About

Alexander E Holroyd is a Professor in the School of Mathematics at the University of Bristol. His research focuses on Probability Theory, Lattices, Integer, Shift of Finite Type, Edge, Stable Matching, Higher Dimensions, and Initial State.

Research topics

• Psychology
• Statistical physics
• Neuroscience
• Quantum mechanics
• Physics
• Mathematics

Selected publications

• Polluted Modified Bootstrap Percolation

  ArXiv.org · 2025-03-19

  preprintOpen access

  In the polluted modified bootstrap percolation model, sites in the square lattice are independently initially occupied with probability $p$ or closed with probability $q$. A site becomes occupied at a subsequent step if it is not closed and has at least one occupied nearest neighbor in each of the two coordinates. We study the final density of occupied sites when $p$ and $q$ are both small. We show that this density approaches $0$ if $q\ge Cp^2/\log p^{-1}$ and $1$ if $q\le p^2/(\log p^{-1})^{1+o(1)}$. Thus we establish a logarithmic correction in the critical scaling, which is known not to be present in the standard model, settling a conjecture of Gravner and McDonald from 1997.

  PublisherOA PDFDOI

• Polluted Modified Bootstrap Percolation

  Electronic Communications in Probability · 2025-01-01

  articleOpen access

  In the polluted modified bootstrap percolation model, sites in the square lattice are independently initially occupied with probability p or closed with probability q. A site becomes occupied at a subsequent step if it is not closed and has at least one occupied nearest neighbor in each of the two coordinates. We study the final density of occupied sites when p and q are both small. We show that this density approaches 0 if q≥Cp2∕logp−1 and 1 if q≤p2∕(logp−1)1+o(1). Thus we establish a logarithmic correction in the critical scaling, which is known not to be present in the standard model, settling a conjecture of Gravner and McDonald from 1997.

  PublisherOA PDFDOI

• Symmetrization for finitely dependent colouring

  Electronic Communications in Probability · 2024-01-01 · 2 citations

  articleOpen access1st authorCorresponding

  We prove the existence of a finitely dependent proper colouring of the integer lattice Z^d that is fully isometry-invariant in law, for all dimensions d. Previously this was known only for d=1, while only translation-invariant examples were known for higher d. Moreover we show that four colours suffice, and that the colouring can be expressed as an isometry-equivariant finitary factor of an i.i.d. process, with exponential tail decay on the coding radius. Our construction starts from known translation-invariant colourings and applies a symmetrization technique of possible broader utility.

  PublisherDOI

• Symmetrization for finitely dependent colouring

  arXiv (Cornell University) · 2023-05-23

  preprintOpen access1st authorCorresponding

  We prove the existence of a finitely dependent proper colouring of the integer lattice Z^d that is fully isometry-invariant in law, for all dimensions d. Previously this was known only for d=1, while only translation-invariant examples were known for higher d. Moreover we show that four colours suffice, and that the colouring can be expressed as an isometry-equivariant finitary factor of an i.i.d. process, with exponential tail decay on the coding radius. Our construction starts from known translation-invariant colourings and applies a symmetrization technique of possible broader utility.

  PublisherOA PDFDOI

• Cyclic Products and Optimal Traps in Cyclic Birth and Death Chains

  The Electronic Journal of Combinatorics · 2023-06-30

  articleOpen access

  A birth-death chain is a discrete-time Markov chain on the integers whose transition probabilities $p_{i,j}$ are non-zero if and only if $|i-j|=1$. We consider birth-death chains whose birth probabilities $p_{i,i+1}$ form a periodic sequence, so that $p_{i,i+1}=p_{i \mod m}$ for some $m$ and $p_0,\ldots,p_{m-1}$. The trajectory $(X_n)_{n=0,1,\ldots}$ of such a chain satisfies a strong law of large numbers and a central limit theorem. We study the effect of reordering the probabilities $p_0,\ldots,p_{m-1}$ on the velocity $v=\lim_{n\to\infty} X_n/n$. The sign of $v$ is not affected by reordering, but its magnitude in general is. We show that for Lebesgue almost every choice of $(p_0,\ldots,p_{m-1})$, exactly $(m-1)!/2$ distinct speeds can be obtained by reordering. We make an explicit conjecture of the ordering that minimises the speed, and prove it for all $m\leq 7$. This conjecture is implied by a purely combinatorial conjecture that we think is of independent interest.

  PublisherOA PDFDOI

• Cyclic products and optimal traps in cyclic birth and death chains

  arXiv (Cornell University) · 2022-03-04

  preprintOpen access

  A birth-death chain is a discrete-time Markov chain on the integers whose transition probabilities $p_{i,j}$ are non-zero if and only if $|i-j|=1$. We consider birth-death chains whose birth probabilities $p_{i,i+1}$ form a periodic sequence, so that $p_{i,i+1}=p_{i \mod m}$ for some $m$ and $p_0,\ldots,p_{m-1}$. The trajectory $(X_n)_{n=0,1,\ldots}$ of such a chain satisfies a strong law of large numbers and a central limit theorem. We study the effect of reordering the probabilities $p_0,\ldots,p_{m-1}$ on the velocity $v=\lim_{n\to\infty} X_n/n$. The sign of $v$ is not affected by reordering, but its magnitude in general is. We show that for Lebesgue almost every choice of $(p_0,\ldots,p_{m-1})$, exactly $(m-1)!/2$ distinct speeds can be obtained by reordering. We make an explicit conjecture of the ordering that minimises the speed, and prove it for all $m\leq 7$. This conjecture is implied by a purely combinatorial conjecture that we think is of independent interest.

