the supercritical regime $u$6, the offspring distribution has finite exponential moments, and the maximum degree in the conditioned tree is logarithmic. If

$u$7

then for fixed $u$8,

$u$9

Thus the entire block structure becomes tree-like in a strong extremal sense (Fleurat et al., 2023).

A broader universality theorem shows that this trichotomy is not specific to general maps decomposed into $u_C=9/5$0-connected blocks. Across eight decomposition schemes—including loopless maps into simple blocks, bipartite variants, and triangulations into irreducible blocks—the same phase diagram reappears. The coefficient asymptotics of the weighted generating function satisfy

$u_C=9/5$1

and the largest blocks are respectively linear, $u_C=9/5$2-scale, and logarithmic (Salvy, 2024).

4. Distances, diameters, and metric scaling limits

The block phase transition is simultaneously a geometric phase transition. Let $u_C=9/5$3 be the block tree of $u_C=9/5$4 or $u_C=9/5$5, endowed with graph distance and the uniform vertex measure. In the supercritical regime $u_C=9/5$6, the conditioned block tree is critical with finite variance $u_C=9/5$7, and after rescaling by $u_C=9/5$8 it converges in the Gromov–Hausdorff–Prokhorov topology to the Brownian continuum random tree. Distances in maps and quadrangulations are asymptotically additive along consecutive blocks and concentrate around positive constants $u_C=9/5$9 and $u_C$0, so the measured metric spaces themselves converge to constant multiples of the Brownian tree (Fleurat et al., 2023).

At criticality, the offspring law is in the domain of attraction of a $u_C$1-stable law. The rescaled block tree, maps, and quadrangulations satisfy

$u_C$2

Hence the critical regime is not Brownian but genuinely stable-tree in its global geometry (Fleurat et al., 2023).

The subcritical regime is different again. For quadrangulations, the unique macroscopic simple component has diameter $u_C$3, while all other components have size $u_C$4 and diameter $u_C$5. Consequently,

$u_C$6

where $u_C$7 is the Brownian sphere. An analogous statement for maps is expected, but the required GHP convergence for uniform $u_C$8-connected maps is not yet available (Fleurat et al., 2023).

The proofs combine tree asymptotics with geometric control inside blocks. If $u_C$9 is a uniform $u_C$0-connected map with $u_C$1 edges and $u_C$2 a uniform simple quadrangulation with $u_C$3 faces, then for every $u_C$4,

$u_C$5

are stretched-exponential in $u_C$6. Together with spine decompositions and moderate deviation estimates for sums of distances across many blocks, these bounds make it possible to transfer scaling limits from the block tree to the full map (Fleurat et al., 2023).

A common misconception is that “subcritical” should mean “tree-like.” In this model the opposite happens: the supercritical regime is tree-like, whereas the subcritical regime is dominated by a giant core block and, for quadrangulations, lies in the Brownian-sphere universality class (Fleurat et al., 2023).

5. Analytic-combinatorial framework and structural universality

The general mechanism can be formulated as a Lagrangian implicit equation. For a large class of block decompositions one starts from

$u_C$7

or

$u_C$8

and rewrites the system as

$u_C$9

This produces a canonical tree representation in which the block tree is a Galton–Watson tree with offspring law

$2$00

The phase transition is then equivalent to the usual Galton–Watson threshold $2$01 (Salvy, 2024).

This unified framework covers eight decomposition schemes: loopless maps into simple blocks, general maps into $2$02-connected blocks, $2$03-connected maps into $2$04-connected simple blocks, bipartite maps into bipartite simple blocks, bipartite maps into bipartite $2$05-connected blocks, bipartite $2$06-connected maps into bipartite $2$07-connected simple blocks, loopless triangulations into simple triangulations, and simple triangulations into irreducible triangulations. Across these families, the critical exponents $2$08, $2$09, and $2$10 are universal at the enumerative level, while the explicit values of $2$11, $2$12, $2$13, and $2$14 depend on the scheme (Salvy, 2024).

