arXiv subjects

Alexander E. Holroyd

Publications and source records attributed to Alexander E. Holroyd.

At least 73 records · Page 4Linked to original sources

arXiv2010

Rotor Walks and Markov Chains

The rotor walk is a derandomized version of the random walk on a graph. On successive visits to any given vertex, the walker is routed to each of the neighboring vertices in some fixed cyclic order, rather than to a random sequence of neighbors. The concept generalizes naturally to Markov chains on a countable state space. Subject to general conditions, we prove that many natural quantities associated with the rotor walk (including normalized hitting frequencies, hitting times and occupation frequencies) concentrate around their expected values for the random walk. Furthermore, the concentration is stronger than that associated with repeated runs of the random walk, with discrepancy at most C/n after n runs (for an explicit constant C), rather than c/sqrt n.

math.PR↗

arXiv2010

Percolation in invariant Poisson graphs with i.i.d. degrees

Let each point of a homogeneous Poisson process in R^d independently be equipped with a random number of stubs (half-edges) according to a given probability distribution mu on the positive integers. We consider translation-invariant schemes for perfectly matching the stubs to obtain a simple graph with degree distribution mu. Leaving aside degenerate cases, we prove that for any mu there exist schemes that give only finite components as well as schemes that give infinite components. For a particular matching scheme that is a natural extension of Gale-Shapley stable marriage, we give sufficient conditions on mu for the absence and presence of infinite components.

math.PR↗

arXiv2009

Random Subnetworks of Random Sorting Networks

A sorting network is a shortest path from 12...n to n...21 in the Cayley graph of S_n generated by nearest-neighbor swaps. For m<=n, consider the random m-particle sorting network obtained by choosing an n-particle sorting network uniformly at random and then observing only the relative order of m particles chosen uniformly at random. We prove that the expected number of swaps in location j in the subnetwork does not depend on n, and we provide a formula for it. Our proof is probabilistic, and involves a Polya urn with non-integer numbers of balls. From the case m=4 we obtain a proof of a conjecture of Warrington. Our result is consistent with a conjectural limiting law of the subnetwork as n->infinity implied by the great circle conjecture Angel, Holroyd, Romik and Virag.

math.PR↗

arXiv2009

Deterministic Thinning of Finite Poisson Processes

Let Pi and Gamma be homogeneous Poisson point processes on a fixed set of finite volume. We prove a necessary and sufficient condition on the two intensities for the existence of a coupling of Pi and Gamma such that Gamma is a deterministic function of Pi, and all points of Gamma are points of Pi. The condition exhibits a surprising lack of monotonicity. However, in the limit of large intensities, the coupling exists if and only if the expected number of points is at least one greater in Pi than in Gamma.

math.PR↗

arXiv2009

Geometric Properties of Poisson Matchings

Suppose that red and blue points occur as independent Poisson processes of equal intensity in R^d, and that the red points are matched to the blue points via straight edges in a translation-invariant way. We address several closely related properties of such matchings. We prove that there exist matchings that locally minimize total edge length in d=1 and d>=3, but not in the strip R x [0,1]. We prove that there exist matchings in which every bounded set intersects only finitely many edges in d>=2, but not in d=1 or in the strip. It is unknown whether there exists a matching with no crossings in d=2, but we prove positive answers to various relaxations of this question. Several open problems are presented.

math.PR↗

arXiv2009

A Nonmeasurable Set from Coin Flips

In this note we give an example of a nonmeasurable set in the probability space for an infinite sequence of coin flips. The example arises naturally from the notion of an equivariant function, and serves as a pedagogical illustration of the need for measure theory.

math.PR↗

arXiv2008

Local Bootstrap Percolation

We study a variant of bootstrap percolation in which growth is restricted to a single active cluster. Initially there is a single active site at the origin, while other sites of Z^2 are independently occupied with small probability p, otherwise empty. Subsequently, an empty site becomes active by contact with 2 or more active neighbors, and an occupied site becomes active if it has an active site within distance 2. We prove that the entire lattice becomes active with probability exp[alpha(p)/p], where alpha(p) is between -pi^2/9 + c sqrt p and pi^2/9 + C sqrt p (-log p)^3. This corrects previous numerical predictions for the scaling of the correction term.

