arXiv subjects

Omer Angel

Publications and source records attributed to Omer Angel.

At least 37 records · Page 2Linked to original sources

arXiv2021

Pairwise optimal coupling of multiple random variables

We generalize the optimal coupling theorem to multiple random variables: Given a collection of random variables, it is possible to couple all of them so that any two differ with probability comparable to the total-variation distance between them. In a number of cases we show that the disagreement probability we achieve is the best possible. The proofs of sharpness rely on new results in extremal combinatorics, which may be of independent interest.

math.PR↗

arXiv2021

A tale of two balloons

From each point of a Poisson point process start growing a balloon at rate 1. When two balloons touch, they pop and disappear. Is every point contained in balloons infinitely often or not? We answer this for the Euclidean space, the hyperbolic plane and regular trees. The result for the Euclidean space relies on a novel 0-1 law for stationary processes. Towards establishing the results for the hyperbolic plane and regular trees, we prove an upper bound on the density of any well-separated set in a regular tree which is a factor of an i.i.d. process.

math.PR↗

arXiv2021

Scaling limits of the three-dimensional uniform spanning tree and associated random walk

We show that the law of the three-dimensional uniform spanning tree (UST) is tight under rescaling in a space whose elements are measured, rooted real trees, continuously embedded into Euclidean space. We also establish that the relevant laws actually converge along a particular scaling sequence. The techniques that we use to establish these results are further applied to obtain various properties of the intrinsic metric and measure of any limiting space, including showing that the Hausdorff dimension of such is given by $3/β$, where $β\approx 1.624\dots$ is the growth exponent of three-dimensional loop-erased random walk. Additionally, we study the random walk on the three-dimensional uniform spanning tree, deriving its walk dimension (with respect to both the intrinsic and Euclidean metric) and its spectral dimension, demonstrating the tightness of its annealed law under rescaling, and deducing heat kernel estimates for any diffusion that arises as a scaling limit.

math.PR↗

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↗

arXiv2020

The number of spanning clusters of the uniform spanning tree in three dimensions

Let ${\mathcal U}_δ$ be the uniform spanning tree on $δ\mathbb{Z}^{3}$. A spanning cluster of ${\mathcal U}_δ$ is a connected component of the restriction of ${\mathcal U}_δ$ to the unit cube $[0,1]^{3}$ that connects the left face $\{ 0 \} \times [0,1]^{2}$ to the right face $\{ 1 \} \times [0,1]^{2}$. In this note, we will prove that the number of the spanning clusters is tight as $δ\to 0$, which resolves an open question raised by Benjamini (1999).

math.PR↗

arXiv2020

On the tail of the branching random walk local time

Consider a critical branching random walk on $\mathbb{Z}^d$, $d\geq 1$, started with a single particle at the origin, and let $L(x)$ be the total number of particles that ever visit a vertex $x$. We study the tail of $L(x)$ under suitable conditions on the offspring distribution. In particular, our results hold if the offspring distribution has an exponential moment.

math.PR↗

arXiv2019

Markov chains with exponential return times are finitary

Consider an ergodic Markov chain on a countable state space for which the return times have exponential tails. We show that the stationary version of any such chain is a finitary factor of an i.i.d. process. A key step is to show that any stationary renewal process whose jump distribution has exponential tails and is not supported on a proper subgroup of $\mathbb{Z}$ is a finitary factor of an i.i.d. process.

math.PR↗

arXiv2019

Long monotone trails in random edge-labelings of random graphs

Given a graph $G$ and a bijection $f : E(G)\rightarrow \{1, 2, \ldots,e(G)\}$, we say that a trail/path in $G$ is $f$-\emph{increasing} if the labels of consecutive edges of this trail/path form an increasing sequence. More than 40 years ago Chvátal and Komlós raised the question of providing the worst-case estimates of the length of the longest increasing trail/path over all edge orderings of $K_n$. The case of a trail was resolved by Graham and Kleitman, who proved that the answer is $n-1$, and the case of a path is still widely open. Recently Lavrov and Loh proposed to study the average case of this problem in which the edge ordering is chosen uniformly at random. They conjectured (and it was proved by Martinsson) that such an ordering with high probability (whp) contains an increasing Hamilton path. In this paper we consider random graph $G=G(n,p)$ and its edge ordering chosen uniformly at random. In this setting we determine whp the asymptotics of the number of edges in the longest increasing trail. In particular we prove an average case of the result of Graham and Kleitman, showing that the random edge ordering of $K_n$ has whp an increasing trail of length $(1-o(1))en$ and this is tight. We also obtain an asymptotically tight result for the length of the longest increasing path for random Erdő-Renyi graphs with $p=o(1)$.

math.CO↗

arXiv2019

The Local Limit of Random Sorting Networks

A sorting network is a geodesic path from $12 \cdots n$ to $n \cdots 21$ in the Cayley graph of $S_n$ generated by adjacent transpositions. For a uniformly random sorting network, we establish the existence of a local limit of the process of space-time locations of transpositions in a neighbourhood of $an$ for $a\in[0,1]$ as $n\to\infty$. Here time is scaled by a factor of $1/n$ and space is not scaled. The limit is a swap process $U$ on $\mathbb{Z}$. We show that $U$ is stationary and mixing with respect to the spatial shift and has time-stationary increments. Moreover, the only dependence on $a$ is through time scaling by a factor of $\sqrt{a(1-a)}$. To establish the existence of $U$, we find a local limit for staircase-shaped Young tableaux. These Young tableaux are related to sorting networks through a bijection of Edelman and Greene.