  PublisherOA PDFDOI

• Minimal matchings of point processes

  Probability Theory and Related Fields · 2022-07-13 · 4 citations

  articleOpen access1st authorCorresponding

  Abstract Suppose that red and blue points form independent homogeneous Poisson processes of equal intensity in $${{\mathbb {R}}}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> . For a positive (respectively, negative) parameter $$\gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>γ</mml:mi> </mml:math> we consider red-blue matchings that locally minimize (respectively, maximize) the sum of $$\gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>γ</mml:mi> </mml:math> th powers of the edge lengths, subject to locally minimizing the number of unmatched points. The parameter can be viewed as a measure of fairness. The limit $$\gamma \rightarrow -\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>→</mml:mo> <mml:mo>-</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> is equivalent to Gale-Shapley stable matching. We also consider limits as $$\gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>γ</mml:mi> </mml:math> approaches 0, $$1-$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> </mml:mrow> </mml:math> , $$1+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> </mml:mrow> </mml:math> and $$\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>∞</mml:mi> </mml:math> . We focus on dimension $$d=1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . We prove that almost surely no such matching has unmatched points. (This question is open for higher d ). For each $$\gamma &lt;1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> we establish that there is almost surely a unique such matching, and that it can be expressed as a finitary factor of the points. Moreover, its typical edge length has finite r th moment if and only if $$r&lt;1/2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . In contrast, for $$\gamma =1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> there are uncountably many matchings, while for $$\gamma &gt;1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> there are countably many, but it is impossible to choose one in a translation-invariant way. We obtain existence results in higher dimensions (covering many but not all cases). We address analogous questions for one-colour matchings also.

  PublisherOA PDFDOI

• The spinor linkage – a mechanical implementation of the plate trick

  Journal of Mathematics and the Arts · 2022-04-03 · 2 citations

  articleOpen access1st authorCorresponding

  The plate trick or belt trick is a striking physical demonstration of properties of the double cover of the three-dimensional rotation group by the sphere of unit quaternions or spinors. The two ends of a flexible object are continuously rotated with respect to each other. Surprisingly, the object can be manipulated so as to avoid accumulating twists. We present a new mechanical linkage that implements this task. It consists of a sequence of rigid bodies connected by hinge joints, together with a purely mechanical control mechanism. It has one degree of freedom, and the motion is generated by simply turning a handle. A video is available at https://www.youtube.com/watch?v=oRPCoEq05Zk.

  PublisherOA PDFDOI

• Constrained percolation in two dimensions

  Annales de l’Institut Henri Poincaré D Combinatorics Physics and their Interactions · 2021 · 8 citations

  1st authorCorresponding
  • Statistical physics
  • Mathematics
  • Physics

  We prove absence of infinite clusters and contours in a class of critical constrained percolation models on the square lattice. The percolation configuration is assumed to satisfy certain hard local constraints, but only weak symmetry and ergodicity conditions are imposed on its law. The proofs use new combinatorial techniques exploiting planar duality. Applications include absence of infinite clusters of diagonal edges for critical dimer models on the square-octagon lattice, as well as absence of infinite contours and infinite clusters for critical XOR Ising models on the square grid. We also prove that there exists at most one infinite contour for high-temperature XOR Ising models, and no infinite contour for low-temperature XOR Ising model.

  PublisherDOI

• The spinor linkage -- a mechanical implementation of the plate trick

  arXiv (Cornell University) · 2021-07-04

  preprintOpen access1st authorCorresponding

  The plate trick or belt trick is a striking physical demonstration of properties of the double cover of the three-dimensional rotation group by the sphere of unit quaternions or spinors. The two ends of a flexible object are continuously rotated with respect to each other. Surprisingly, the object can be manipulated so as to avoid accumulating twists. We present a new mechanical linkage that implements this task. It consists of a sequence of rigid bodies connected by hinge joints, together with a purely mechanical control mechanism. It has one degree of freedom, and the motion is generated by simply turning a handle.

  PublisherOA PDFDOI

Frequent coauthors

• Omer Angel

  University of British Columbia

  50 shared

• Yuval Peres

  Beijing Institute of Mathematical Sciences and Applications

  40 shared

• Geoffrey Grimmett

  38 shared

• James B. Martin

  Deakin University

  25 shared

• Janko Gravner

  20 shared

• Rick Durrett

  18 shared

• Pietro Caputo

  18 shared

• David Aldous

  18 shared

Education

• B.A.

  University of Bristol

• Other

  University of Bristol

• M.A.

  University of Bristol

• Ph.D.

  University of Bristol