Recursive Separable Permutons

• The paper demonstrates that recursive separable permutons arise as the stationary measure for scaling limits of integrable up–down chains on permutations.
• It employs explicit spectral analysis and triangularity in eigenfunction construction to derive convergence rates and separation distance formulas in the permutation setting.
• The study links recursive separability to ergodic Feller diffusions, offering concrete tools for analyzing mixing times and pattern densities in random permutation limits.

A recursive separable permuton is the stationary measure for a class of Feller diffusions on the compact space of permutons, arising as the scaling limit of explicit integrable up–down Markov chains on permutations. These objects lie at the intersection of combinatorial probability, stochastic processes on permutons, and the spectral theory of Markov chains, encapsulating a universal ergodic structure in random permutation limits governed by recursive separability. Their construction and properties are tightly connected to the theory of up–down chains, intertwining relations, and the diagonalizability of Markov semigroups, with explicit connections to separation distances and special functions such as the Dedekind eta function.

1. Definition and Construction

Recursive separable permutons emerge as stationary distributions of scaling limits of certain integrable up–down chains on finite permutations. An up–down chain is a Markov chain defined by composition of a "size-increasing" ("up") kernel $\mathsf U_n$ from $S_n$ to $S_{n+1}$ and an "erasing" ("down") kernel $\mathsf D_{n+1}$ from $S_{n+1}$ back to $S_n$, so that one step on $S_n$ is $P_n = \mathsf U_n\mathsf D_{n+1}$, i.e.,

$$P_n(x, x') = \sum_{y \in S_{n+1}} \mathsf U_n(x, y) \; \mathsf D_{n+1}(y, x').$$

The family is termed integrable if for all $n\geq1$,

$$\mathsf U_n\mathsf D_{n+1} = \beta_n \mathsf D_n \mathsf U_{n-1} + (1 - \beta_n) I_n,$$

for explicit constants $\beta_n\in(0,1)$ and identity $I_n$. In the permutation setting, the canonical up–down chain uses "inflate" (random insert) and "delete" (random remove) kernels, with

$\beta_n = \frac{n-1}{n+1}.$

Under natural embedding, the scaling limit of these chains as $n\to\infty$ defines a Feller diffusion on the space of permutons, the so-called permuton-valued diffusion. The stationary measure—identified as the recursive separable permuton—is obtained as the unique invariant measure of this limiting semigroup (Féray et al., 23 Dec 2025).

2. Algebraic and Spectral Structure

The spectral theory of integrable up–down chains is highly explicit due to triangularity and a Pieri-type rule for densities. The eigenfunctions of $P_n$ are constructed using pattern densities $d_s(u)$ and pseudo-up operators $U_n$. There are explicit formulas for all eigenvalues and multiplicities: $$P_n h_s |_{S_n} = \Big(1 - \frac{c_{|s|-1}}{c_n}\Big) h_s |_{S_n},$$ for $c_n = \beta_1^{-1}\cdots \beta_n^{-1}$, and $h_s$ as sums over patterns via the up operators. For the permutation chain, the spectrum is completely diagonalizable, enabling closed formulas for the rate of convergence to stationarity. The separation distance for the permutation chain is determined as (see also Fulman): $$\Delta_n(m) = \sum_{k=1}^n (-1)^{k-1}(2k+1)\, \frac{(n-1)!\,n!}{(n-1-k)!\,(n+k)!} \left(1-\frac{k(k+1)}{n(n+1)}\right)^m.$$ In continuous time, this becomes

$$\Delta_n^*(t) = \sum_{k=1}^{n} (-1)^{k-1}(2k+1)\, e^{-t\,k(k+1)}.$$

As $n\to\infty$, these expressions yield precise estimates for convergence in the scaling limit (Féray et al., 23 Dec 2025).

3. Scaling Limits and Permuton-Valued Diffusions

Upon rescaling the up–down chains and embedding in the space of permutons, the scaling limit is a Feller process whose semigroup acts diagonally on the density extensions associated with permutation patterns. Formally, under a suitable injection $\iota$ from finite permutations to the permuton space $\mathcal{E}$, the chain $\iota(X_n(\lfloor c_n t\rfloor))$ converges in distribution (in $D([0,\infty),\mathcal{E})$) to a continuous process $X(t)$ governed by

$Q(t) d_s^o = e^{-t\,c_{|s|-1}} d_s^o,$

where $d_s^o$ denotes the extension of the finite density $d_s$ to the permuton compactum. The generator $\mathcal{L}$ acts by

$$\mathcal{L} d_s^o = -c_{|s|-1}\left(d_s^o - \sum_{|r|=|s|-1} d_r^o \mathsf U_{|r|}(r, s)\right).$$

The unique stationary measure for this semigroup is the recursive separable permuton (Féray et al., 23 Dec 2025).

4. Characterization and Properties

The recursive separable permuton is characterized by the following properties:

• Stationarity: It is the unique invariant measure of the limiting diffusion.
• Ergodicity: Proven via explicit spectral gap calculations. All positive times yield convergence in law to the recursive separable permuton from any initial condition.
• Recursion and Separability: The measure arises as the infinite-size limit of uniformly random separable permutations generated by successively inflating and deleting, reflecting recursive decomposability into smaller separable structures.
• Pattern Densities: The law of the recursive separable permuton is determined by explicit pattern densities, which satisfy a recursive relation dictated by the up–down chain's kernels.
• Explicit Separation Distance: The maximal distance to stationarity (in separation) is given in terms of the eigenvalues, connecting to the Dedekind eta function in continuous scaling.

5. Connections to Related Structures

Recursive separable permutons are linked to a wider landscape of "recursive" or "additive" structures:

• Recursive Cographons: For up–down chains on graphs (cographs), an analogous scaling limit yields recursive cographon-valued Feller diffusions, with stationary law the recursive cographon (Féray et al., 23 Dec 2025).
• Partitions, Trees, Graphs: Similar constructions exist for up–down chains on other combinatorial structures, such as Young diagrams (partitions), unlabelled cladograms, and more, each with their own recursive stationary measure in the scaling limit.
• Intertwiners and Boundary Theory: The transition semigroups for the finite up–down chains intertwine via the down-operators, leading (via the Borodin–Olshanski method) to a Markov process on the boundary (the permutons), with the recursive separable permuton the boundary's distinguished ergodic law (Féray et al., 23 Dec 2025).
• Integrable System Techniques: The Markov chain integrability manifests in explicit spectral analysis, combinatorial density expansions, and semigroup diagonalization.

6. Explicit Examples and Applications

A primary explicit example is the Markov chain on permutations with "inflate-delete" dynamics, for which the recursive separable permuton is the scaling limit. In practice, this measure governs the asymptotic random structure of large separable permutations built by recursive inflation and deletion, with:

• Diagonalizable transition kernel;
• Explicit stationary law;
• Quantitative mixing time estimates.

Applications span from the study of large random permutations and their limiting shapes, to the analysis of ergodic Feller diffusions on spaces of measures, and quantitative bounds for convergence to equilibrium in separable or recursive symmetry classes. The explicit separation distance formulas provide concrete tools for mixing time analysis and Stein's method for permutation-valued observables (Féray et al., 23 Dec 2025).