Universal Graphon and Permuton Limits

• Universal graphon and permuton limiting objects are analytic and probabilistic structures that capture the asymptotic behavior of large graphs and permutations through subgraph and pattern densities.
• They are characterized by a universal mapping property and unique constructions up to measure-preserving transformations, ensuring consistent limits for convergent sequences.
• Applications span dense combinatorial structures, including scaling limits, phase transitions, and diffusive dynamics, bridging insights from probability, combinatorics, and mathematical physics.

Universal graphon and permuton limiting objects are analytic and probabilistic structures that capture the asymptotic behavior of large graphs and permutations, respectively. These objects provide universal parametrizations for dense combinatorial structures via observables such as subgraph densities or pattern densities. Their universality refers both to their role as terminal limit objects in model-theoretic frameworks and to their appearance as scaling limits in a broad spectrum of probabilistic models, including striking universality phenomena in substitution closed classes, extremal combinatorics, phase transitions, and heavy-tailed regimes.

1. Formal Definitions and Universal Properties

A graphon is a symmetric measurable function $W:[0,1]^2 \to [0,1]$, defined up to measure-preserving transformations, representing a continuum limit of finite graphs. The space of graphons modulo measure-preserving bijections forms a compact metric space under the cut-metric $\delta_\square$ (Féray et al., 2017, Coregliano et al., 2019).

A permuton is a Borel probability measure $\mu$ on $[0,1]^2$ with uniform marginals. Permutons capture the global limit of permutations by encoding convergence of normalized empirical diagrams or pattern densities (Féray et al., 2017, Coregliano et al., 2019).

Both structures are universal limit objects for their respective classes:

• Any convergent sequence of graphs (resp. permutations) admits a unique graphon (resp. permuton) limit up to the relevant equivalence.
• These limits possess a universal mapping property in the model-theoretic sense: every (weak) limit object is an image of the universal one under some measurable, structure-preserving map.
• Uniqueness is characterized by the convergence of all observable densities and is up to measure-preserving transformations (Coregliano et al., 2019).

2. Canonical Constructions and Characterizations

Graphons

Fundamental results due to Lovász and Szegedy and extended in model-theoretic work show:

• For any convergent sequence $(G_n)$ of dense graphs, there exists a unique graphon $W$ such that for every finite graph $H$, the subgraph density $t(H,G_n)$ converges to $t(H,W)$, where

$$t(H,W) = \int_{[0,1]^{|V(H)|}} \prod_{\{u,v\}\in E(H)} W(x_u,x_v) dx_1 \ldots dx_{|V(H)|}$$

• Existence and uniqueness extend to more general dense combinatorial objects through the framework of first-order theories and exchangeable array cryptomorphisms (Coregliano et al., 2019).

Permutons

For permutations:

• Any sequence of permutations whose pattern densities converge admits a unique permuton limit (Féray et al., 2017, Coregliano et al., 2019).
• Pattern density $t_\sigma(\mu)$ for a pattern $\sigma$ of size $k$ is defined as

$t_\sigma(\mu) = \Pr[\text{%%%%0%%%% i.i.d. %%%%1%%%% induce %%%%2%%%%}]$

• Similar model-theoretic and cryptomorphic uniqueness characterizations apply.

3. Universal and Natural Limiting Objects

Several classes of combinatorial structures exhibit universal limiting behavior, which is captured by specific graphon and permuton distributions:

Biased Brownian Separable Permuton

For proper substitution-closed permutation classes, the limiting object is the biased Brownian separable permuton $\mu^{(p)}$ (Bassino et al., 2017, Borga, 2021). This object arises universally under mild analytic assumptions on the generating series of simple permutations:

• For each class, the bias parameter $p$ is computed from the structural generating series and pattern counts:

$$p = \frac{(1+\kappa)^3\mathrm{Occ}_{12}(\kappa)+1}{(1+\kappa)^3(\mathrm{Occ}_{12}(\kappa)+\mathrm{Occ}_{21}(\kappa))+2}$$

where $\kappa$ solves $S'(\kappa) = 2/(1+\kappa)^2-1$.

