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arxiv: 2605.06471 · v1 · submitted 2026-05-07 · 🧮 math.CO · math.PR

Leap generators for composition schemes

\'Eric Fusy , Carine Pivoteau This is my paper

Pith reviewed 2026-05-08 08:14 UTC · model grok-4.3

classification 🧮 math.CO math.PR
keywords leap generatorscomposition schemesrandom generationasymptotic uniformitysupercritical regimePólya treesplanar mapsexact-size sampling
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The pith

Leap generators extend exact-size random sampling from sequences to supercritical composition schemes while achieving asymptotic uniformity.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends leap generators, first introduced for sequence constructions, to general supercritical composition schemes of the form C equals A composed with B. These generators produce exact-size samples from C in linear time, provided the component generators for A and B meet basic efficiency conditions. The resulting distribution, called the leap distribution, is not perfectly uniform but approaches the uniform distribution on structures of size n, with total variation distance decaying as (c plus little-o of 1) times n to the minus one-half for an explicit constant c. The method applies directly to several families of walks and trees, including Pólya trees, and extends in modified form to certain critical composition schemes arising in planar maps, where the distance instead behaves like c times n to the minus one-third.

Core claim

Leap generators for supercritical composition schemes C = A ∘ B produce exact-size random structures according to a leap distribution whose total variation distance to the uniform distribution on C_n is (c + o(1)) n^{-1/2} for an explicit constant c, while retaining linear time complexity under conditions on the sampling complexity of A and B.

What carries the argument

The leap generator, which adapts recursive sampling by allowing controlled leaps over intermediate sizes in the composition A ∘ B.

If this is right

  • Exact-size sampling becomes practical for Pólya trees and other tree families that admit supercritical decompositions.
  • Linear-time generation carries over to many classes of walks and trees built by composition.
  • Certain critical composition schemes for planar maps also admit leap generators, but with total variation distance decaying like n to the power -1/3.
  • The generators remain simple to implement once sub-generators for A and B are available.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same leap mechanism could be tested on nested compositions beyond a single outer layer to see whether the error rate remains controlled.
  • For applications where n is large, the n^{-1/2} error term may already be negligible enough that leap generators replace exact uniform samplers without visible bias.
  • The explicit constant c in the error bound offers a way to compare the quality of leap generators across different combinatorial classes.

Load-bearing premise

The composition scheme must be supercritical and the generators for the inner and outer classes must themselves run in linear or better time.

What would settle it

A concrete supercritical composition scheme in which the total variation distance between the leap distribution and the uniform distribution on size-n objects fails to decay proportionally to n to the power -1/2.

Figures

Figures reproduced from arXiv: 2605.06471 by Carine Pivoteau, \'Eric Fusy.

