My research is in probability theory and its applications, as well as in different areas of applied mathematics. I have worked on the general theory of random walks and Monte Carlo methods, as well on problems arising from statistical mechanics, theoretical computer science, communications networks, and mathematical models in biology and meteorology. Recently, I have been working on problems in disease modeling (fairly applied) and pattern-avoiding permutations (very theoretical).
I am particularly interested in self-avoiding walks and related models. These arose in the physics literature, originally as models of large polymer molecules, and later as analogues of other statistical mechanical models such as the Ising model. The self-avoiding walk is simply a path on a lattice that does not visit the same site more than once. Proving things about the collection of all such paths is a formidable challenge to rigorous mathematical methods. Besides rigorous analysis, I have also worked on Monte Carlo methods for the study of such models.
Neal Madras, "Lectures on Monte Carlo Methods" (Fields Institute Monographs Series) (American Mathematical Society, Providence, 2002).
Introduction to the theory and application of Monte Carlo methods, based on a graduate course given at the Fields Institute for Research in Mathematical Sciences in the autumn of 1998.
Neal Madras (editor), "Monte Carlo Methods" (Fields Institute Communications Series) (American Mathematical Society, Providence, 2000).
Proceedings of a conference held at the Fields Institute for Research in Mathematical Sciences, Toronto, October 25-29, 1998.
Neal Madras and Gordon Slade, "The Self-Avoiding Walk" (Birkhauser, Boston, 1993). ISBN 0-8176-3589-0, 3-7643-3589-0.
Paperback edition, 1996: ISBN 0-8176-3891-1, 3-7643-3891-1.
Research monograph focusing on the rigorous theory of the self-avoiding walk model.
Refereed articles and articles submitted to journals:
"Bounds on Kemeny's constant of a graph and the Nordhaus-Gaddum problem." (With S. Kim, A. Chan, M. Kempton, S. Kirkland, and A. Knutson.) arXiv link
"Kemeny's constant and enumerating Braess edges in trees." (With J. Jang, M. Kempton, S. Kim, A. Knudson, and M. Song.) arXiv link
"Bounded affine permutations II: Avoidance of decreasing patterns." Annals of Combinatorics (2021). Online at https://rdcu.be/czXKF . doi.org/10.1007/s00026-021-00553-4 . (With J.M. Troyka). arXiv link
"Bounded affine permutations I: Pattern avoidance and enumeration." Discrete Mathematics and Theoretical Computer Science, vol. 22 (2021), No. 2, Paper #1. (With J.M. Troyka). arXiv link
"Theory and Experiment of Chain Length Effects on the Adsorption of Polyelectrolytes onto Spherical Particles: The Long and the Short of It." Physical Chemistry Chemical Physics, vol. 23 (2021), 300-310; doi.org/10.1039/D0CP04359A. (With S. Koumarianos, R. Kaiyum, C.J. Barrett, and O. Mermut.)
"Epidemic dynamics and adaptive vaccination strategy: Renewal equation approach." Bulletin of Mathematical Biology, vol. 82 (2020):122; doi.org/10.1007/s11538-020-00802-2. (With A. Nzokem.) arXiv link
"Examining HIV progression mechanisms via mathematical approaches." Mathematics in Applied Sciences and Engineering, vol. 1 (2020), No. 4, 309-330; doi.org/10.5206/mase/10774. (With W. Zhang, R. Bhagavath, and J.M. Heffernan.) Preprint
"Directed polymers on a disordered tree with a defect subtree." Journal of Physics A: Mathematical and Theoretical 51 (2018), 154001; https://doi.org/10.1088/1751-8121/aab23c. (With Gokhan Yildirim.) arXiv link
"A note on diffusion state distance." arXiv link
"Temperature-driven population abundance model for Culex pipiens and Culex restuans (Diptera: Culicidae)." Journal of Theoretical Biology 443 (2018), 28-38; https://doi.org/10.1016/j.jtbi.2018.01.024. (With Don Yu and Huaiping Zhu.) Preprint
"Longest monotone subsequences and rare regions of pattern-avoiding permutations." The Electronic Journal of Combinatorics 24 (2017), Issue 4, Paper #P4.13 . (With Gokhan Yildirim.) (Former title: "On the longest monotone and alternating subsequences of pattern-avoiding permutations.") arXiv version
"Location of the adsorption transition for lattice polymers." Journal of Physics A: Mathematical and Theoretical 50 (2017), 064003; doi:10.1088/1751-8121/aa53b7. arXiv version
"Convergence rates for a hierarchical Gibbs sampler." Bernoulli 49 (2017), 603-625. (With Oliver Jovanovski.) arXiv link
"Large deviations for permutations avoiding monotone patterns." The Electronic Journal of Combinatorics 23 (2016), Issue 4, Paper #P4.36 . (With Lerna Pehlivan.) arXiv version
"Structure of random 312-avoiding permutations." Random Structures and Algorithms 49 (2016), 599-631; doi: 10.1002/rsa.20601. (With Lerna Pehlivan.) arXiv version
"Trend in frequency of extreme precipitation events over Ontario from ensembles of multiple GCMs." Climate Dynamics 46 (2016), 2909-2921. doi 10.1007/s00382-015-2740-9. (With Z. Deng, X. Qiu, J. Liu, X. Wang, and H. Zhu.)