• When $\mathrm{Occ}_{12} = \mathrm{Occ}_{21}$, $p=1/2$ and the limit is the Brownian separable permuton, universal for symmetric classes.
• The empirical measure of a random permutation in these classes converges in distribution to $\mu^{(p)}$ (Bassino et al., 2017, Borga, 2021).

Other Universal Permuton Limits

• Baxter Permuton: Scaling limits of Baxter permutations converge to the Baxter permuton, constructed via a flow induced from a two-dimensional Brownian excursion of correlation $-1/2$ (Borga, 2021).
• Skew Brownian Permuton: This two-parameter family $\mu_{\rho,q}$ encompasses diverse pattern-avoiding classes and is realized as a scaling limit in settings connected to SLE and Liouville quantum gravity (LQG) (Borga et al., 2024).
• Meandric Permuton: The conjectural universal limit for meandric permutations, represented as the permuton of independent whole-plane SLE$_8$ on certain LQG spheres (Borga et al., 2024).

Poisson–Dirichlet Graphons and Permutons

A new universal two-parameter family arises in heavy-tailed compositional regimes:

• PD(α,θ; L_H, L_C) Graphon/Permuton: Built by sampling block sizes from PD(α,θ), assigning structures from law $L_C$ to diagonal blocks and $L_H$ or $μ_H$ to off-diagonal blocks or coordinate maps. These objects interpolate between purely atomic and purely continuous limits, generalizing the Brownian separable regime (Stufler, 20 Jan 2026).

4. Structural Universality and Extremal Consequences

Universality also manifests in the theory of finitely forcible graphons:

• Every graphon is a subgraphon of some finitely forcible graphon occupying a fixed fraction of the unit square (Cooper et al., 2017).
• Finitely forcible graphons, though topologically meager, may encode arbitrarily complex structure. As optimizers of finite extremal problems, they provide universal "carriers" for graphon-data.
• This result dismisses the conjecture that finitely forcible graphons always have simple or low-dimensional structure.

Limiting Object	Regime/Class	Construction Principle
$\mu^{(p)}$ (Biased Brownian sep.)	Substitution-closed, proper classes	CRT + sign-labeled subtree encoding, generating series data
PD$(\alpha,\theta;L_H,L_C)$ graphon	Heavy-tailed supergraph/superpermutation regime	Poisson–Dirichlet block process with combinatorial glueing
Finitely forcible graphons	Arbitrary graphon as subgraphon	Forcing by subgraph densities, partitioned tile encoding
Skew Brownian $\mu_{\rho,q}$	Pattern-avoidance/SLE-LQG scaling limits	Planar SLE, LQG, Brownian-type SDEs
Baxter permuton	Baxter permutations	Brownian excursion SDE, flows

5. Scaling Limits, Dynamics, and Diffusive Universality

Beyond static limits, Feller diffusions on the spaces of graphons and permutons provide dynamic universality (Féray et al., 23 Dec 2025):

• Markov up-down chains on finite permutations or graphs yield scaling limits which are Feller diffusions in the permuton or graphon space.
• The stationary laws are the recursive separable permuton and recursive cographon, which connect to the static universal objects in classical theory.
• These diffusions are integrable, with explicit diagonalizable generators in the test algebra (pattern or subgraph densities) and explicit expressions for rates of convergence to equilibrium.
• The separation distances to stationarity reveal connections to the Dedekind eta function and modular forms.

6. Generalizations, Context, and Open Directions

Universal graphons and permutons are instances of more general mod-Gaussian moduli spaces, providing a unified fluctuation theory for random combinatorial objects (Féray et al., 2017). They serve as parameters indexing natural probability spaces for large-scale combinatorial models. Further generalizations encompass:

• Weak and strong limit objects for a broad class of first-order structures ("theons"), including hypergraphs, colored graphs, posets, digraphs, etc. (Coregliano et al., 2019).
• Existence, uniqueness, and removal lemmas extend uniformly across these domains by leveraging exchangeability and algebraic representation theorems.
• Open problems include the classification of singular loci (parameters where generic fluctuation theory breaks down), the extension to more general model-theoretic and dynamical settings, and the full identification of universality classes for new combinatorial regimes.

Universal graphon and permuton limits provide a robust, unifying framework for analyzing, representing, and classifying both deterministic and random large combinatorial structures, synthesizing insights from probability, combinatorics, model theory, and mathematical physics.