The methods combine analytic combinatorics, singularity analysis, and conditioned Galton–Watson theory. In this perspective, heavy-tailed offspring laws lead to stable limits and condensation, while exponential tails lead to CRT scaling. Probabilistically, the framework is connected to enriched-tree methods of Stufler and to stable-tree and condensation results associated with Kortchemski and Janson, all of which are explicitly cited as part of the methodological backbone of the subject (Fleurat et al., 2023, Salvy, 2024).

Decorations can change the universality class without destroying the block-tree paradigm. Tree-rooted maps weighted by their number of $2$15-connected blocks provide a notable example. There the same block decomposition yields a phase transition, but the singular exponents become $2$16, $2$17, and $2$18, the largest blocks are of order $2$19 at criticality rather than $2$20, and the scaling limit is the Brownian CRT for all $2$21, with rescalings $2$22 at criticality and $2$23 in the supercritical regime (Albenque et al., 2024). This suggests that block weighting is robust as a mechanism, but the decoration of blocks can alter both singular exponents and global geometry.

6. Duality, critical block proliferation, and related directions

A later line of work interprets the critical point of block weighting as a discrete realization of Liouville quantum duality. In the universal substitution form

$2$24

assume that the single-block generating function has a dominant singularity of exponent $2$25. Then the string susceptibility exponents are

$2$26

and they satisfy

$2$27

For quadrangulations decomposed into simple blocks, $2$28, $2$29, $2$30, and $2$31. In that critical regime, global distances scale as $2$32, while distances inside a block scale as $2$33 (Duplantier et al., 16 Jul 2025).

This duality program extends beyond the original map/quadrangulation model to Hamiltonian cycles on cubic or bicubic planar maps decomposed into irreducible blocks and to meandric systems. In each case the block substitution produces the same subcritical/critical/supercritical trichotomy, while the critical exponents match the KPZ relations and the $2$34 duality predicted by Liouville quantum gravity (Duplantier et al., 16 Jul 2025).

Part II develops the critical regime further by deriving conditional laws for the root block. At the dual critical point $2$35, the root-block size given total size $2$36 is finite in the simply rooted model, but in the doubly rooted model it has a macroscopic scaling window of order $2$37 with a universal Wright-function density. Conversely, if the root block size is fixed at $2$38, then the total size scales like $2$39, with Laplace transform

$2$40

The same constant $2$41 governs dual/direct partition-function ratios with punctures and the convolution formula relating the critical block-distance profile to the single-block profile. For quadrangulations, stuffed quartic maps, and bicubic maps, the universal structure is identical and only the non-universal constant $2$42 changes (Duplantier et al., 27 Apr 2026).

Related models clarify what block weighting is not. Face-degree weighted Boltzmann bipartite planar maps, analyzed via the BDG bijection and infinite mobiles, are weighted on faces rather than on $2$43-connected blocks; they have their own local limits, recurrence theory, and one-infinite-face condensation regime, but they belong to a distinct weighting scheme (Björnberg et al., 2013). Likewise, block-weighted planar graphs exhibit the same $2$44, $2$45, and $2$46 enumerative trichotomy, yet differ sharply from maps in the subcritical regime: because the graph decomposition is organized around cut-vertices decorated by sets of derived blocks, subcritical planar graphs have a unique giant $2$47-connected block of asymptotic fraction $2$48, whereas subcritical block-weighted planar maps do not (Kang et al., 27 Feb 2026).

Taken together, these results establish block-weighted random planar maps as a family in which decomposition theory, singularity analysis, conditioned Galton–Watson trees, stable processes, and continuum random geometry interact in a particularly explicit way. The model is simultaneously combinatorial, probabilistic, and geometric: the same parameter $2$49 controls enumeration, extremal block sizes, tree structure, and scaling limits, while the critical point organizes both the discrete phase diagram and its continuum interpretation (Fleurat et al., 2023, Salvy, 2024, Duplantier et al., 16 Jul 2025, Duplantier et al., 27 Apr 2026).