Connected subgraphs of Z^2 obtained from a reinforcement process (governed by a parameter a) In this picture a=5 and all connected components have diameter at most 3. 
Mark Holmes Statistics Dept., U. Auckland Private Bag 92019 Auckland 1142, New Zealand Phone: +649 3737599 x88679 Email:

Coupled connected clusters of the origin for the 2dimensional orthant model as p changes from 0 to 1. The (random) set of vertices that the origin can reach is green. 
Research interests: random walks, random media (e.g. percolation models), interacting particle systems, reinforcement models, measurevalued processes, and applications of probability. 
Preprints
Holmes, M., and Kleptsyn, V. Proof of the WARM whisker conjecture for neuronal connections (pdf)
Holmes, M., and Salisbury, T.S. Ballisticity and invariance principle for random walk in nonelliptic random environment. (pdf)
Holmes, M., and Salisbury, T.S. Forward clusters for degenerate random environments. Combinatorics, Probability, and Computing 25(5):744765 (2016). (pdf)
Holmes, M. Backbone scaling for critical lattice trees in high dimensions. Journal of Physics A: Mathematical and Theoretical, 49 (2016) 314001 (26pp) (Special issue in honour of Tony Guttmann)(pdf)
Hofstad, R.v.d., Holmes, M., Kuznetsov, A., and Ruszel, W. Strongly reinforced Polya urns with graphbased competition. Annals of Applied Probability 26(4):24942539 (2016). (pdf)
Holmes, M. On strict monotonicity of the speed for excited random walks in one dimension Electron. Commun. Probab. 20:17 (2015) (pdf)
Holmes, M., Mohylevskyy, Y., and Newman, M. The voter model chordal interface in two dimensions. Journal of Statistical Physics 159:937957, (2015) (pdf)
Hofstad, R.v.d., Holmes, M., and Perkins, E. A criterion for convergence to superBrownian motion on path space. Annals of Probability  to appear(pdf)
Holmes, M., and Salisbury, T.S. Random walks in degenerate random environments . Canad. J. Math. 66:10501077, (2014)(pdf)
v.d. Hofstad, R. and Holmes, M., The survival probability and rpoint functions in high dimensions. Annals of Mathematics 178(2): 665685, (2013). (pdf)
Holmes, M., Kojadinovic, I., Quessy, J.F. Nonparametric tests for changepoint detection a la Gombay and Horvath. (pdf) J. of Multivariate Analysis 115, pages 1632, 2013
Holmes, M., and Salisbury, T.S. Degenerate random environments. Random Structures and Algorithms 45:111–137, (2014) (pdf)
Holmes, M., Sun, R. A monotonicity property for random walk in a partially random environment. Stochastic Processes and their Applications 122: 13691396, 2012. (pdf)
Galbraith, S., and Holmes, M., A nonuniform birthday problem with applications to discrete logarithms. Discrete Applied Mathematics 160:15471560, 2012. (pdf)
Holmes, M., and Salisbury, T.S. A combinatorial result with applications to selfinteracting random walks. Journal of Combinatorial Theory, Series A 119:460475, 2012(pdf)
Holmes, M., Excited against the tide: A random walk with competing drifts. Annales de l'Institut Henri Poincare Probab. Statist. 48: 745773, 2012. (pdf)
Wang, Y., Ziedins, I., Holmes, M. and Challands, N. Treestructured Models for Difference and Change Detection in a Complex Environment. Annals of Applied Statistics, 6, 11621184. 2012.(pdf)
Chen, Y., Holmes, M., and Ziedins, I. Monotonicity properties of user equilibrium policies for parallel batch systems. Queueing Systems 70: 81103, 2012.(pdf)
van der Hofstad, R., and Holmes, M.P., An expansion for selfinteracting random walks. Brazilian Journal of Probability and Statistics 26:155, 2012(pdf)
Kojadinovic, I., Yan, J. and Holmes, M. Fast largesample goodnessoffit tests for copulas. Statistica Sinica 21:841871, 2011. (pdf)
van der Hofstad, R., and Holmes, M.P., Monotonicity for excited random walk in high dimensions. Probability Theory and Related Fields 147:333348, 2010(pdf)
Holmes, M.P. The scaling limit of senile reinforced random walk. Electronic Communications in Probability 14:104115 (2009).(pdf)
Kojadinovic, I. and Holmes, M. Tests of independence among continuous random vectors based on Cramervon Mises functionals of the empirical copula process.. J. of Multivariate Analysis 100:11371154 (2009)(pdf)
van der Hofstad, R. Holmes, M.P. and Slade, G. An extension of the inductive approach to the lace expansion. Electronic Communications in Probability 13:291301, (2008).(pdf)
Holmes, M.P., Convergence of lattice trees to superBrownian motion above the critical dimension. Electronic J. of Probability 13 (2008) pp671755(pdf)
Holmes, M.P., Sakai, A., Senile reinforced random walks. Stochastic Processes and their Applications 117 (2007) pp15191539 (pdf)
Holmes, M.P., Perkins, E., Weak convergence of measurevalued processes and rpoint functions. Annals of Probability 35 (2007) pp17691782 (pdf)
Holmes, M.P., Jarai, A.A., Sakai, A., and Slade, G. High dimensional graphical networks of selfavoiding walks. Canad. J. Math. 56:77114, (2004).(pdf)
Other
van der Hofstad, R., Holmes, M., and Slade, G., Extension of the generalized inductive approach to the lace expansion (full version in pdf)
Holmes, M., and Salisbury, T.S., Speed calculations for random walks in degenerate random environments (pdf)
An interesting discussion on what student evaluations actually measure
Videos
A random walk in a random galaxy far far away
A supercritical senile reinforced random walk with probability one, this walk gets stuck on a single (random) edge.
Interactive
Site Percolation Defined by a single parameter, p. Bond percolation does not percolate at the critical point in 2 dimensions and above 19 dimensions, what about 3 dimensions? (use arrow keys to change p)
Urn model Defined by a single parameter, p. Starting from one ball of each colour, after you have chosen a red ball n times then there will be (n+1)^p balls of that colour in the urn. Is there a difference between p<1, p>1 and p=1?
The voter model in two dimensions An interacting particle system in which each voter is influenced by their neighbour.
If I have seen less far than others, it is because there are %$#@ing giants with people on their shoulders blocking my view.
ZZZ = the slumber of the beast
The problem with quotes on the internet is that it's hard to verify their authenticity (Abraham Lincoln)
A witch goes into a shop on knockturn alley to purchase a bag of thumbs. ``I reaaaally don't think you want thumbs'' says the shopkeeper, handing over the bag. The next day the witch returns to the shop and says ``It turns out that you were right, the thumbs were useless''. Looking sheepish the shopkeeper replies ``I hate to say I sold you toe, but I sold you toe''.
Three literalists walk into a bar. They all sustain minor injuries.
3.14 mathematicians walk into a bakery.
Construction worker: ``I think we need to cover this drain.'' Council official: ``Sounds like a grate idea to me.''
If we ever travel thousands of light years to a planet inhabited by intelligent life, let's just make patterns in their crops and leave.
Two rules to live by: 1) Never tell all that you know