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Benedikt Stufler

Publications and source records attributed to Benedikt Stufler.

At least 19 recordsLinked to original sources
arXiv2026

Non-bijective scaling limits and phase transitions of planar maps

We prove that the uniform random non-separable planar map with $n$ edges admits the Brownian sphere as Gromov--Hausdorff--Prokhorov scaling limit as $n$ tends to infinity. Our proof introduces a non-bijective ``common-core transfer method'' that constitutes a novel and universal proof strategy for scaling limits of random discrete structures. As an application, we complete the phase diagram for limiting shapes of block-weighted planar maps by Stufler~(2019). We describe phases with limits given by the Brownian sphere, stable trees, and Brownian sphere decorated stable trees recently introduced by S{é}nizergues, Stef{á}nsson and Stufler~(2023).

math.PR↗
arXiv2026

Limit laws of random simplex tree-child networks

We prove that the longer and shorter Sackin indices of a uniformly random simplex tree-child network with $n$ taxa admit joint distributional limits after rescaling by $n^{-7/4}$. The limiting distributions are described by functionals of a Brownian excursion. We also identify the limiting law of the height after rescaling by $n^{-3/4}$, thereby answering a question of Zhang~(2022). Moreover, we establish sharp tail bounds for the height, which imply convergence of all moments in the above distributional limits. We further obtain a scaling limit for the entire height profile of the leaves. Finally, we determine the local limits of large simplex networks around the fixed root, a uniformly random vertex, and a uniformly random leaf.

math.CO↗
arXiv2026

Uniform integrability of the distance to the nearest leaf in random trees

We study the distance from the root to the nearest leaf, the analogous quantity for a uniformly chosen vertex, and its protection number, in size-conditioned simply generated trees. We prove a uniform exponential tail bound for each of these quantities, valid for arbitrary offspring distributions. As a consequence, these random variables are uniformly integrable of every order. This yields convergence of all moments to those of the corresponding local limit. The argument is probabilistic and unified across the three quantities.

math.PR↗
arXiv2026

Scaling limits of multitype Bienaymé trees

We consider critical multitype Bienaymé trees that are either irreducible or possess a critical irreducible component with attached subcritical components. These trees are studied under two distinct conditioning frameworks: first, conditioning on the value of a linear combination of the numbers of vertices of given types; and second, conditioning on the precise number of vertices belonging to a selected subset of types. We prove that, under a finite exponential moment condition, the scaling limit as the tree size tends to infinity is given by the Brownian Continuum Random Tree. Additionally, we establish strong nonasymptotic tail bounds for the height of such trees. Our main tools include a flattening operation applied to multitype trees and sharp estimates regarding the structure of monotype trees with a given sequence of degrees.

math.PR↗
arXiv2026

Poisson-Dirichlet graphons and permutons

We introduce classes of supergraphs and superpermutations with novel universal graphon and permuton limiting objects whose construction involves the two-parameter Poisson-Dirichlet process introduced by Pitman and Yor (1997). We demonstrate the universality of these limiting objects through general invariance principles in a heavy-tailed regime and establish a comprehensive phase diagram for the asymptotic shape of superstructures.

math.PR↗
arXiv2026

Probabilistic enumeration and equivalence of nonisomorphic trees

We present a new probabilistic proof of Otter's asymptotic formula for the number of unlabelled trees with a given number of vertices. We additionally prove a new approximation result, showing that the total variation distance between random Pólya trees and random unlabelled trees tends to zero when the number of vertices tends to infinity. In order to demonstrate that our approach is not restricted to trees we extend our results to tree-like classes of graphs.

math.CO↗
arXiv2025

Gibbs partitions and lattice paths

This work is devoted to the analysis of a Gibbs partition model, also known as a composition scheme. We consider a natural new condition on the component weights. It leads to a new behavior for the total number of components. We discover a condensation phenomenon, producing a unique giant component comprising almost the entire mass. Additionally, we prove a point process limit describing the asymptotic size of the non-maximal components exhibiting a sublinear power-law growth. A particular motivation for our article stems from applications, ranging from simple random walks in the cube, over lattice paths models in the plane, pairs of directed random walks, over to urn models and card guessing games.

math.CO↗
arXiv2025

The scaling limit of random 2-connected series-parallel maps

A finite graph embedded in the plane is called a series-parallel map if it can be obtained from a finite tree by repeatedly subdividing and doubling edges. We study the scaling limit of weighted random two-connected series-parallel maps with $n$ edges and show that under some integrability conditions on these weights, the maps with distances rescaled by a factor $n^{-1/2}$ converge to a constant multiple of Aldous' continuum random tree (CRT) in the Gromov--Hausdorff sense. The proof relies on a bijection between a set of trees with $n$ leaves and a set of series-parallel maps with $n$ edges, which enables one to compare geodesics in the maps and in the corresponding trees via a Markov chain argument introduced by Curien, Haas and Kortchemski (2015).