Figure 1
Figure 1. Figure 1: The leap generator process. For γ drawn under P C x , the expectation µ(x) and standard deviation σ(x) of the random variable |γ| are given by (4) µ(x) = xC′ (x) C(x) , σ(x) = p xµ′(x). A Boltzmann sampler [16] is a procedure ΓC(x) that draws a random object in C under the distribution P C x , for any fixed x > 0 such that C(x) converges. It is said to be of linear time complexity if the runtime of every g… view at source ↗
Figure 2
Figure 2. Figure 2: Left: a rooted tree (labels not indicated) endowed with an automorphism. Right: decomposition into a core (the subtree made of the fixed vertices) where at each vertex is attached a rooted tree endowed with an automorphism fixing only its root-vertex. 4.2.1. P´olya trees. Recall that a P´olya tree (resp. a rooted Cayley tree) is a rooted unlabeled (resp. labeled) tree where the children of nodes are not or… view at source ↗
Figure 3
Figure 3. Figure 3: Algorithm ΓB(x) for P´olya trees Algorithm ΓB(x) : if Bern(x/B(x)) then return Z else γ ← ΓA˜(x 2 ) return [Z, [γ, γ]] view at source ↗
Figure 4
Figure 4. Figure 4: Algorithm ΓB(x) for phylogenetic trees P´olya theory ensures that B(z) is the exponential generating function of B; in￾deed the cycle index sum of sets endowed with a permutation with no fixed point is exp(P i≥2 1 i si). The composition scheme is known to be admissible, with ρC ≈ 0.338 and ρB = ρ 1/2 C . The Boltzmann sampler ΓB(x) (to be called at ρC in the leap generation process) is deduced from the kno… view at source ↗
Figure 5
Figure 5. Figure 5: Left: a rooted map γ. Middle: γ is decomposed into a core (block containing the root-edge) where at each cor￾ner is attached a (possibly empty) rooted map. Right: the col￾lection of blocks of γ. The block sizes in decreasing order are 5, 4, 3, 2, 2, 1, 1, 1, 1, 1. of (possibly empty) rooted maps, one attached at each corner of the root-block, see view at source ↗
Figure 6
Figure 6. Figure 6: Plot of dn := dTV(πn, π′ n ) (red) and d rej n := dTV(πn, πrej n ) (blue) for Motzkin walks for n from 10 to 1000. 5. Experiments We have conducted some experiments to verify closeness of the leap distribution to the uniform distribution in practice. We start with some exact experiments on Motzkin walks, for which total vari￾ance distance can be exactly computed up to large sizes (in particular it takes ad… view at source ↗
Figure 7
Figure 7. Figure 7: (a) Plot of n 1/2dn for Motzkin walks, for n from 100 to 5000. (b) The part of the plot for n from 900 to 1000, showing fluctuations of period 3 view at source ↗
Figure 8
Figure 8. Figure 8: Plot of n drej n for Motzkin walks, for n from 100 to 5000. The horizontal axis is at the height 0.09074 predicted by Proposition 9 as the limit view at source ↗
Figure 9
Figure 9. Figure 9: Core-size of Motzkin walks of size n = 5000: for k in the vicinity of n/µ = 5000/3, (a) shows the plots of πn(k) (blue), π ′ n (k) (red) and π rej n (k) (gray, indistinguishable from πn(k)), su￾perimposed with the gaussian limit density function, (b) the plot of dn,k − 1, (c) the plot of π ′ n (k) − πn(k), whose total area equals 2 dn, (d) the plot of d rej n,k − 1, (e) the plot of π rej n (k) − πn(k), who… view at source ↗
Figure 10
Figure 10. Figure 10: Total variation distance for the height of Motzkin walks: plots of ˆdn := dTV(ˆπn, πˆ ′ n ) (red) and ˆd rej n := dTV(ˆπn, πˆ rej n ) (blue), for n from 10 to 1000. (a) (b) view at source ↗
Figure 11
Figure 11. Figure 11: Total variation distance for the height of Motzkin walks: (a) plot of n ˆdn, (b) plot of n 2 ˆd rej n , both for n from 10 to 1000 view at source ↗
Figure 12
Figure 12. Figure 12: Height in random Motzkin walks of size n = 1000: (a) gives the (nearly indistinguishable) plots of ˆπn(h) (blue), ˆπ ′ n (h) (red), and ˆπ rej n (h) (gray), together with the limit curve; (b) the plot of ˆπ ′ n (h) − πˆn(h), whose total area equals 2 ˆdn (c) the plot of πˆ rej n (h) − πˆn(h), whose total area equals 2 ˆd rej n view at source ↗
Figure 13
Figure 13. Figure 13: Histogram (size 1000 with 106 samples) for the height of random P´olya trees under the uniform distribution (red) and the leap distribution (blue), superimposed with the limit curve. The difference between the two histograms is shown below. On Motzkin walks we can also compute the total variation distance for the height parameter, using the fact that the height of a Motzkin walk equals the height of its D… view at source ↗
Figure 14
Figure 14. Figure 14: Histogram (size 1000 with 106 samples) for the height of random phylogenetic trees under the uniform distribution (red) and the leap distribution (blue), superimposed with the limit curve. The difference between the two histograms is shown below view at source ↗
Figure 15
Figure 15. Figure 15: Same as view at source ↗
Figure 16
Figure 16. Figure 16: Histograms for the height in random phylogenetic trees under the leap distribution, in sizes 500, 1000, 5000, 10000 from top to bottom, with 106 samples in each case. The non￾cumulative (resp. cumulative) histograms are given in the left (resp. right) column. In each case, the histogram is superimposed with the limit density (resp. cumulative) function colored red view at source ↗
Figure 17
Figure 17. Figure 17: Histograms for the average path-length in random P´olya trees under the uniform distribution (blue) and the leap dis￾tribution (red), in size 103 with 106 samples, superimposed with the limit density function. Samples are grouped into buckets ac￾cording to the floor of the average path length. The difference between the two histograms is shown below view at source ↗
Figure 18
Figure 18. Figure 18: Histograms for the number of leaves in random P´olya trees (top) and cherries in random phylogenetic trees (bottom) of size 103 with 106 samples, for the uniform distribution (blue) and the leap distribution (red), superimposed with the gaussian limit density function view at source ↗
read the original abstract

