Research Interests

Some of my areas of current research are random spatial networks, non-homogeneous random walks, and martingales. For some brief descriptions (including some pictures) of a random selection of current and past research interests, click on one of the images below or follow a link: non-homogeneous random walks, random spatial networks, self-interacting random walks, stochastic billiards, random walks in random environments.

[Convex hull of random walk] [Rank-driven process] [Simple 
random walk and its centre-of-mass] [Random walk 
with barycentric self-interaction] [A non-homogeneous random walk] [A non-homogeneous random walk] [Random spatial networks] [Stochastic 
billiards] [Simple harmonic urn] [Poisson earthquake]

On this page, jump to: co-authors, Ph.D. students, publications, talks, or posters.

Co-authors

Francis Comets Edward Crane James Cruise Richard Darling Nicholas Georgiou Michael Grinfeld Ostap Hryniv Jonathan Jordan Philip Knight Chak Hei Lo Iain MacPhee
James McRedmond Mikhail Menshikov Aleksandar Mijatović Mathew Penrose Dimitri Petritis Serguei Popov Marina Vachkovskaia Stanislav Volkov Robert Waters Chang Xu Sandy Zabell

Ph.D. students

Publications

There is a list of most of my published work at MathSciNet or Zentralblatt. Preprint versions of most of my papers are available for download from arXiv.

Pre-publication versions of some of my papers are available via Durham Research Online.

You can also look at my researcher profiles at google scholar, ResearcherID, ResearchGate, or ORCiD.

Here is a list of papers. Follow the "Abstract" links for full bibliographic details.

Preprints/submitted work:

[CMW] Random walks avoiding their convex hull with a finite memory with Francis Comets and Mikhail Menshikov. Submitted. Abstract. ArXiv.
[GMPW] Markov chains with heavy-tailed increments and asymptotically zero drift with Nicholas Georgiou, Mikhail Menshikov, and Dimitri Petritis. Submitted. Abstract. ArXiv.

Book:

[MPoW]     Non-homogeneous Random Walks with Mikhail Menshikov and Serguei Popov. Cambridge University Press, December 2016. Errata.

Refereed journal articles:

