arXiv subjects

Alexander E. Holroyd

Publications and source records attributed to Alexander E. Holroyd.

At least 55 records · Page 3Linked to original sources

arXiv2013

Drawing Permutations with Few Corners

A permutation may be represented by a collection of paths in the plane. We consider a natural class of such representations, which we call tangles, in which the paths consist of straight segments at 45 degree angles, and the permutation is decomposed into nearest-neighbour transpositions. We address the problem of minimizing the number of crossings together with the number of corners of the paths, focusing on classes of permutations in which both can be minimized simultaneously. We give algorithms for computing such tangles for several classes of permutations.

cs.DM↗

arXiv2013

Insertion and Deletion Tolerance of Point Processes

We develop a theory of insertion and deletion tolerance for point processes. A process is insertion-tolerant if adding a suitably chosen random point results in a point process that is absolutely continuous in law with respect to the original process. This condition and the related notion of deletion-tolerance are extensions of the so-called finite energy condition for discrete random processes. We prove several equivalent formulations of each condition, including versions involving Palm processes. Certain other seemingly natural variants of the conditions turn out not to be equivalent. We illustrate the concepts in the context of a number of examples, including Gaussian zero processes and randomly perturbed lattices, and we provide applications to continuum percolation and stable matching.

math.PR↗

arXiv2012

A pattern theorem for random sorting networks

A sorting network is a shortest path from 12..n to n..21 in the Cayley graph of the symmetric group S(n) generated by nearest-neighbor swaps. A pattern is a sequence of swaps that forms an initial segment of some sorting network. We prove that in a uniformly random n-element sorting network, any fixed pattern occurs in at least cn^2 disjoint space-time locations, with probability tending to 1 exponentially fast as n tends to infinity. Here c is a positive constant which depends on the choice of pattern. As a consequence, the probability that the uniformly random sorting network is geometrically realizable tends to 0.

math.PR↗

arXiv2012

Lattice embeddings in percolation

Does there exist a Lipschitz injection of $\mathbb{Z}^d$ into the open set of a site percolation process on $\mathbb{Z}^D$, if the percolation parameter p is sufficiently close to 1? We prove a negative answer when d=D and also when $d\geq2$ if the Lipschitz constant M is required to be 1. Earlier work of Dirr, Dondl, Grimmett, Holroyd and Scheutzow yields a positive answer for d<D and M=2. As a result, the above question is answered for all d, D and M. Our proof in the case d=D uses Tucker's lemma from topological combinatorics, together with the aforementioned result for d<D. One application is an affirmative answer to a question of Peled concerning embeddings of random patterns in two and more dimensions.

math.PR↗

arXiv2012

Edge Routing with Ordered Bundles

Edge bundling reduces the visual clutter in a drawing of a graph by uniting the edges into bundles. We propose a method of edge bundling drawing each edge of a bundle separately as in metro-maps and call our method ordered bundles. To produce aesthetically looking edge routes it minimizes a cost function on the edges. The cost function depends on the ink, required to draw the edges, the edge lengths, widths and separations. The cost also penalizes for too many edges passing through narrow channels by using the constrained Delaunay triangulation. The method avoids unnecessary edge-node and edge-edge crossings. To draw edges with the minimal number of crossings and separately within the same bundle we develop an efficient algorithm solving a variant of the metro-line crossing minimization problem. In general, the method creates clear and smooth edge routes giving an overview of the global graph structure, while still drawing each edge separately and thus enabling local analysis.

cs.CG↗

arXiv2012

The Phase Transition for Dyadic Tilings

A dyadic tile of order n is any rectangle obtained from the unit square by n successive bisections by horizontal or vertical cuts. Let each dyadic tile of order n be available with probability p, independently of the others. We prove that for p sufficiently close to 1, there exists a set of pairwise disjoint available tiles whose union is the unit square, with probability tending to 1 as n->infinity, as conjectured by Joel Spencer in 1999. In particular we prove that if p=7/8, such a tiling exists with probability at least 1-(3/4)^n. The proof involves a surprisingly delicate counting argument for sets of unavailable tiles that prevent tiling.

