arXiv subjects

Brett Kolesnik

Publications and source records attributed to Brett Kolesnik.

At least 19 recordsLinked to original sources

arXiv2026

The critical activation density in graph bootstrap percolation

In graph bootstrap percolation, edges of an Erd\H{o}s-R\'enyi random graph ${\mathcal G}_{n,p}$ are initially active, and activation spreads to other edges of $K_n$ via the combinatorics of a fixed graph $H$: an edge becomes active whenever it is the unique inactive edge in a copy of $H$. The process $H$-percolates if all edges of $K_n$ are eventually activated. While classical cases such as $H=K_3$ (connectivity) and $H=K_4$ (related to $2$-neighbor bootstrap percolation) have been studied extensively, general graphs $H$ can exhibit wildly different behaviors. In this work, we determine the critical $H$-percolation threshold $p_c(n,H)$ for every graph $H$, fully resolving a longstanding open question of Balogh, Bollob\'as, and Morris. The location of $p_c(n,H)$ is governed by a new, universal parameter $\rho(H)$, which measures the maximal efficiency of witness graphs that activate an edge. To achieve this, we introduce a novel framework based on the unfolding and refolding of witness graphs. While previous works were restricted to specific families of $H$, our approach provides a unified strategy for all $H$. Inspired by algebraic topology, we lift witness graphs to covering graphs and algorithmically embed folded versions into ${\mathcal G}_{n,p}$ via a sequence of extensions. Crucially, this allows us to incorporate highly efficient witness graphs of unbounded size, which are potentially far larger than ${\mathcal G}_{n,p}$ itself. Beyond resolving $p_c(n,H)$, our framework recovers and strengthens several existing bounds in the literature. Finally, we initiate the study of the universal density parameter $\rho(H)$ and pose central open questions regarding its computability and its exact correspondence with the sharpness of the $H$-percolation threshold.

math.PR↗

arXiv2025

Sharp Fuss-Catalan thresholds in graph bootstrap percolation

We study graph bootstrap percolation on the Erd\H{o}s-R\'enyi random graph ${\mathcal G}_{n,p}$. For all $r \ge 5$, we locate the sharp $K_r$-percolation threshold $p_c \sim (\gamma n)^{-1/\lambda}$, solving a problem of Balogh, Bollob\'as and Morris. The case $r=3$ is the classical graph connectivity threshold, and the threshold for $r=4$ was found using strong connections with the well-studied $2$-neighbor dynamics from statistical physics. When $r \ge 5$, such connections break down, and the process exhibits much richer behavior. The constants $\lambda=\lambda(r)$ and $\gamma=\gamma(r)$ in $p_c$ are determined by a class of $\left({r\choose2}-1\right)$-ary tree-like graphs, which we call $K_r$-tree witness graphs. These graphs are associated with the most efficient ways of adding a new edge in the $K_r$-dynamics, and they can be counted using the Fuss-Catalan numbers. Also, in the subcritical setting, we determine the asymptotic number of edges added to ${\mathcal G}_{n,p}$, showing that the edge density increases only by a constant factor, whose value we identify.

math.PR↗

arXiv2025

On the threshold for triangulations inside convex polygons

Start with a large convex polygon and add all other edges inside independently with probability $p$. At what critical threshold $p_c$ do triangulations of the polygon begin to appear? The first author and Gravner asked this question, and observed that $p_c=\Theta(1)$, using the relationship with the Catalan numbers and a coupling with oriented site percolation on ${\mathbb Z}^2$. More recently, Archer, Hartarsky, the first author, Olesker-Taylor, Schapira and Valesin proved that $1/4<p_c<p_c^o$, where $1/4$ is the Catalan exponential growth rate and $p_c^o$ is the critical threshold for oriented percolation. The upper bound is strict, but non-quantitative, and follows by a renormalization argument. We show that $p_c<1/2$ using a simple ear clipping algorithm, which can be analyzed using the gambler's ruin problem. This bound is closer to the truth (perhaps near $0.4$) and shows that most configurations of edges inside large convex polygons contain triangulations.