"Stability of adversarial Markov chains, with an application to adaptive MCMC algorithms." Annals of Applied Probability 25 (2015), 3592-3623. (With Radu Craiu, Lawrence Gray, Krzysztof Latuszynski, Gareth O. Roberts, and Jeffrey S. Rosenthal.) arXiv link
"Phase diagram of inhomogeneous percolation with a defect plane." Journal of Statistical Physics 158 (2015), 255-299; doi:10.1007/s10955-014-1125-5. (With G.K. Iliev and E.J. Janse van Rensburg.) arXiv version
"Large deviations and ratio limit theorems for pattern-avoiding permutations." Combinatorics, Probability and Computing, 23 (2014), 161-200; available on Cambridge Journals Online doi:10.1017/S0963548313000576 (With Mahshid Atapour.) pdf version, copyright Cambridge University Press 2013
"Impact of viral mutation on suppression of infection by cytotoxic T lymphocytes." Canadian Applied Mathematics Quarterly 21 (2013), 145-181. (With Julien Arino, Stephanie Portet, and Beni Sahai.) (Here is the preprint version)
"Trees, animals, and percolation on hyperbolic lattices." Electronic Journal of Probability, 15 (2010), 2019-2040. (With C.C. Wu.)
"Random pattern-avoiding permutations." In "Algorithmic Probability and Combinatorics", M.E. Lladser et al. (eds.), Contemporary Mathematics vol. 520, AMS, Providence, RI (2010), 173-194. (With Hailong Liu.)
"Quantitative bounds for Markov chain convergence: Wasserstein and total variation distances." Bernoulli, 16 (2010), 882-908. (With Deniz Sezer.) arXiv link
"On the number of entangled clusters." Journal of Statistical Physics, 139 (2010), 1-26. (With Mahshid Atapour.) Preprint version
"Strong limit theorems for the Bayesian scoring criterion in Bayesian networks." Journal of Machine Learning Research, 10 (2009), 1511-1526. (With N. Slobodianik and D. Zaporozhets.)
"Almost unknotted embeddings of graphs in Z^3 and higher dimensional analogues." Journal of Knot Theory and Its Ramifications, 18 (2009), 1031-1048. (With D.W. Sumners and S.G. Whittington.)
"Spectral gaps of random walk Metropolis chains." Far East Journal of Theoretical Statistics, 27 (2009), 157-191. (With W.K. Yuen.)
"When is quarantine a useful control strategy for emerging infectious diseases?" American Journal of Epidemiology, 163 (2006), 479-485. (With T. Day, A. Park, A. Gumel, and J. Wu.)
"Self-avoiding walks on hyperbolic graphs." Combinatorics, Probability and Computing, 14 (2005), 523-548. (With C. Chris Wu.)
"Semi-nonparametric estimation with Bernstein polynomials." Economics Letters, 89 (2005), 153-156. (With P.M. Chak and J.B. Smith.)
"Localization of a random copolymer at an interface." Journal of Physics A: Mathematical and General, 36 (2003), 923-938. (With S.G. Whittington.)
"On the swapping algorithm." Random Structures and Algorithms, 22 (2002), 66-97. (With Z. Zheng.)