math.PR↗
arXiv2024

Poisson-Dirichlet scaling limits of Kemp's supertrees

We determine the Gromov--Hausdorff--Prokhorov scaling limits and local limits of Kemp's $d$-dimensional binary trees and other models of supertrees. The limits exhibit a root vertex with infinite degree and are constructed by rescaling infinitely many independent stable trees or other spaces according to a function of a two-parameter Poisson--Dirichlet process and gluing them together at their roots. We discuss universality aspects of random spaces constructed in this fashion and sketch a phase diagram.

math.PR↗
arXiv2022

Gibbs partitions: a comprehensive phase diagram

We study Gibbs partition models, also known as composition schemes. Our main results comprehensively describe their phase diagram, including a phase transition from the convergent case described in Stufler (2018, Random Structures \& Algorithms) to a new dense regime characterized by a linear number of components with fluctuations of smaller order quantified by an $α$-stable law for $1< α\le 2$. We prove a functional scaling limit for a process whose jumps correspond to the component sizes and discuss applications to extremal component sizes. At the transition we observe a mixture of the two asymptotic shapes. We also treat extended composition schemes and prove a local limit theorem in a dilute regime with the limiting law being related to an $α$-stable law for $0< α< 1$. We describe the asymptotic size of the largest components via a point process limit.

math.PR↗
arXiv2022

Decorated stable trees

We define decorated $α$-stable trees which are informally obtained from an $α$-stable tree by blowing up its branchpoints into random metric spaces. This generalizes the $α$-stable looptrees of Curien and Kortchemski, where those metric spaces are just deterministic circles. We provide different constructions for these objects, which allows us to understand some of their geometric properties, including compactness, Hausdorff dimension and self-similarity in distribution. We prove an invariance principle which states that under some conditions, analogous discrete objects, random decorated discrete trees, converge in the scaling limit to decorated $α$-stable trees. We mention a few examples where those objects appear in the context of random trees and planar maps, and we expect them to naturally arise in many more cases.

math.PR↗
arXiv2022

First-passage percolation on random simple triangulations

We study first-passage percolation on random simple triangulations and their dual maps with independent identically distributed link weights. Our main result shows that the first-passage percolation distance concentrates in an $o_p(n^{1/4})$ window around a constant multiple of the graph distance.

math.PR↗
arXiv2022

The scaling limit of random cubic planar graphs

We study the random simple connected cubic planar graph $\mathsf{C}_n$ with an even number $n$ of vertices. We show that the Brownian map arises as Gromov--Hausdorff--Prokhorov scaling limit of $\mathsf{C}_n$ as $n \in 2 \ndN$ tends to infinity, after rescaling distances by $γn^{-1/4} $ for a specific constant $γ>0$.

math.PR↗
arXiv2022

The Uniform Infinite Cubic Planar Graph

We prove that the random simple cubic planar graph $\mathsf{C}_n$ with an even number $n$ of vertices admits a novel uniform infinite cubic planar graph (UICPG) as quenched local limit. We describe how the limit may be constructed by a series of random blow-up operations applied to the dual map of the type~III Uniform Infinite Planar Triangulation established by Angel and Schramm (Comm. Math. Phys., 2003). Our main technical lemma is a contiguity relation between $\mathsf{C}_n$ and a model where the networks inserted at the links of the largest $3$-connected component of $\mathsf{C}_n$ are replaced by independent copies of a specific Boltzmann network. We prove that the number of vertices of the largest $3$-connected component concentrates at $κn$ for $κ\approx 0.85085$, with Airy-type fluctuations of order $n^{2/3}$. The second-largest component is shown to have significantly smaller size $O_p(n^{2/3})$.

math.PR↗
arXiv2022

Mass and radius of balls in Gromov-Hausdorff-Prokhorov convergent sequences

We survey some properties of Gromov--Hausdorff--Prokhorov convergent sequences $(\mathsf{X}_n, d_{\mathsf{X}_n}, ν_{\mathsf{X}_n})_{n \ge 1}$ of random compact metric spaces equipped with Borel probability measures. We formalize that if the limit is almost surely non-atomic, then for large $n$ each open ball in $\mathsf{X}_n$ with small radius must have small mass. Conversely, if the limit is almost surely fully supported, then each closed ball in $\mathsf{X}_n$ with small mass must have small radius. We do not claim any new results, but justifications are provided for properties for which we could not find explicit references.

math.PR↗