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 L\'evy 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 L\'evy process follows the one big jump principle.

math.PR↗

arXiv2025

The snake in the Brownian sphere

The Brownian sphere is a random metric space, homeomorphic to the two-dimensional sphere, which arises as the universal scaling limit of many types of random planar maps. The direct construction of the Brownian sphere is via a continuous analogue of the Cori--Vauquelin--Schaeffer (CVS) bijection. The CVS bijection maps labeled trees to planar maps, and the continuous version maps Aldous' continuum random tree with Brownian labels (the Brownian snake) to the Brownian sphere. In this work, we describe the inverse of the continuous CVS bijection, by constructing the Brownian snake as a measurable function of the Brownian sphere. Special care is needed to work with the orientation of the Brownian sphere.

math.PR↗

arXiv2024

Tournament score sequences, Erd\H{o}s-Ginzburg-Ziv numbers, and the L\'evy-Khintchine method

We give a short proof of a recent result of Claesson, Dukes, Frankl\'in and Stef\'ansson, connecting the number $S_n$ of score sequences and the Erd\H{o}s-Ginzburg-Ziv numbers $N_n$ from additive number theory. Our proof utilizes the lattice path representation of score sequences by Erd\H{o}s 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 L\'evy-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↗

arXiv2024

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 L\'evy-Khintchine formula. Our main result complements a result of Stanley, that ordered graphical sequences are related to quasi-forests.

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↗

arXiv2024

Catalan percolation

In Catalan percolation, all nearest-neighbor edges $\{i,i+1\}$ along $\mathbb Z$ are initially occupied, and all other edges are open independently with probability $p$. Open edges $\{i,j\}$ are occupied if some pair of edges $\{i,k\}$ and $\{k,j\}$, with $i<k<j$, become occupied. This model was introduced by Gravner and the third author, in the context of polluted graph bootstrap percolation. We prove that the critical $p_{\mathrm c}$ is strictly between that of oriented site percolation on $\mathbb Z^2$ and the Catalan growth rate $1/4$. Our main result shows that an enhanced oriented percolation model, with non-decaying infinite-range dependency, has a strictly smaller critical parameter than the classical model. This is reminiscent of the work of Duminil-Copin, Hil\'ario, Kozma and Sidoravicius on brochette percolation. Our proof differs, however, in that we do not use Aizenman--Grimmett enhancements or differential inequalities. Two key ingredients are the work of Hil\'ario, S\'a, Sanchis and Teixeira on stretched lattices, and the Russo--Seymour--Welsh result for oriented percolation by Duminil-Copin, Tassion and Teixeira.

math.PR↗

arXiv2024

Critical beta-splitting, via contraction

The critical beta-splitting tree, introduced by Aldous, is a Markov branching phylogenetic tree. Aldous and Pittel recently proved, amongst other results, a central limit theorem for the height of a random leaf. We give an alternative proof, via contraction methods for random recursive structures. These methods were developed by Neininger and R\"{u}schendorf, motivated by Pittel's article "Normal convergence problem? Two moments and a recurrence may be the clues." Aldous and Pittel estimated the leading order terms in the first two moments. More recently, Aldous and Janson obtained an asymptotic expansion for the average height. We show that a central limit theorem follows, and bound the distance to normality. Our results also apply to the continuous version of the model, in which branching times are exponential.

math.PR↗

arXiv2024

Sina\u{i} excursions: An analogue of Sparre Andersen's formula for the area process of a random walk

Sina\u{i} 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\u{\i} excursions are a class of random polymer chains exhibiting entropic repulsion.

math.PR↗

arXiv2024

Random walks on Coxeter interchange graphs

A tournament is an orientation of a graph. Vertices are players and edges are games, directed away from the winner. Kannan, Tetali and Vempala and McShine showed that tournaments with given score sequence can be rapidly sampled, via simple random walks on the interchange graphs of Brualdi and Li. These graphs are generated by the cyclically directed triangle, in the sense that traversing an edge corresponds to the reversal of such a triangle in a tournament. We study Coxeter tournaments on Zaslavsky's signed graphs. These tournaments involve collaborative and solitaire games, as well as the usual competitive games. The interchange graphs are richer in complexity, as a variety of other generators are involved. We prove rapid mixing by an intricate application of Bubley and Dyer's method of path coupling, using a delicate re-weighting of the graph metric. Geometric connections with the Coxeter permutahedra introduced by Ardila, Castillo, Eur and Postnikov are discussed.

