Alexander E. Holroyd

Email: holroyd at math dot ubc dot ca
Telephone/Fax: 604 822 6532
Office: Math Annex 1210
Mailing Address: Department of Mathematics, University of British Columbia, 121 - 1984 Mathematics Rd, Vancouver, BC V6T 1Z2, CANADA

Bootstrap Percolation

Stable Marriage Poisson-Lebesgue

Random Sorting Networks

B-M-L Traffic Model

Gallery

Cornell Summer School

UBC Probability

C.V.

Change Ringing

Currently I spend most of my time in the Theory Group of Microsoft Research, Redmond USA

Research Interests:

Probability theory, with emphasis on discrete spatial models, including cellular automata, percolation, matching, coupling.

Papers:

[34] [PS] [PDF] N. Dirr, P. W. Dondl, G. R. Grimmett, A. E. Holroyd & M. Scheutzow: Lipschitz Percolation. Preprint (9 pages).
[33] [PS] [PDF] O. Angel & A. E. Holroyd: Random subnetworks of random sorting networks. Preprint (8 pages).
[32] [PS] [PDF] O. Angel, A. E. Holroyd, J. B. Martin & J. Propp: Discrete low-discrepancy sequences. Preprint (8 pages).
[31] [PS] [PDF] A. E. Holroyd: Geometric Properties of Poisson Matchings. Preprint (21 pages).
[30] [PS] [PDF] A. E. Holroyd, R. Lyons & T. Soo: Poisson Splitting by Factors. Preprint (55 pages).
[29] [PS] [PDF] A. E. Holroyd & J. Propp: Rotor Walks and Markov Chains. Preprint (33 pages).
[28] [PS] [PDF] J. Gravner & A. E. Holroyd: Local Bootstrap Percolation. Electronic Journal of Probability, 2009, Vol 14, Paper 14, 385-399.
[27] [PS] [PDF] O. Angel, A. E. Holroyd & D. Romik: The Oriented Swap Process. The Annals of Probability, to appear (35 pages).
[26] [PS] [PDF] A. E. Holroyd, L. Levine, K. Meszaros, Y. Peres, J. Propp & D. B. Wilson: Chip-Firing and Rotor-Routing on Directed Graphs. In and Out of Equilibrium 2, (eds. Vladas Sidoravicius and Maria Eulalia Vares), 2008, 331-364, Birkhauser.
[25] [PS] [PDF] A. E. Holroyd, R. Pemantle, Y. Peres & O. Schramm: Poisson Matching. Annales de l'Institut Henri Poincare (B). To appear (36 pages).
[24] [PS] [PDF] A. E. Holroyd: Partition Identities and the Coin Exchange Problem. Journal of Combinatorial Theory A, 2008, Vol 115, 1096-1101.
[23] [PS] [PDF] J. Gravner & A. E. Holroyd: Slow Convergence in Bootstrap Percolation. The Annals of Applied Probability, 2008, Volume 18, No 3, 909-928.
[22] [PS] [PDF] A. E. Holroyd: Astonishing Cellular Automata (Expository article). Bulletin du Centre de Recherches Mathematiques, 2007, Vol 13, No 1, 10-13.
[21] [PS] [PDF] A. E. Holroyd & T. Soo: A Non-Measurable Set from Coin-Flips. (Expository note). American Mathematical Monthly, to appear (2 pages).
[20] [PS] [PDF] O. Angel, A. E. Holroyd, D. Romik & B. Virag: Random Sorting Networks. Advances in Mathematics. 2007, Vol 215, Issue 2, 839-868.
[19] [PS] [PDF] C. Cotar, A. E. Holroyd & D. Revelle: A Percolating Hard Sphere Model. Random Structures and Algorithms, 2009, Vol 34, Issue 2, 285-299.
[18] [PS] [PDF] A. E. Holroyd: The Metastability Threshold for Modified Bootstrap Percolation in d Dimensions. Electronic Journal of Probability, 2006, Vol 11, Paper 17, 418-433.
[17] [PS] [PDF] C. Hoffman, A. E. Holroyd & Y. Peres: Tail Bounds for the Stable Marriage of Poisson and Lebesgue. Canadian Journal of Mathematics. To appear. (30 pages).
[16] [PS] [PDF] C. Hoffman, A. E. Holroyd & Y. Peres: A Stable Marriage of Poisson and Lebesgue. The Annals of Probability, 2006, Vol 34, No 4, 1241-1272. [Version without pictures here].
[15] [PS] [PDF] O. Angel, A. E. Holroyd & J. B. Martin: The Jammed Phase of the Biham-Middleton-Levine Traffic Model. Electronic Communications in Probability, 2005, Vol 10, Paper 17, 167-178.
[14] [PS] [PDF] N. Harvey, A. E. Holroyd, Y. Peres & D. Romik: Universal Finitary Codes with Exponential Tails. Proceedings of the London Mathematical Society, 2007 Vol 94, No 2, 475-496.
[13] [PS] [PDF] A. E. Holroyd & Y. Peres: Extra Heads and Invariant Allocations. The Annals of Probability. 2005, Vol 33, no 1, 31-52.
[12] [PS] [PDF] A. E. Holroyd, T. M. Liggett & D. Romik: Integrals, Partitions, and Cellular Automata. Transactions of the American Mathematical Society, 2004, Vol 356, 3349-3368.
[11] [PS] [PDF] A. E. Holroyd & Y. Peres: Trees and Matchings from Point Processes. Electronic Communications in Probability, 2003, Vol 8, Paper 3, 17-27.
[10] [PS] [PDF] A. E. Holroyd: Sharp Metastability Threshold for Two-Dimensional Bootstrap Percolation. Probability Theory and Related Fields, 2003, Vol. 125, No. 2, 195-224.
[9] [PS] [PDF] A. E. Holroyd: Inequalities in Entanglement Percolation. Journal of Statistical Physics, 2002, 109(1) 317-323.
[8] [PS] [PDF] A. E. Holroyd: Knotted Paths in Percolation. Journal of Statistical Physics, 2002, 109(1) 325-330.
[7] [PS] [PDF] A. E. Holroyd: Entanglement and Rigidity in Percolation Models. In and Out of Equilibrium (V. Sidoravicius, ed.), 2002, Birkhauser, 299-307.
[6] [PS] [PDF] A. E. Holroyd & T. M. Liggett: How to Find an Extra Head: Optimal Random Shifts of Bernoulli and Poisson Random Fields. The Annals of Probability., 2001, Vol. 29, No. 4, 1405-1425.
[5] [PS] [PDF] A. E. Holroyd: Rigidity Percolation and Boundary Conditions. The Annals of Applied Probability, 2001, Vol. 11, No. 4, 1063-1078.
[4] [PS] [PDF] G. R. Grimmett & A. E. Holroyd: Entanglement in Percolation. Proceedings of the London Mathematical Society (3), 2000, Vol. 81, No. 2, 485-512.
[3] [PS] [PDF] A. E. Holroyd: Existence of a Phase Transition for Entanglement Percolation. Mathematical Proceedings of the Cambridge Philosophical Society, 2000, Vol. 129, 231-251.
[2] [PS] [PDF] A. E. Holroyd: Existence and Uniqueness of Infinite Components in Generic Rigidity Percolation. The Annals of Applied Probability, 1998, Vol. 8, No. 3, 944-973.
[1] A. E. Holroyd: Percolation Beyond Connectivity. Ph.D. thesis, University of Cambridge, 2000 (unpublished).

