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Serte Donderwinkel

Publications and source records attributed to Serte Donderwinkel.

At least 19 recordsLinked to original sources
arXiv2026

Convergence and compactness of discrete aggregation trees

We study a class of random aggregation trees that generalizes the discrete stick-breaking construction of the uniform labelled tree (Aldous, 1991). To sample the tree of size $n$, vertices are added sequentially, with the $i$th vertex starting a new branch with a prescribed probability $f(n,i)$; otherwise, it extends the current branch. We also define a continuum analogue by replacing the Poisson point process of intensity $tdt$ in the construction of the Brownian continuum random tree by one of intensity $f(t)dt$. We establish scaling limits for two families of discrete aggregation trees in the Gromov-Hausdorff-Prokhorov topology. When $f(n,i)=(i/n)^β$, with $β>0$, after rescaling the graph distance by $n^{-β/(β+1)}$, the random tree converges to the compact aggregation tree with $f(t)=t^β$. This recovers convergence to the Brownian continuum random tree when $β=1$ (Aldous, 1991), as well as scaling limits of choice spanning trees for integer $β$ (Archer and Shalev, 2024). We also prove convergence under rescaling to the compact aggregation tree with $f(t)=\log^γ(1+t)$ for every $γ>1$. Finally, we identify necessary conditions for compactness. Consequently, the threshold $γ>1$ in the logarithmic family is sharp. These results provide insight into an open problem on compactness criteria for random aggregation trees (Curien and Haas, 2014).

math.PR↗
arXiv2026

On the Fragility of Majority Illusions

A majority illusion in a social network occurs when the majority of neighbors of an agent has a certain opinion while the majority of agents in the network has another opinion. We study the fragility of majority illusions, that is, whether illusions persist as a result of changes in the underlying network. We consider two settings. First, we study networks where agents have opinions that change over time and find that majority illusions disappear under majority updates. Second, we study sequences of large random graphs of which the size increases, and show that the likelihood of majority illusions goes to zero.

cs.SI↗
arXiv2026

The number and structure of connected graphs with a fixed degree sequence

We study connected graphs with a fixed degree sequence, in the sparse setting where the number of edges grows linearly in the number of vertices. Using the relation to the configuration model, we identify the number of such connected graphs up to the exponential order. We do this by viewing a connected graph with a given degree distribution as the realization of the giant component in a larger configuration model, and carefully choosing the degree distribution of the larger graph so that it is likely that its giant component has the required degree distribution. To ensure that the connected graph has exactly the correct degrees, we use a switching argument. Additionally, we obtain results on rare event probabilities and describe the local structure of a uniform connected graph with a fixed degree sequence.

math.CO↗
arXiv2025

The largest common subtree of two random trees

We study the size and structure of the largest common subtree (LCS) between two independent Bienaymé trees conditioned to have size $n$. When the trees are critical with finite $2$nd and $(2+κ)$th moment respectively for some $κ>0$, we prove that the LCS has size of order $\sqrt{n}$, and is approximated by the length of three paths meeting at a central node. Moreover, we show that the largest common subtree between two critical independent Bienaymé trees with size $n$ and finite second moments may be much larger than $\sqrt{n}$, implying that our result is tight. We also pose a number of open questions and suggestions for future research.

math.PR↗
arXiv2025

Evolution of recursive trees with limited memory

Motivated by questions in social networks, distributed computing and probabilistic combinatorics, the last few years have seen increasing interest in network evolution models where new vertices entering the system need to make decisions based on a partial snapshot of the current state of the network. This paper considers a specific variant of the classical random recursive tree dynamics, where a vertex at time $n+1$ has information only on those vertices that have arrived in the interval $[j(n), n]$ for a sequence $j(n) \uparrow \infty$, and connects to vertices uniformly at random amongst this set. We consider two different regimes on the density information, termed macroscopic and mesoscopic regimes, which respectively correspond to $j(n)=θn$ for some $θ\in (0,1)$, and $j(n)=n-n^β$ for some $β\in (0,1)$. Our main interest is in studying asymptotics of various local and global functionals of the network. We show that in the macroscopic regime, the local limit is expressed in terms of an associated continuous time branching process that depends on the parameter $θ$, while it is a $\mathrm{Poisson}(1)$-branching process in the mesoscopic regime for any $β\in (0,1)$. Furthermore, the height of the macroscopic tree is logarithmic, which we prove exploiting a connection with scaled-attachment random recursive trees (SARRTs) as studied by Devroye, Fawzi and Fraiman (RSA 2011), while it is polynomial in the mesoscopic regime; our argument in this latter case relies on a differential equation approach to track the ancestor indices of late-coming vertices, together with a multiscale analysis. Further, we develop an exploration algorithm to simultaneously reveal the ancestral path of youngest vertices. Using this algorithm, we show that in the mesoscopic regime, the global structure experiences a phase transition at $β=1/2$.