math.PR↗

arXiv2012

Stochastic Domination and Comb Percolation

There exists a Lipschitz embedding of a d-dimensional comb graph (consisting of infinitely many parallel copies of Z^{d-1} joined by a perpendicular copy) into the open set of site percolation on Z^d, whenever the parameter p is close enough to 1 or the Lipschitz constant is sufficiently large. This is proved using several new results and techniques involving stochastic domination, in contexts that include a process of independent overlapping intervals on Z, and first-passage percolation on general graphs.

math.PR↗

arXiv2011

Poisson splitting by factors

Given a homogeneous Poisson process on ${\mathbb{R}}^d$ with intensity $λ$, we prove that it is possible to partition the points into two sets, as a deterministic function of the process, and in an isometry-equivariant way, so that each set of points forms a homogeneous Poisson process, with any given pair of intensities summing to $λ$. In particular, this answers a question of Ball [Electron. Commun. Probab. 10 (2005) 60--69], who proved that in $d=1$, the Poisson points may be similarly partitioned (via a translation-equivariant function) so that one set forms a Poisson process of lower intensity, and asked whether the same is possible for all $d$. We do not know whether it is possible similarly to add points (again chosen as a deterministic function of a Poisson process) to obtain a Poisson process of higher intensity, but we prove that this is not possible under an additional finitariness condition.

math.PR↗

arXiv2011

Correction of Teledyne Acoustic Doppler Current Profiler (ADCP) Bottom-Track Range Measurements for Instrument Pitch and Roll

The Workhorse Acoustic Doppler Current Profiler (ADCP) manufactured by Teledyne RD Instruments (RDI) uses a "Bottom-Tracking" algorithm to yield data intended to represent distance to the bottom. However, current RDI software processing does not take into account the pitch and roll of the instrument. This technical note outlines post-deployment computations required to correct the reported Bottom-Track Ranges for instrument pitch and roll in the scenario where the ADCP is upward-looking with Bottom-Tracking being used to estimate distance to the surface.

physics.ao-ph↗

arXiv2011

Stable Poisson Graphs in One Dimension

Let each point of a homogeneous Poisson process on $\RR$ independently be equipped with a random number of stubs (half-edges) according to a given probability distribution $μ$ on the positive integers. We consider schemes based on Gale-Shapley stable marriage for perfectly matching the stubs to obtain a simple graph with degree distribution $μ$. We prove results on the existence of an infinite component and on the length of the edges, with focus on the case $μ(\{2\})=1$. In this case, for the random direction stable matching scheme introduced by Deijfen and Meester we prove that there is no infinite component, while for the stable matching of Deijfen, Häggström and Holroyd we prove that existence of an infinite component follows from a certain statement involving a {\em finite} interval, which is overwhelmingly supported by simulation evidence.

math.PR↗

arXiv2011

Some circumstances where extra updates can delay mixing

Peres and Winkler proved a "censoring" inequality for Glauber dynamics on monotone spins systems such as the Ising model. Specifically, if, starting from a constant-spin configuration, the spins are updated at some sequence of sites, then inserting another site into this sequence brings the resulting configuration closer in total variation to the stationary distribution. We show by means of simple counterexamples that the analogous statements fail for Glauber dynamics on proper colorings of a graph, and for lazy transpositions on permutations, answering two questions of Peres. It is not known whether the censoring property holds in other natural settings such as the Potts model.

math.PR↗

arXiv2011

Recurrent Rotor-Router Configurations

We prove the existence of recurrent initial configurations for the rotor walk on many graphs, including Z^d, and planar graphs with locally finite embeddings. We also prove that recurrence and transience of rotor walks are invariant under changes in the starting vertex and finite changes in the initial configuration.

