Coupled connected clusters of the origin for the orthant model |
Mark Holmes Statistics Dept., University of Auckland Private Bag 92019 Auckland 1142, New Zealand Phone: +649 3737599 x88679 Email: mholmes(at)stat.auckland.ac.nz
Links |
The set of vertices that the origin can reach is green. |
If I have seen less far than others, it is because there are %$#@ing giants with people on their shoulders blocking my view.
Research interests.
My current interests lie in the realm of self-interacting random walks, including random walks in random media. I am also interested in measure-valued processes and the lace expansion
Holmes, M.P., Jarai, A.A., Sakai, A., and Slade, G. High dimensional graphical networks of self-avoiding walks. Canad. J. Math. 56:77--114, (2004).(pdf)
Holmes, M.P., Perkins, E., Weak convergence of measure-valued processes and r-point functions. Annals of Probability 35 (2007) pp1769--1782 (pdf)
Holmes, M.P., Sakai, A., Senile reinforced random walks. Stochastic Processes and their Applications 117 (2007) pp1519-1539 (pdf)
Holmes, M.P., Convergence of lattice trees to super-Brownian motion above the critical dimension. Electronic J. of Probability 13 (2008) pp671-755(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 13:291--301, (2008).(pdf)
Holmes, M.P. The scaling limit of senile reinforced random walk. Electronic Communications in Probability 14:104-115 (2009).(pdf)
van der Hofstad, R., and Holmes, M.P., Monotonicity for excited random walk in high dimensions To appear, Probability Theory and Related Fields (pdf)
Kojadinovic, I. and Holmes, M. Tests of independence among continuous random vectors based on Cramer-von Mises functionals of the empirical copula process. J. of Multivariate Analysis 100:1137-1154 (2009)(pdf)
Kojadinovic, I., Yan, J. and Holmes, M. Fast large-sample goodness-of-fit tests for copulas. To appear, Statistica Sinica (pdf)
Preprints
Holmes, M.P., and Salisbury, T.S. Degenerate random environments (pdf)
Holmes, M.P., A monotonicity property for a random walk in a partially random environment (pdf)
Holmes, M.P., Excited against the tide: A random walk with competing drifts (pdf)
van der Hofstad, R., and Holmes, M.P., An expansion for self-interacting random walks (pdf)
Other
van der Hofstad, R., Holmes, M.P., and Slade, G., Extension of the generalized inductive approach to the lace expansion (full version in pdf)
aussois slides
An interesting discussion on what student evaluations actually measure
Videos
A random walk in a random galaxy far far away
drift for ORRW (red) on a tree coupled with SRW (black) this video shows the distance of the coupled walks from the root of the regular binary tree
The failure of coupling of ORRW on a tree with different reinforcement parameters (red>black) in this video there are times where the walk with larger reinforcement is further from the root
Traffic model with 3 types of particles (joint work with B. Fralix and A. Lopker) particles of 3 different speeds arrive as Poisson processes and cannot pass each other
An excited random walk with parameter beta=1
A supercritical senile reinforced random walk (joint work with A. Sakai) with probability one, this walk gets stuck on a single (random) edge.
A 2-d random walk perturbed at extrema
A 2-d VRRW with parameter 1 "obviously" recurrent, but this has not been proved
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?