[GMW2] Invariance principle for non-homogeneous random walks with Nicholas Georgiou and Aleksandar Mijatović. To appear in Electronic Journal of Probability. Abstract. ArXiv.
[LW2]         On the centre of mass of a random walk with Chak Hei Lo. To appear in Stochastic Processes and their Applications. Abstract. Article. ArXiv.
[CW] The critical greedy server on the integers is recurrent with James Cruise. Annals of Applied Probability 29 (2019) 1233–1261. Abstract. Article. ArXiv.
[McRW] The convex hull of a planar random walk: perimeter, diameter, and shape with James McRedmond. Electronic Journal of Probability 23 (2018) paper no. 131. Abstract. Article. ArXiv.
[GMW1] A radial invariance principle for non-homogeneous random walks with Nicholas Georgiou and Aleksandar Mijatović. Electronic Communications in Probability 23 (2018) paper no. 56. Abstract. Article. ArXiv.
[MPeW]     Heavy-tailed random walks on complexes of half-lines with Mikhail Menshikov and Dimitri Petritis. Journal of Theoretical Probability 31 (2018) 1819–1859. Abstract. Article. ArXiv.
[LW1] Non-homogeneous random walks on a half strip with generalized Lamperti drifts with Chak Hei Lo. Markov Processes and Related Fields 23 (2017) 125–146. Abstract. ArXiv.
[GMMW] Anomalous recurrence properties of many-dimensional zero-drift random walks with Nicholas Georgiou, Mikhail Menshikov and Aleksandar Mijatović. Advances in Applied Probability 48A (2016) 99–118. Abstract. Article. ArXiv.
[WX2] Convex hulls of random walks and their scaling limits with Chang Xu. Stochastic Processes and their Applications 125 (2015) 4300–4320. Abstract. Article. ArXiv.
[JW] Phase transitions for random geometric preferential attachment graphs with Jonathan Jordan. Advances in Applied Probability 47 (2015) 565–588. Abstract. Article. ArXiv.
[GVW] Convergence in a multidimensional randomized Keynesian beauty contest with Michael Grinfeld and Stanislav Volkov. Advances in Applied Probability 47 (2015) 57–82. Abstract. Article. ArXiv.
[WX1] Convex hulls of planar random walks with drift with Chang Xu. Proceedings of the American Mathematical Society 143 (2015) 433–445. Abstract. Article. ArXiv.
[DPWZ] Rank deficiency in sparse random GF[2] matrices with Richard Darling, Mathew Penrose and Sandy Zabell. Electronic Journal of Probability 19 (2014) paper no. 83. Abstract. Article. ArXiv.
[GW] Non-homogeneous random walks on a semi-infinite strip with Nicholas Georgiou. Stochastic Processes and their Applications 124 (2014) 3179–3205. Abstract. Article. ArXiv.
[HMW2] Random walk in mixed random environment without uniform ellipticity with Ostap Hryniv and Mikhail Menshikov. Proceedings of the Steklov Institute of Mathematics 282 (2013) 106–123. Abstract. Article. ArXiv.
[HMW1] Excursions and path functionals for stochastic processes with asymptotically zero drifts with Ostap Hryniv and Mikhail Menshikov. Stochastic Processes and their Applications 123 (2013) 1891–1921. Abstract. Article. ArXiv.
[MMW2] Moments of exit times from wedges for non-homogeneous random walks with asymptotically zero drifts with Iain MacPhee and Mikhail Menshikov. Journal of Theoretical Probability 26 (2013) 1–30. Abstract. Article. ArXiv.
[HMMW] Non-homogeneous random walks with non-integrable increments and heavy-tailed random walks on strips with Ostap Hryniv, Iain MacPhee and Mikhail Menshikov. Electronic Journal of Probability 17 (2012) paper no. 59. Abstract. Article. ArXiv.
[GKW2] Rank-driven Markov processes with Michael Grinfeld and Philip Knight. Journal of Statistical Physics 146 (2012) 378–407. Abstract. Article. ArXiv.
[CGVWW] The simple harmonic urn with Edward Crane, Nicholas Georgiou, Stanislav Volkov and Robert Waters. Annals of Probability 39 (2011) 2119–2177. Abstract. Article. ArXiv.
[GKW1] Bak-Sneppen-type models and rank-driven processes with Michael Grinfeld and Philip Knight. Physical Review E 84 (2011) 041124. Abstract. Article. ArXiv.
[CMVW] Random walk with barycentric self-interaction with Francis Comets, Mikhail Menshikov and Stanislav Volkov. Journal of Statistical Physics 143 (2011) 855–888. Abstract. Article. ArXiv.
[MMVW] Passage-time moments and hybrid zones for the exclusion-voter model with Iain MacPhee, Mikhail Menshikov and Stanislav Volkov. Bernoulli 16 (2010) 1312–1342. Abstract. Article. ArXiv.
[PW6] Limit theorems for random spatial drainage networks with Mathew Penrose. Advances in Applied Probability 42 (2010) 659–688. Abstract. Article. ArXiv.
[MW3] Rate of escape and central limit theorem for the supercritical Lamperti problem with Mikhail Menshikov. Stochastic Processes and their Applications 120 (2010) 2078–2099. Abstract. Article. ArXiv.
[MMW1] Angular asymptotics for multi-dimensional non-homogeneous random walks with asymptotically zero drift with Iain MacPhee and Mikhail Menshikov. Markov Processes and Related Fields 16 (2010) 351–388. Abstract. Article. ArXiv.
[W2] Asymptotic theory for the multidimensional random on-line nearest-neighbour graph. Stochastic Processes and their Applications 119 (2009) 1889–1911. Abstract. Article. ArXiv.
[MVW] Asymptotic behaviour of randomly reflecting billiards in unbounded tubular domains with Mikhail Menshikov and Marina Vachkovskaia. Journal of Statistical Physics 132 (2008) 1097–1133. Abstract. Article. ArXiv.
[PW4] Multivariate normal approximation in geometric probability with Mathew Penrose. Journal of Statistical Theory and Practice 2 (2008) 293–326. Abstract. Article. ArXiv.
[MW2] Logarithmic speeds for one-dimensional perturbed random walk in random environment with Mikhail Menshikov. Stochastic Processes and their Applications 118 (2008) 389–416. Abstract. Article. ArXiv.
[PW3] Limit theory for the random on-line nearest-neighbor graph with Mathew Penrose. Random Structures and Algorithms 32 (2008) 125–156. Abstract. Article. ArXiv.
[W1] Explicit laws of large numbers for random nearest-neighbour-type graphs. Advances in Applied Probability 39 (2007) 326–342. Abstract. Article. ArXiv.
[MW1] Random walk in random environment with asymptotically zero perturbation with Mikhail Menshikov. Journal of the European Mathematical Society 8 (2006) 491–513. Abstract. Article. ArXiv.
[PW2] On the total length of the random minimal directed spanning tree with Mathew Penrose. Advances in Applied Probability 38 (2006) 336–372. Abstract. Article. Extended ArXiv version.
[PW1] Random minimal directed spanning trees and Dickman-type distributions with Mathew Penrose. Advances in Applied Probability 36 (2004) 691–714. Abstract. Article. Preprint.

Contributions to edited books:

[CHW]       Stochastic processes with James Cruise and Ostap Hryniv. Chapter 1, pp. 3–38 in Mathematical Tools for Physicists, 2nd ed., Ed. M. Grinfeld, Wiley, November 2014. Abstract. Available as a chapter of the Encyclopedia of Applied Physics.
[PW5]    Random directed and on-line networks with Mathew Penrose. Section II.7, pp. 248–274 in New Perspectives in Stochastic Geometry, Eds. W.S. Kendall and I. Molchanov, Oxford University Press, November 2009. Abstract.

PhD thesis:

[W0]          Limiting Behaviour of Random Spatial Graphs and Asymptotically Homogeneous RWRE. PhD thesis, University of Durham, 2005.

Selected recent and not-so-recent talks

See here for a list of my presentations.

Posters

Click on the thumbnails for details and larger images.

[A self-interacting random walk model for polymer chains]     
[Limit theory for random spatial graph models for drainage networks and network evolution]     
[Limit theory for the random on-line nearest-neighbour graph]     
[Random walk in one-dimensional perturbed random environment]     
[Random minimal directed spanning trees and Dickman-type distributions]     
[Nearest-neighbour graphs]     
[The mixed exclusion-voter model by Iain MacPhee]     

Miscellaneous


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