On monochromatic arm exponents for 2D critical percolation

We investigate the so-called monochromatic arm exponents for critical percolation in two dimensions. These exponents, describing the probability of observing j disjoint macroscopic paths, are shown to exist and to form a different family from the (now well understood) polychromatic exponents. More specifically, our main result is that the monochromatic j-arm exponent is strictly between the polychromatic j-arm and (j+1)-arm exponents.Comment: Published in at http://dx.doi.org/10.1214/10-AOP581 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Near-critical percolation with heavy-tailed impurities, forest fires and frozen percolation

Consider critical site percolation on a "nice" planar lattice: each vertex is occupied with probability $p = p_c$, and vacant with probability $1 - p_c$. Now, suppose that additional vacancies ("holes", or "impurities") are created, independently, with some small probability, i.e. the parameter $p_c$ is replaced by $p_c - \varepsilon$, for some small $\varepsilon > 0$. A celebrated result by Kesten says, informally speaking, that on scales below the characteristic length $L(p_c - \varepsilon)$, the connection probabilities remain of the same order as before. We prove a substantial and subtle generalization to the case where the impurities are not only microscopic, but allowed to be "mesoscopic". This generalization, which is also interesting in itself, was motivated by our study of models of forest fires (or epidemics). In these models, all vertices are initially vacant, and then become occupied at rate $1$. If an occupied vertex is hit by lightning, which occurs at a (typically very small) rate $\zeta$, its entire occupied cluster burns immediately, so that all its vertices become vacant. Our results for percolation with impurities turn out to be crucial for analyzing the behavior of these forest fire models near and beyond the critical time (i.e. the time after which, in a forest without fires, an infinite cluster of trees emerges). In particular, we prove (so far, for the case when burnt trees do not recover) the existence of a sequence of "exceptional scales" (functions of $\zeta$). For forests on boxes with such side lengths, the impact of fires does not vanish in the limit as $\zeta \searrow 0$.Comment: 67 pages, 15 figures (some small corrections and improvements, one additional figure); version to be submitte

Two-dimensional volume-frozen percolation: exceptional scales

We study a percolation model on the square lattice, where clusters "freeze" (stop growing) as soon as their volume (i.e. the number of sites they contain) gets larger than N, the parameter of the model. A model where clusters freeze when they reach diameter at least N was studied in earlier papers. Using volume as a way to measure the size of a cluster - instead of diameter - leads, for large N, to a quite different behavior (contrary to what happens on the binary tree, where the volume model and the diameter model are "asymptotically the same"). In particular, we show the existence of a sequence of "exceptional" length scales.Comment: 20 pages, 2 figure

A percolation process on the binary tree where large finite clusters are frozen

We study a percolation process on the planted binary tree, where clusters freeze as soon as they become larger than some fixed parameter N. We show that as N goes to infinity, the process converges in some sense to the frozen percolation process introduced by Aldous. In particular, our results show that the asymptotic behaviour differs substantially from that on the square lattice, on which a similar process has been studied recently by van den Berg, de Lima and Nolin.Comment: 11 page

Multiples bris communs en variance et en moyenne des panels de séries temporelles

Rapport de recherche présenté à la Faculté des arts et des sciences en vue de l'obtention du grade de Maîtrise en sciences économiques

Master of Science

thesisThe goal of this project was to study a control system and prototype for a wearable lift assist device. The purpose of this device is to support the upper body during torso flexion and extension to reduce the erector spinae muscle activity and decrease back compressive force. It could also be used in rehabilitation or return to work scenarios after back injury or surgery. The proposed device is actively controlled and able to provide support based on the wearer's position, anthropometry, and desired level of assist. The upper body is supported by direct current (DC) motors and bilateral torsion springs. The assist level is a function of the percentage of torque needed to statically support the torso for a given angle and can range from 0 to 100%. To develop and test the control system, an electromechanical system was developed to simulate the human torso during lifting. The torso and support mechanism were driven by separate DC motors and drive trains. The torso and support of the mechanism were operated independently, as would be required of an actual assist device. The support control used a position sensor to approximate torso angle and a load cell to determine if the correct amount of torque was applied. The load cell also served to measure the amount of actual assistance provided and gave feedback to the control system. The support control system and mechanism were evaluated using three angular trajectories, derived from actual lifting data. The measured trajectory of the torso with no assist was compared at each level (20 - 100%, in increments of 10%). Ten percent was not analyzed because the support system was unstable at such a low assist level. The torso's trajectory, load cell measurements, and torque produced by the motor driving the torso were analyzed to determine performance. There was a statistical difference when comparing the torso's trajectory with no assist to each level of assist (t 0.05). The torque produced by the torso motor was reduced for all trials and were statistically different (t < 0.0001). This study demonstrates that a mechanism can be actively controlled to provide assistance during lifting. A feasible control system was developed and tested. System stability and mechanical losses in the apparatus were the primary sources of error. Further work on active lifting assist devices and studies to determine actual benefits to users is warranted. The next step is to develop a wearable prototype and explore the benefits of using it in rehabilitation, and low back pain relief