math.CO↗

arXiv2023

Coxeter interchange graphs

Brualdi and Li introduced tournament interchange graphs. In such a graph, each vertex represents a tournament. Traversing an edge corresponds to reversing a cyclically directed triangle. Such a triangle is neutral, in that its reversal does not affect the score sequence. An interchange graph encodes the combinatorics of the set of tournaments with a given score sequence, or equivalently, of a given fiber of the classical permutahedron from discrete geometry. Coxeter tournaments were introduced by the first author and Sanchez, in relation to the Coxeter permutahedra in Ardila, Castillo, Eur and Postnikov. Coxeter tournaments have collaborative and solitaire games, in addition to the usual competitive games in classical tournaments. We introduce Coxeter interchange graphs. These graphs are more intricate, as there are multiple neutral structures at play, which interact with one another. Our main result shows that the Coxeter interchange graphs are regular, and we describe the degree geometrically, in terms of distances in the Coxeter permutahedra. We also characterize the set of score sequences of Coxeter tournaments, generalizing a classical result of Landau.

math.CO↗

arXiv2023

Annihilating branching Brownian motion

We study an interacting system of competing particles on the real line. Two populations of positive and negative particles evolve according to branching Brownian motion. When opposing particles meet, their charges neutralize and the particles annihilate, as in an inert chemical reaction. We show that, with positive probability, the two populations coexist and that, on this event, the interface is asymptotically linear with a random slope. A variety of generalizations and open problems are discussed.

math.PR↗

arXiv2023

$H$-percolation with a random $H$

In $H$-percolation, we start with an Erd\H{o}s--R\'enyi graph ${\mathcal G}_{n,p}$ and then iteratively add edges that complete copies of $H$. The process percolates if all edges missing from ${\mathcal G}_{n,p}$ are eventually added. We find the critical threshold $p_c$ when $H={\mathcal G}_{k,1/2}$ is uniformly random, solving a problem of Balogh, Bollob\'as and Morris.

math.CO↗

arXiv2023

Coxeter tournaments

We describe the Coxeter permutahedra, recently studied by Ardila, Castillo, Eur and Postnikov, in terms of random Coxeter tournaments, which involve cooperative and solitaire games, as well as the usual competitive games in graph tournaments. In this way, we establish a Coxeter version of Moon's theorem on random tournaments. We present a geometric proof by the Mirsky--Thompson generalized Birkhoff's theorem, a probabilistic proof by Strassen's coupling theorem, and an algorithmic proof by a Coxeter analogue of the Havel--Hakimi algorithm. These proofs have interpretations in terms of players choosing competitors/collaborators with respect to relative weakness/strength. We also introduce a natural Coxeter analogue of the Bradley--Terry model, from the statistical theory of paired comparisons.

math.CO↗

arXiv2022

The asymptotic number of score sequences

A tournament on a graph is an orientation of its edges. The score sequence lists the in-degrees in non-decreasing order. Works by Winston and Kleitman (1983) and Kim and Pittel (2000) showed that the number $S_n$ of score sequences on the complete graph $K_n$ satisfies $S_n=\Theta(4^n/n^{5/2})$. By combining a recent recurrence relation for $S_n$ in terms of the Erd\H{o}s--Ginzburg--Ziv numbers $N_n$ with the limit theory for discrete infinitely divisible distributions, we observe that $n^{5/2}S_n/4^n\to e^\lambda/2\sqrt{\pi}$, where $\lambda=\sum_{k=1}^\infty N_k/k4^k$. This limit agrees numerically with the asymptotics of $S_n$ conjectured by Tak\'acs (1986). We also identify the asymptotic number of strong score sequences, and show that the number of irreducible subscores in a random score sequence converges in distribution to a shifted negative binomial with parameters $r=2$ and $p=e^{-\lambda}$.

math.CO↗

arXiv2020

Large deviations of the greedy independent set algorithm on sparse random graphs

We study the greedy independent set algorithm on sparse Erd\H{o}s-R\'enyi random graphs ${\mathcal G}(n,c/n)$. This range of $p$ is of interest due to the threshold at $c=e$, beyond which it appears that greedy algorithms are affected by a sudden change in the independent set landscape. A large deviation principle was recently established by Bermolen et al. (2020), however, the proof and rate function are somewhat involved. Upper bounds for the rate function were obtained earlier by Pittel (1982). By discrete calculus, we identify the optimal trajectory realizing a given large deviation and obtain the rate function in a simple closed form. In particular, we show that Pittel's bounds are sharp. The proof is brief and elementary. We think the methods presented here will be useful in analyzing the tail behavior of other random growth and exploration processes.

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