math.PR↗
arXiv2025

Revisiting scaling limits for critical inhomogeneous random graphs with finite third moments

We consider the rank-1 inhomogeneous random graph in the Brownian regime in the critical window. Aldous studied the weights of the components, and showed that this ordered sequence converges in the $\ell^2$-topology to the ordered excursions of a Brownian motion with parabolic drift when appropriately rescaled (http://doi.org/10.1214/aop/1024404421), as the number of vertices $n$ tends to infinity. We show that, under the finite third moment condition, the same conclusion holds for the ordered component sizes. This in particular proves a result claimed by Bhamidi, Van der Hofstad and Van Leeuwaarden (https://doi.org/10.1214/EJP.v15-817). We also show that, for the large components, the ranking by component weights coincides with the ranking by component sizes with high probability as $n \to \infty$.

math.PR↗
arXiv2025

To see the forest for the trees: On the infinite divisibility of unlabeled forests

Inspired by Stufler's recent probabilistic proof of Otter's asymptotic number of unlabeled trees, we revisit work of Palmer and Schwenk, and study unlabeled forests from a probabilistic point of view. We show that the number of trees in a random forest converges, with all of its moments, to a shifted compound Poisson. We also find the asymptotic proportion of forests that are trees. The key fact is that the number of trees $t_n$ and forests $f_n$ are related by a Lévy process. As such, the results by Palmer and Schwenk follow by an earlier and far-reaching limit theory by Hawkes and Jenkins. We also show how this limit theory implies results by Schwenk and by Meir and Moon, related to degrees in large random trees. Our arguments apply, more generally, to the enumeration of sub-exponentially weighted integer partitions, or, in fact, any setting where the underlying Lévy process follows the one big jump principle.

math.PR↗
arXiv2025

Discrete snakes with globally centered displacements

We prove a scaling limit for globally centered discrete snakes on size-conditioned critical Bienaymé trees. More specifically, under a global finite variance condition, we prove convergence in the sense of random finite-dimensional distributions of the head of the discrete snake (suitably rescaled) to the head of the Brownian snake driven by a Brownian excursion. When the third moment of the offspring distribution is finite, we further prove uniform functional convergence under a necessary tail condition on the displacements. We also consider displacement distributions with heavier tails, for which we instead obtain convergence to a variant of the hairy snake introduced by Janson and Marckert. We further give two applications of our main result. Firstly, we obtain a scaling limit for the difference between the height process and the Łukasiewicz path of a size-conditioned critical Bienaymé tree. Secondly, we obtain a scaling limit for the difference between the height process of a size-conditioned critical Bienaymé tree and the height process of its associated looptree.

math.PR↗
arXiv2025

Sinaĭ excursions: An analogue of Sparre Andersen's formula for the area process of a random walk

Sinaĭ initiated the study of random walks with persistently positive area processes, motivated by shock waves in solutions to the inviscid Burgers' equation. We find the precise asymptotic probability that the area process of a random walk bridge is an excursion. A key ingredient is an analogue of Sparre Andersen's classical formula. The asymptotics are related to von Sterneck's subset counting formulas from additive number theory. Our results sharpen bounds by Aurzada, Dereich and Lifshits and respond to a question of Caravenna and Deuschel, which arose in their study of the wetting model. In this context, Sina\uı excursions are a class of random polymer chains exhibiting entropic repulsion.

math.PR↗
arXiv2025

Critical trees are neither too short nor too fat

We establish lower tail bounds for the height, and upper tail bounds for the width, of critical size-conditioned Bienaymé trees. Our bounds are optimal at this level of generality. We also obtain precise asymptotics for offspring distributions within the domain of attraction of a Cauchy distribution, under a local regularity assumption. Finally, we pose some questions on the possible asymptotic behaviours of the height and width of critical size-conditioned Bienaymé trees.

math.PR↗
arXiv2025

Temporal connectivity of Random Geometric Graphs

A temporal random geometric graph is a random geometric graph in which all edges are endowed with a uniformly random time-stamp, representing the time of interaction between vertices. In such graphs, paths with increasing time stamps indicate the propagation of information. We determine a threshold for the existence of monotone increasing paths between all pairs of vertices in temporal random geometric graphs. The results reveal that temporal connectivity appears at a significantly larger edge density than simple connectivity of the underlying random geometric graph. This is in contrast with Erdős-Rényi random graphs in which the thresholds for temporal connectivity and simple connectivity are of the same order of magnitude. Our results hold for a family of "soft" random geometric graphs as well as the standard random geometric graph.

math.PR↗
arXiv2025

Tournament score sequences, Erdős-Ginzburg-Ziv numbers, and the Lévy-Khintchine method