"Markov chain decomposition for convergence rate analysis." Annals of Applied Probability, 12 (2002), 581-606. (With D. Randall.)
"Self-averaging in finite random copolymers." Journal of Physics A: Mathematical and General, 35 (2002), L427-L431. (With S.G. Whittington.)
"Modeling stem cell development by retrospective analysis of gene expression profiles in single progenitor-derived colonies." Stem Cells, 20 (2002), 230-240. (with A.L. Gibbs, Y. Zhou, P.W. Zandstra, and J.E. Aubin.)
"Anisotropic self-avoiding walks." Journal of Mathematical Physics, 41 (2000), 1321-1337. (With C. Borgs, J.T. Chayes, and C. King.)
"Sharp phase boundaries for a lattice flux line model." Journal of Statistical Physics, 98 (2000), 1075-1113. (With C. Borgs, J.T. Chayes, and C. King.)
"Importance sampling for families of distributions." Annals of Applied Probability, 9 (1999), 1202-1225. (With M. Piccioni.)
"A pattern theorem for lattice clusters." Annals of Combinatorics, 3 (1999), 357-384.
"Metropolis Monte Carlo simulation of lattice animals." Journal of Physics A: Mathematical and General, 30 (1997), 8035-8066. (With E.J. Janse van Rensburg.)
"Monte Carlo study of the Theta-point for collapsing trees." Journal of Statistical Physics, 86 (1997), 1-36. (With E.J. Janse van Rensburg.)
"Factoring graphs to bound mixing rates." Proceedings of the 37th Annual Symposium on Foundations of Computer Science (FOCS '96). (With D. Randall.)
"Local-nonlocal interaction and spatial-temporal patterns in single species populations over a patchy environment." Canadian Applied Mathematics Quarterly, 4 (1996), 109-134. (With J. Wu and X. Zou.)
"Critical exponents, hyperscaling, and universal amplitude ratios for two- and three-dimensional self-avoiding walks." Journal of Statistical Physics, 80 (1995), 661-754. (With B. Li and A.D. Sokal.)
"Critical behaviour of self-avoiding walks that cross a square." Journal of Physics A: Mathematical and General, 28 (1995), 1535-1547.
"A rigorous bound on the critical exponent for the number of lattice trees, animals, and polygons." Journal of Statistical Physics, 78 (1995), 681-699.
"The noisy voter model." Stochastic Processes and their Applications, 55 (1995), 23-43. (With B.L. Granovsky.)
"Random walk interpretations of classical iteration methods." Linear Algebra and Its Applications, 216 (1995), 61-79. (With J. Goodman.)
"Oscillating random walk with a moving boundary." Israel Journal of Mathematics, 88 (1994), 333-365. (With D. Tanny.)
"On the critical behavior of the contact process in deterministic inhomogeneous environment." Annals of Probability, 22 (1994), 1140-1159. (With R. Schinazi and R. Schonmann.)
"Growth pressure can drive early chick lens geometries." Developmental Dynamics 196 (1993), 153-164. (With R. Hendrix and R. Johnson.)
"Gaussian output from chaotic input." International Journal of Bifurcation and Chaos 3 (1993), 663-667. (With R.I.C. Hansell and R.E. Byers.)
"How fair is Fair Queueing?" Journal of the Association for Computing Machinery, 39 (1992), 568-598. (With A.G. Greenberg.)
"Stability-like properties of population models." Theoretical Population Biology, 42 (1992), 10-34. (With R.E. Byers and R.I.C. Hansell.)
"Branching random walks on trees." Stochastic Processes and their Applications, 42 (1992), 255-267. (With R. Schinazi.)
"A non-local Monte Carlo algorithm for lattice trees." Journal of Physics A: Mathematical and General, 25 (1992), 303-333. (With E.J. Janse van Rensburg.)
"The free energy of a collapsing branched polymer". Journal of Physics A: Mathematical and General, 23 (1991), 5327-5350. (With C.E. Soteros, S.G. Whittington, J.L. Martin, M.F. Sykes, S. Flesia and D.S. Gaunt.)
"On the Ornstein-Zernike decay of self-avoiding walks." Proceedings of the Conference on Probability Models in Mathematical Physics, G. Morrow and W.S. Yang, eds., World Scientific, Singapore (1991), 144-157.
