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Brett Kolesnik

Publications and source records attributed to Brett Kolesnik.

28 records · Page 2Linked to original sources
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 $S_n=Θ(4^n/n^{5/2})$. By combining a recent recurrence relation for $S_n$ in terms of the Erdős--Ginzburg--Ziv numbers $N_n$ with the limit theory for discrete infinitely divisible distributions, we observe that $n^{5/2}S_n/4^n\to e^λ/2\sqrtπ$, where $λ=\sum_{k=1}^\infty N_k/k4^k$. This limit agrees numerically with the asymptotics of $S_n$ conjectured by Takács (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 $p=e^{-λ}$.

math.CO↗
arXiv2021

Large deviations for subcritical bootstrap percolation on the random graph

We study atypical behavior in bootstrap percolation on the Erdős-Rényi random graph. Initially a set $S$ is infected. Other vertices are infected once at least $r$ of their neighbors become infected. Janson et al. (2012) locates the critical size of $S$, above which it is likely that the infection will spread almost everywhere. Below this threshold, a central limit theorem is proved for the size of the eventually infected set. In this note, we calculate the rate function for the event that a small set $S$ eventually infects an unexpected number of vertices, and identify the least-cost trajectory realizing such a large deviation.

math.PR↗
arXiv2021

Sharp threshold for $K_4$-percolation

We locate the critical threshold $p_c$ at which it becomes likely that the complete graph $K_n$ can be obtained from the Erdős-Rényi graph ${\cal G}_{n,p}$ by iteratively completing copies of $K_4$ minus an edge. This refines work of Balogh, Bollobás and Morris that bounds the threshold up to multiplicative constants.

math.PR↗
arXiv2020

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

We study the greedy independent set algorithm on sparse Erdős-Rényi 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.

math.PR↗
arXiv2020

The geometry of random tournaments

A tournament is an orientation of a graph. Each edge represents a match, directed towards the winner. The score sequence lists the number of wins by each team. Landau (1953) characterized score sequences of the complete graph. Moon (1963) showed that the same conditions are necessary and sufficient for mean score sequences of random tournaments. We present short and natural proofs of these results that work for any graph using zonotopes from convex geometry. A zonotope is a linear image of a cube. Moon's Theorem follows by identifying elements of the cube with distributions and the linear map as the expectation operator. Our proof of Landau's Theorem combines zonotopal tilings with the theory of mixed subdivisions. We also show that any mean score sequence can be realized by a tournament that is random within a subforest, and deterministic otherwise.

math.CO↗
arXiv2018

To stay discovered: On tournament mean score sequences and the Bradley--Terry model

On being told that a piece of work he thought was his discovery had duplicated an earlier mathematician's work, Larry Shepp once replied "Yes, but when {\em I} discovered it, it {\em stayed} discovered". In this spirit we give discussion and probabilistic proofs of two related known results (Moon 1963, Joe 1988) on random tournaments which seem surprisingly unknown to modern probabilists. In particular our proof of Moon's theorem on mean score sequences seems more constructive than previous proofs. This provides a comparatively concrete introduction to a longstanding mystery, the lack of a canonical construction for a joint distribution in the representation theorem for convex order.

math.PR↗
arXiv2016

Stability of geodesics in the Brownian map

The Brownian map is a random geodesic metric space arising as the scaling limit of random planar maps. We strengthen the so-called confluence of geodesics phenomenon observed at the root of the map, and with this, reveal several properties of its rich geodesic structure. Our main result is the continuity of the cut locus at typical points. A small shift from such a point results in a small, local modification to the cut locus. Moreover, the cut locus is uniformly stable, in the sense that any two cut loci coincide outside a closed, nowhere dense set of zero measure. We obtain similar stability results for the set of points inside geodesics to a fixed point. Furthermore, we show that the set of points inside geodesics of the map is of first Baire category. Hence, most points in the Brownian map are endpoints. Finally, we classify the types of geodesic networks which are dense. For each $k\in\{1,2,3,4,6,9\}$, there is a dense set of pairs of points which are joined by networks of exactly $k$ geodesics and of a specific topological form. We find the Hausdorff dimension of the set of pairs joined by each type of network. All other geodesic networks are nowhere dense.

math.PR↗
arXiv2016

Thresholds for contagious sets in random graphs

For fixed $r\geq 2$, we consider bootstrap percolation with threshold $r$ on the Erdős-Rényi graph ${\cal G}_{n,p}$. We identify a threshold for $p$ above which there is with high probability a set of size $r$ which can infect the entire graph. This improves a result of Feige, Krivelevich and Reichman, which gives bounds for this threshold, up to multiplicative constants. As an application of our results, we also obtain an upper bound for the threshold for $K_4$-bootstrap percolation on ${\cal G}_{n,p}$, as studied by Balogh, Bollobás and Morris. We conjecture that our bound is asymptotically sharp. These thresholds are closely related to the survival probabilities of certain time-varying branching processes, and we derive asymptotic formulae for these survival probabilities which are of interest in their own right.

math.PR↗
arXiv2015

Lower bounds for the isoperimetric numbers of random regular graphs

The vertex isoperimetric number of a graph $G=(V,E)$ is the minimum of the ratio $|\partial_{V}U|/|U|$ where $U$ ranges over all nonempty subsets of $V$ with $|U|/|V|\le u$ and $\partial_{V}U$ is the set of all vertices adjacent to $U$ but not in $U$. The analogously defined edge isoperimetric number---with $\partial_{V}U$ replaced by $\partial_{E}U$, the set of all edges with exactly one endpoint in $U$---has been studied extensively. Here we study random regular graphs. For the case $u=1/2$, we give asymptotically almost sure lower bounds for the vertex isoperimetric number for all $d\ge3$. Moreover, we obtain a lower bound on the asymptotics as $d\to\infty$. We also provide asymptotically almost sure lower bounds on $|\partial_{E}U|/|U|$ in terms of an upper bound on the size of $U$ and analyse the bounds as $d\to\infty$.

math.CO↗