We give a short proof of a recent result of Claesson, Dukes, Franklín and Stefánsson, connecting the number $S_n$ of score sequences and the Erdős-Ginzburg-Ziv numbers $N_n$ from additive number theory. Our proof utilizes the lattice path representation of score sequences by Erdős and Moser, and remarks by Kleitman added to an article of Moser regarding cyclic shifts of such paths. The connection between $S_n$ and $N_n$ is an instance of the Lévy-Khintchine formula from probability theory. We highlight the utility of such formulas, by giving a short proof of Moser's conjecture that $S_n\sim C4^n/n^{5/2}$, where $C$ is described in terms of $N_n$.

math.CO↗
arXiv2025

Graphical sequences and plane trees

Balister, the second author, Groenland, Johnston and Scott recently showed that there are asymptotically $C4^n/n^{3/4}$ many unordered sequences that occur as degree sequences of graphs. Combining limit theory for infinitely divisible distributions with a new bijective connection between a class of random walk trajectories and a subset counting formula from additive number theory, we describe $C$ in terms of Walkup's number of rooted plane trees. The bijection is related to an instance of the Lévy-Khintchine formula. Our main result complements a result of Stanley, that ordered graphical sequences are related to quasi-forests.

math.CO↗
arXiv2024

Tight universal bounds on the height times the width of random trees

We obtain assumption-free, non-asymptotic, uniform bounds on the product of the height and the width of uniformly random trees with a given degree sequence, conditioned Bienaymé trees and simply generated trees. We show that for a tree of size $n$, this product is $O(n \log n)$ in probability, answering a question by Addario-Berry (2019). The order of this bound is tight in this generality.

math.PR↗
arXiv2024

Counting graphic sequences via integrated random walks

Given an integer $n$, let $G(n)$ be the number of integer sequences $n-1\ge d_1\ge d_2\ge\dotsb\ge d_n\ge 0$ that are the degree sequence of some graph. We show that $G(n)=(c+o(1))4^n/n^{3/4}$ for some constant $c>0$, improving both the previously best upper and lower bounds by a factor of $n^{1/4}$ (up to polylog-factors). Additionally, we answer a question of Royle, extend the values of $n$ for which the exact value of $G(n)$ is known from $n\le290$ to $n\le 1651$ and determine the asymptotic probability that the integral of a (lazy) simple symmetric random walk bridge remains non-negative.

math.CO↗
arXiv2024

Refined Horton-Strahler numbers I: a discrete bijection

The Horton-Strahler number of a rooted tree $T$ is the height of the tallest complete binary tree that can be homeomorphically embedded in $T$. The number of full binary trees with $n$ internal vertices and Horton-Strahler number $s$ is known to be the same as the number of Dyck paths of length $2n$ whose height $h$ satisfies $\lfloor \log_2(1+h)\rfloor=s$. In this paper, we present a new bijective proof of the above result, that in fact strengthens and refines it as follows. We introduce a sequence of trees $(τ_i,i \ge 0)$ which "interpolates" the complete binary trees, in the sense that $τ_{2^h-1}$ is the complete binary tree of height $h$ for all $h \ge 0$, and $τ_{i+1}$ strictly contains $τ_i$ for all $i \ge 0$. Defining $\mathcal{S}(T)$ to be the largest $i$ for which $τ_i$ can be homeomorphically embedded in $T$, we then show that the number of full binary trees $T$ with $n$ internal vertices and with $\mathcal{S}(T)=h$ is the same as the number of Dyck paths of length $2n$ with height $h$. (We call $\mathcal{S}(T)$ the refined Horton-Strahler number of $T$.) Our proof is bijective and relies on a recursive decomposition of binary trees (resp. Dyck paths) into subtrees with strictly smaller refined Horton-Strahler number (resp. subpaths with strictly smaller height). In a subsequent paper, we will show that the bijection has a continuum analogue, which transforms a Brownian continuum random tree into a Brownian excursion and under which (a continuous analogue of) the refined Horton-Strahler number of the tree becomes the height of the excursion.

math.CO↗
arXiv2024

Tournaments and random walks

We study the relationship between tournaments and random walks. This connection was first observed by Erdős and Moser. Winston and Kleitman came close to showing that $S_n=Θ(4^n/n^{5/2})$. Building on this, and works by Takács, these asymptotic bounds were confirmed by Kim and Pittel. In this work, we verify Moser's conjecture that $S_n\sim C4^n/n^{5/2}$, using limit theory for integrated random walk bridges. Moreover, we show that $C$ can be described in terms of random walks. Combining this with a recent proof and number-theoretic description of $C$ by the second author, we obtain an analogue of Louchard's formula, for the Laplace transform of the squared Brownian excursion/Airy area measure. Finally, we describe the scaling limit of random score sequences, in terms of the Kolmogorov excursions, studied recently by Bär, Duraj and Wachtel. Our results can also be interpreted as answering questions related to a class of random polymers, which began with influential work of Sinaĭ. From this point of view, our methods yield the precise asymptotics of a persistence probability, related to the pinning/wetting models from statistical physics, that was estimated up to constants by Aurzada, Dereich and Lifshits, as conjectured by Caravenna and Deuschel.

math.PR↗