Leap generators have been introduced in [Duchon et al.'04] for exact-size random generation of structures in a class of the form $\mathcal{C}=\mathrm{Seq}(\mathcal{B})$ (sequence construction), in the supercritical case. We extend these generators to supercritical composition schemes $\mathcal{C}=\mathcal{A}\circ\mathcal{B}$. Compared to the sequence construction, the obtained exact-size random generator for $\mathcal{C}$ still has linear time complexity (under conditions on the sampling complexity in $\mathcal{A}$ and $\mathcal{B}$), but perfect uniformity of the distribution is lost in general. However the distribution on $\mathcal{C}_n$, called leap distribution, is asymptotically uniform, the total variation distance from the uniform distribution being $(c+o(1))n^{-1/2}$ for an explicit constant $c$. These generators are simple to implement and can be applied to several classes of walks and trees, in particular P\'olya trees. Leap generators can also be given for certain critical composition schemes, those relating planar map families, where this time the total variation distance to the uniform distribution is $\sim c\,n^{-1/3}$ for an explicit constant $c$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript extends leap generators, previously defined for supercritical sequence schemes Seq(B), to general supercritical composition schemes C = A ∘ B. The resulting exact-size generators for C_n run in linear time (under conditions on the sampling oracles for A and B) and produce the leap distribution, which is shown to be asymptotically uniform on C_n with total variation distance (c + o(1)) n^{-1/2} to the uniform distribution for an explicit constant c. Applications to walks, trees, and Pólya trees are given, together with an extension to selected critical composition schemes arising in planar maps, where the distance becomes ∼ c n^{-1/3}.

Significance. If the proofs hold, the work supplies a practical, nearly uniform linear-time sampler for a broad family of combinatorial classes defined by composition, together with an explicit error term. The explicit constant c and the concrete applications (especially Pólya trees) are clear strengths; the restriction to the supercritical regime (plus selected critical map cases) is stated cleanly.

minor comments (3)
  1. [Abstract] The abstract asserts an 'explicit constant c' but neither states its closed form nor points to the equation or theorem where it is derived; adding this reference would improve immediate readability.
  2. [Section 2] The precise hypotheses on the sampling complexities of the oracles for A and B are invoked repeatedly but are not collected in a single displayed statement; a boxed list of the required complexity bounds would help readers verify the linear-time claim for the listed applications.
  3. [Section 5] In the applications to Pólya trees, the verification that the required sampling oracles satisfy the stated complexity conditions is only sketched; a short explicit check (e.g., citing the known linear-time Boltzmann sampler for Pólya trees) would strengthen the claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, the recognition of the practical value of the leap generators, the explicit error term, and the applications (particularly to Pólya trees). The recommendation for minor revision is noted. No major comments appear in the report, so we have no point-by-point rebuttals to offer. Any minor issues will be handled in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via singularity analysis

full rationale

The paper extends the leap generator construction from the sequence case in the external reference [Duchon et al. 2004] to supercritical composition schemes C = A ∘ B. The claimed linear-time exact-size sampling and the total-variation bound (c + o(1)) n^{-1/2} to uniformity are obtained by applying standard singularity analysis and local limit theorems to the bivariate generating functions of the composition; these steps do not reduce to any self-definition, fitted parameter renamed as prediction, or load-bearing self-citation. The uniformity result is explicitly restricted to the supercritical regime (and selected critical map cases), and the complexity claim is conditioned on the oracles for A and B, keeping the argument independent of its own outputs. No ansatz is smuggled and no known empirical pattern is merely renamed.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, new entities, or non-standard axioms are stated. The work relies on the analytic-combinatorics framework of the cited prior paper.

axioms (1)
  • domain assumption Standard assumptions of analytic combinatorics for supercritical and critical composition schemes
    The extension and error bounds presuppose the growth and singularity analysis developed in Duchon et al. 2004 and related literature.

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