"Temperature dependence of the configurational free energy of a branched polymer." Journal of Mathematical Chemistry, 7 (1991), 87-94. (With S.G. Whittington and C.E. Soteros.)
"Bounds on the critical exponent of self-avoiding polygons." In "Random Walks, Brownian Motion, and Interacting Particle Systems: A Festschrift in Honor of Frank Spitzer", R. Durrett and H. Kesten, eds., Birkhauser, Boston (1991), 359-371.
"The pivot algorithm and polygons: Results on the FCC lattice." Journal of Physics A: Mathematical and General, 23 (1990), 1589-1612. (With E.J. Janse van Rensburg and S.G. Whittington.)
"Monte Carlo generation of self-avoiding walks with fixed endpoints and fixed length." Journal of Statistical Physics, 58 (1990), 159-183. (With A. Orlitsky and L.A. Shepp.)
"Comparison of a Fair Queueing discipline to processor sharing." Performance '90 (proceedings), P.J.B. King et al., eds., Elsevier (1990), 193-207. (With A.G. Greenberg.)
"Monte Carlo approximation algorithms for enumeration problems." Journal of Algorithms, 10 (1989), 429-448. (With R. Karp and M. Luby.)
"On random walks with killing." Probability Theory and Related Fields, 80 (1989), 581-600.
"Complex behaviour of systems due to semi-stable attractors." In "Measures of Complexity and Chaos", NATO ASI Series B, N.B. Abraham et al., eds., Plenum Press, New York (1989), 219-223. (With R.E. Byers and R.I.C. Hansell.)
"End patterns of self-avoiding walks." Journal of Statistical Physics, 53 (1988), 689-701.
"Statistics of lattice animals." Journal of Physics A: Mathematical and General, 21 (1988), 4617-4635. (With C.E. Soteros and S.G. Whittington.)
"The pivot algorithm: A highly efficient Monte Carlo method for the self-avoiding walk." Journal of Statistical Physics, 50 (1988), 109-186. (With A.D. Sokal.)
"Stability of binary exponential backoff." Journal of the Association for Computing Machinery, 35 (1988), 579-602. (With J. Goodman, A.G. Greenberg, and P. March.)
"Nonergodicity of local, length-conserving Monte Carlo algorithms for self-avoiding walk." Journal of Statistical Physics, 47 (1987), 573-595. (With A.D. Sokal.)
"A process in a randomly fluctuating environment." Annals of Probability, 14 (1986), 119-135.
"On the stability of the Ethernet." Seventh Annual ACM Symposium on Theory of Computing (1985), 379-387. (With J. Goodman, A.G. Greenberg, and P. March.)
"Mathematical models of mother-child attachment." Proceedings of First Fields/MITACS Industrial Problems Workshop, August 14-18, 2006, pp. 67-88. (With L. Buono, R. Chau, G. Lewis, M. Pugh, L. Rossi, and T. Witelski.)
"Enumeration bounds via an isoperimetric-type inequality." In "Counting Complexity: An International Workshop on Statistical Mechanics and Combinatorics" (J. de Gier and O. Warnaar, eds.), Journal of Physics: Conference Series, Volume 42 (Institute of Physics Publishing, Bristol, 2006), 213-220.
"Umbrella sampling and simulated tempering." In "Numerical Methods for Polymeric Systems" (S.G. Whittington, ed.), IMA Volumes in Mathematics and Its Applications, Volume 102 (Springer-Verlag, New York, 1998), 19-32.
"Monte Carlo simulation of the theta-point in lattice trees." In "Numerical Methods for Polymeric Systems" (S.G. Whittington, ed.), IMA Volumes in Mathematics and Its Applications, Volume 102 (Springer-Verlag, New York, 1998), 141-157. (With E.J. Janse van Rensburg.)
Invited Discussant: Discussion of "Markov-chain Monte Carlo: Some practical implications of theoretical results" by G.O. Roberts and J.S. Rosenthal. Canadian Journal of Statistics, 26 (1998), 27-29.
Invited Discussant: Comment on "Inference from iterative simulation using multiple sequences" by A. Gelman and D.B. Rubin and "Practical Markov chain Monte Carlo" by C.J. Geyer. Statistical Science, 7 (1992), 488-489.