math.CO↗

arXiv2010

A sharper threshold for bootstrap percolation in two dimensions

Two-dimensional bootstrap percolation is a cellular automaton in which sites become 'infected' by contact with two or more already infected nearest neighbors. We consider these dynamics, which can be interpreted as a monotone version of the Ising model, on an n x n square, with sites initially infected independently with probability p. The critical probability p_c is the smallest p for which the probability that the entire square is eventually infected exceeds 1/2. Holroyd determined the sharp first-order approximation: p_c \sim π^2/(18 log n) as n \to \infty. Here we sharpen this result, proving that the second term in the expansion is -(log n)^{-3/2+ o(1)}, and moreover determining it up to a poly(log log n)-factor. The exponent -3/2 corrects numerical predictions from the physics literature.

math.PR↗

arXiv2010

Rotor walks on general trees

The rotor walk on a graph is a deterministic analogue of random walk. Each vertex is equipped with a rotor, which routes the walker to the neighbouring vertices in a fixed cyclic order on successive visits. We consider rotor walk on an infinite rooted tree, restarted from the root after each escape to infinity. We prove that the limiting proportion of escapes to infinity equals the escape probability for random walk, provided only finitely many rotors send the walker initially towards the root. For i.i.d. random initial rotor directions on a regular tree, the limiting proportion of escapes is either zero or the random walk escape probability, and undergoes a discontinuous phase transition between the two as the distribution is varied. In the critical case there are no escapes, but the walker's maximum distance from the root grows doubly exponentially with the number of visits to the root. We also prove that there exist trees of bounded degree for which the proportion of escapes eventually exceeds the escape probability by arbitrarily large o(1) functions. No larger discrepancy is possible, while for regular trees the discrepancy is at most logarithmic.

math.CO↗

arXiv2010

Plaquettes, Spheres, and Entanglement

The high-density plaquette percolation model in d dimensions contains a surface that is homeomorphic to the (d-1)-sphere and encloses the origin. This is proved by a path-counting argument in a dual model. When d=3, this permits an improved lower bound on the critical point p_e of entanglement percolation, namely p_e >= μ^-2 where μis the connective constant for self-avoiding walks on Z^3. Furthermore, when the edge density p is below this bound, the radius of the entanglement cluster containing the origin has an exponentially decaying tail.

math.PR↗

arXiv2010

Escape of resources in distributed clustering processes

In a distributed clustering algorithm introduced by Coffman, Courtois, Gilbert and Piret \cite{coffman91}, each vertex of $\mathbb{Z}^d$ receives an initial amount of a resource, and, at each iteration, transfers all of its resource to the neighboring vertex which currently holds the maximum amount of resource. In \cite{hlrnss} it was shown that, if the distribution of the initial quantities of resource is invariant under lattice translations, then the flow of resource at each vertex eventually stops almost surely, thus solving a problem posed in \cite{berg91}. In this article we prove the existence of translation-invariant initial distributions for which resources nevertheless escape to infinity, in the sense that the the final amount of resource at a given vertex is strictly smaller in expectation than the initial amount. This answers a question posed in \cite{hlrnss}.

math.PR↗

arXiv2010

Geometry of Lipschitz percolation

We prove several facts concerning Lipschitz percolation, including the following. The critical probability p_L for the existence of an open Lipschitz surface in site percolation on Z^d with d\ge 2 satisfies the improved bound p_L \le 1-1/[8(d-1)]. Whenever p > p_L, the height of the lowest Lipschitz surface above the origin has an exponentially decaying tail. The lowest surface is dominated stochastically by the boundary of a union of certain independent, identically distributed random subsets of Z^d. As a consequence, for p sufficiently close to 1, the connected regions of Z^{d-1} above which the surface has height 2 or more exhibit stretched-exponential tail behaviour.

math.PR↗

arXiv2010

Discrete low-discrepancy sequences

Holroyd and Propp used Hall's marriage theorem to show that, given a probability distribution pi on a finite set S, there exists an infinite sequence s_1,s_2,... in S such that for all integers k >= 1 and all s in S, the number of i in [1,k] with s_i = s differs from k pi(s) by at most 1. We prove a generalization of this result using a simple explicit algorithm. A special case of this algorithm yields an extension of Holroyd and Propp's result to the case of discrete probability distributions on infinite sets.

math.CO↗