math.PR↗

arXiv2008

Poisson Matching

Suppose that red and blue points occur as independent homogeneous Poisson processes in R^d. We investigate translation-invariant schemes for perfectly matching the red points to the blue points. For any such scheme in dimensions d=1,2, the matching distance X from a typical point to its partner must have infinite d/2-th moment, while in dimensions d>=3 there exist schemes where X has finite exponential moments. The Gale-Shapley stable marriage is one natural matching scheme, obtained by iteratively matching mutually closest pairs. A principal result of this paper is a power law upper bound on the matching distance X for this scheme. A power law lower bound holds also. In particular, stable marriage is close to optimal (in tail behavior) in d=1, but far from optimal in d>=3. For the problem of matching Poisson points of a single color to each other, in d=1 there exist schemes where X has finite exponential moments, but if we insist that the matching is a deterministic factor of the point process then X must have infinite mean.

math.PR↗

arXiv2007

Partition Identities and the Coin Exchange Problem

The number of partitions of n into parts divisible by a or b equals the number of partitions of n in which each part and each difference of two parts is expressible as a non-negative integer combination of a or b. This generalizes identities of MacMahon and Andrews. The analogous identities for three or more integers (in place of a,b) hold in certain cases.

math.CO↗

arXiv2007

Slow Convergence in Bootstrap Percolation

In the bootstrap percolation model, sites in an L by L square are initially infected independently with probability p. At subsequent steps, a healthy site becomes infected if it has at least 2 infected neighbours. As (L,p)->(infinity,0), the probability that the entire square is eventually infected is known to undergo a phase transition in the parameter p log L, occurring asymptotically at lambda = pi^2/18. We prove that the discrepancy between the critical parameter and its limit lambda is at least Omega((log L)^(-1/2)). In contrast, the critical window has width only Theta((log L)^(-1)). For the so-called modified model, we prove rigorous explicit bounds which imply for example that the relative discrepancy is at least 1% even when L = 10^3000. Our results shed some light on the observed differences between simulations and rigorous asymptotics.

math.PR↗

arXiv2006

A stable marriage of Poisson and Lebesgue

Let $Ξ$ be a discrete set in ${\mathbb{R}}^d$. Call the elements of $Ξ$ centers. The well-known Voronoi tessellation partitions ${\mathbb{R}}^d$ into polyhedral regions (of varying sizes) by allocating each site of ${\mathbb{R}}^d$ to the closest center. Here we study ``fair'' allocations of ${\mathbb{R}}^d$ to $Ξ$ in which the regions allocated to different centers have equal volumes. We prove that if $Ξ$ is obtained from a translation-invariant point process, then there is a unique fair allocation which is stable in the sense of the Gale--Shapley marriage problem. (I.e., sites and centers both prefer to be allocated as close as possible, and an allocation is said to be unstable if some site and center both prefer each other over their current allocations.) We show that the region allocated to each center $ξ$ is a union of finitely many bounded connected sets. However, in the case of a Poisson process, an infinite volume of sites are allocated to centers further away than $ξ$. We prove power law lower bounds on the allocation distance of a typical site. It is an open problem to prove any upper bound in $d>1$.

math.PR↗

arXiv2006

Random Sorting Networks

A sorting network is a shortest path from 12...n to n...21 in the Cayley graph of S_n generated by nearest-neighbour swaps. We prove that for a uniform random sorting network, as n->infinity the space-time process of swaps converges to the product of semicircle law and Lebesgue measure. We conjecture that the trajectories of individual particles converge to random sine curves, while the permutation matrix at half-time converges to the projected surface measure of the 2-sphere. We prove that, in the limit, the trajectories are Holder-1/2 continuous, while the support of the permutation matrix lies within a certain octagon. A key tool is a connection with random Young tableaux.

math.PR↗

arXiv2006

A Percolating Hard Sphere Model

Given a homogeneous Poisson point process in R^d, Haggstrom and Meester asked whether it is possible to place spheres (of differing radii) centred at the points, in a translation-invariant way, so that the spheres do not overlap but there is an unbounded component of touching spheres. We prove that the answer is yes in sufficiently high dimension.