Co-authors:

Omer Angel, Codina Cotar, Janko Gravner, Geoffrey R. Grimmett, Nate Harvey, Christopher Hoffman, Thomas M. Liggett, James B. Martin, Robin Pemantle, Yuval Peres, David Revelle, Dan Romik, Oded Schramm, Terry Soo, Balint Virag

Sequels:

A selection of relevant later articles by others.

Bootstrap percolation:
- P. De Gregorio, A. Lawlor, P. Bradley & K. A. Dawson: Clarification of the Bootstrap Percolation Paradox. Phys. Rev. Lett. 93, 025501 (2004). [Investigation of the apparent discrepancy between the rigorous results in [10] and simulations.]
- Henrik Eriksson: A Tricky Integral [An elementary solution of the integral in [12].]
- George E. Andrews: Partitions with short sequences and mock theta functions. PNAS, vol. 102, no. 13 (2005) (inaugural article). [Greatly extends the partition-theoretic results of [12].]
- L. Grey: A Mathematician Looks at Wolfram's a New Kind of Science. Notices Amer. Math. Soc. 50, 200-211 (2003). [Book review including dicussion of bootstrap percolation].
Factors and Homomorphisms:
- Adam Timar: Tree and Grid Factors for General point Processes. Elec. Comm. in Probab. 9, 53-59 (2004). [Solves many of the open problems in [11]].
- Karen Ball: Poisson Thinning by Monotone Factors. Elec. Comm. in Probab. 9, 53-59 (2004). [Related to [11] and [14]].
Stable Marriage and Extra Heads:
- Sara Robinson: Intriguing Mathematical Picture Merits Second Look. SIAM News, Vol 38, No 6, (2005). [Press article on [16] and [17]].
- Marcelo Freire, Serguei Popov and Marina Vachkovskaia: Percolation for the stable marriage of Poisson and Lebesgue. [Partly solves one of the open problems in [16]].
Entanglement and Rigidity Percolation: [See [7]]
- O. Haggstrom: Uniqueness of infinite rigid components in percolation models: the case of nonplanar lattices, Probab. Th. Relat. Fields 127, 513-534 (2003).
- O. Haggstrom: Uniqueness in two-dimensional rigidity percolation, Math. Proc. Cambridge Phil. Soc. 130, 175-188 (2001).
- O. Haggstrom: Uniqueness of the infinite entangled component in three-dimensional bond percolation, Ann. Probab. 29, 127-136 (2001).