math.PR↗

arXiv2018

Mallows permutations as stable matchings

We show that the Mallows measure on permutations of $1,\ldots,n$ arises as the law of the unique Gale-Shapley stable matching of the random bipartite graph conditioned to be perfect, where preferences arise from a total ordering of the vertices but are restricted to the (random) edges of the graph. We extend this correspondence to infinite intervals, for which the situation is more intricate. We prove that almost surely every stable matching of the random bipartite graph obtained by performing Bernoulli percolation on the complete bipartite graph $K_{\mathbb{Z},\mathbb{Z}}$ falls into one of two classes: a countable family $(σ_n)_{n\in\mathbb{Z}}$ of tame stable matchings, in which the length of the longest edge crossing $k$ is $O(\log |k|)$ as $k\to\pm \infty$, and an uncountable family of wild stable matchings, in which this length is $\exp Ω(k)$ as $k\to +\infty$. The tame stable matching $σ_n$ has the law of the Mallows permutation of $\mathbb{Z}$ (as constructed by Gnedin and Olshanski) composed with the shift $k\mapsto k+n$. The permutation $σ_{n+1}$ dominates $σ_{n}$ pointwise, and the two permutations are related by a shift along a random strictly increasing sequence.

math.PR↗

arXiv2018

Hyperbolic and Parabolic Unimodular Random Maps

We show that for infinite planar unimodular random rooted maps, many global geometric and probabilistic properties are equivalent, and are determined by a natural, local notion of average curvature. This dichotomy includes properties relating to amenability, conformal geometry, random walks, uniform and minimal spanning forests, and Bernoulli bond percolation. We also prove that every simply connected unimodular random rooted map is sofic, that is, a Benjamini-Schramm limit of finite maps.

math.PR↗

arXiv2018

Perfect shuffling by lazy swaps

We characterize the minimum-length sequences of independent lazy simple transpositions whose composition is a uniformly random permutation. For every reduced word of the reverse permutation there is exactly one valid way to assign probabilities to the transpositions. It is an open problem to determine the minimum length of such a sequence when the simplicity condition is dropped.

math.PR↗

arXiv2017

Counterexamples for percolation on unimodular random graphs

We construct an example of a bounded degree, nonamenable, unimodular random rooted graph with $p_c=p_u$ for Bernoulli bond percolation, as well as an example of a bounded degree, unimodular random rooted graph with $p_c<1$ but with an infinite cluster at criticality. These examples show that two well-known conjectures of Benjamini and Schramm are false when generalised from transitive graphs to unimodular random rooted graphs.

math.PR↗

arXiv2017

The string of diamonds is nearly tight for rumour spreading

For a rumour spreading protocol, the spread time is defined as the first time that everyone learns the rumour. We compare the synchronous push&pull rumour spreading protocol with its asynchronous variant, and show that for any $n$-vertex graph and any starting vertex, the ratio between their expected spread times is bounded by $O \left({n}^{1/3}{\log^{2/3} n}\right)$. This improves the $O(\sqrt n)$ upper bound of Giakkoupis, Nazari, and Woelfel (in Proceedings of ACM Symposium on Principles of Distributed Computing, 2016). Our bound is tight up to a factor of $O(\log n)$, as illustrated by the string of diamonds graph. We also show that if for a pair $α,β$ of real numbers, there exists infinitely many graphs for which the two spread times are $n^α$ and $n^β$ in expectation, then $0\leqα\leq 1$ and $α\leq β\leq \frac13 + \frac23 α$; and we show each such pair $α,β$ is achievable.

math.PR↗

arXiv2017

Local max-cut in smoothed polynomial time

In 1988, Johnson, Papadimitriou and Yannakakis wrote that "Practically all the empirical evidence would lead us to conclude that finding locally optimal solutions is much easier than solving NP-hard problems". Since then the empirical evidence has continued to amass, but formal proofs of this phenomenon have remained elusive. A canonical (and indeed complete) example is the local max-cut problem, for which no polynomial time method is known. In a breakthrough paper, Etscheid and Röglin proved that the smoothed complexity of local max-cut is quasi-polynomial, i.e., if arbitrary bounded weights are randomly perturbed, a local maximum can be found in $n^{O(\log n)}$ steps. In this paper we prove smoothed polynomial complexity for local max-cut, thus confirming that finding local optima for max-cut is much easier than solving it.

cs.DS↗

arXiv2017

Limit laws for self-loops and multiple edges in the configuration model

We consider self-loops and multiple edges in the configuration model as the size of the graph tends to infinity. The interest in these random variables is due to the fact that the configuration model, conditioned on being simple, is a uniform random graph with prescribed degrees. Simplicity corresponds to the absence of self-loops and multiple edges. We show that the number of self-loops and multiple edges converges in distribution to two independent Poisson random variables when the second moment of the empirical degree distribution converges. We also provide an estimate on the total variation distance between the number of self-loops and multiple edges and the Poisson limit of their sum. This revisits previous works of Bollobás, of Janson, of Wormald and others. The error estimates also imply sharp asymptotics for the number of simple graphs with prescribed degrees. The error estimates follow from an application of Stein's method for Poisson convergence, which is a novel method for this problem. The asymptotic independence of self-loops and multiple edges follows from a Poisson version of the Cramér-Wold device using thinning, which is of independent interest. When the degree distribution has infinite second moment, our general results break down. We can, however, prove a central limit theorem for the number of self-loops, and for the multiple edges between vertices of degrees much smaller than the square root of the size of the graph, or when we truncate the degrees similarly. Our results and proofs easily extend to directed and bipartite configuration models.

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↗