math.PR↗

arXiv2006

The Metastability Threshold for Modified Bootstrap Percolation in d Dimensions

In the modified bootstrap percolation model, sites in the cube {1,...,L}^d are initially declared active independently with probability p. At subsequent steps, an inactive site becomes active if it has at least one active nearest neighbour in each of the d dimensions, while an active site remains active forever. We study the probability that the entire cube is eventually active. For all d>=2 we prove that as L\to\infty and p\to 0 simultaneously, this probability converges to 1 if L=exp^{d-1} (lambda+epsilon)/p, and converges to 0 if L=exp^{d-1} (lambda-epsilon)/p, for any epsilon>0. Here exp^n denotes the n-th iterate of the exponential function, and the threshold lambda equals pi^2/6 for all d.

math.PR↗

arXiv2005

Tail Bounds for the Stable Marriage of Poisson and Lebesgue

Let Ξbe a discrete set in R^d. Call the elements of Ξcenters. The well-known Voronoi tessellation partitions R^d into polyhedral regions (of varying volumes) by allocating each site of R^d to the closest center. Here we study allocations of R^d to Ξin which each center attempts to claim a region of equal volume α. We focus on the case where Ξarises from a Poisson process of unit intensity. It was proved in math.PR/0505668 that there is a unique allocation which is stable in the sense of the Gale-Shapley marriage problem. We study the distance X from a typical site to its allocated center in the stable allocation. The model exhibits a phase transition in the appetite α. In the critical case α=1 we prove a power law upper bound on X in dimension d=1. It is an open problem to prove any upper bound in d\geq 2. (Power law lower bounds were proved in math.PR/0505668 for all d). In the non-critical cases α<1 and α>1 we prove exponential upper bounds on X.

math.PR↗

arXiv2005

Extra heads and invariant allocations

Let Πbe an ergodic simple point process on R^d and let Π^* be its Palm version. Thorisson [Ann. Probab. 24 (1996) 2057-2064] proved that there exists a shift coupling of Πand Π^*; that is, one can select a (random) point Y of Πsuch that translating Πby -Y yields a configuration whose law is that of Π^*. We construct shift couplings in which Y and Π^* are functions of Π, and prove that there is no shift coupling in which Πis a function of Π^*. The key ingredient is a deterministic translation-invariant rule to allocate sets of equal volume (forming a partition of R^d) to the points of Π. The construction is based on the Gale-Shapley stable marriage algorithm [Amer. Math. Monthly 69 (1962) 9-15]. Next, let Γbe an ergodic random element of {0,1}^{Z^d} and let Γ^* be Γconditioned on Γ(0)=1. A shift coupling X of Γand Γ^* is called an extra head scheme. We show that there exists an extra head scheme which is a function of Γif and only if the marginal E[Γ(0)] is the reciprocal of an integer. When the law of Γis product measure and d\geq3, we prove that there exists an extra head scheme X satisfying E\exp c\|X\|^d<\infty; this answers a question of Holroyd and Liggett [Ann. Probab. 29 (2001) 1405-1425].

math.PR↗

arXiv2005

Universal finitary codes with exponential tails

In 1977, Keane and Smorodinsky showed that there exists a finitary homomorphism from any finite-alphabet Bernoulli process to any other finite-alphabet Bernoulli process of strictly lower entropy. In 1996, Serafin proved the existence of a finitary homomorphism with finite expected coding length. In this paper, we construct such a homomorphism in which the coding length has exponential tails. Our construction is source-universal, in the sense that it does not use any information on the source distribution other than the alphabet size and a bound on the entropy gap between the source and target distributions. We also indicate how our methods can be extended to prove a source-specific version of the result for Markov chains.

math.PR↗

arXiv2003

Integrals, Partitions, and Cellular Automata

We prove that $$\int_0^1\frac{-\log f(x)}xdx=\frac{π^2}{3ab}$$ where $f(x)$ is the decreasing function that satisfies $f^a-f^b=x^a-x^b$, for $0<a<b$. When $a$ is an integer and $b=a+1$ we deduce several combinatorial results. These include an asymptotic formula for the number of integer partitions not having $a$ consecutive parts, and a formula for the metastability thresholds of a class of threshold growth cellular automaton models related to bootstrap percolation.

math.PR↗