Browsing by Author "Apell, Kasperi"

Let L_N denote the maximal number of points in a rate 1 Poisson process on the plane which a piecewise linear increasing path can pass through while traversing from (0, 0) to (N, N). It is well-known that the expectation of L_N / N converges to 2 as N increases without bound. A perturbed version of this system can be introduced by superimposing an independent one-dimensional rate c > 0 Poisson process on the main diagonal line {x = y} of the plane. Given this modification, one asks whether and if so, how, the above limit might be affected. A particular question of interest has been whether this modified system exhibits a nontrivial phase transition in c. That is, whether there exists a threshold value c_0 > 0 such that the limit is preserved for all c < c_0 but lost outside this interval. Let L^c_N denote the maximal number of points in the system perturbed by c > 0 which an increasing piecewise linear path can pass through while traversing from (0, 0) to (N, N). In 2014, Basu, Sidoravicius, and Sly showed that there is no such phase transition and that, for all c > 0, the expectation of L^c_N / N converges to a number strictly greater than 2 as N increases without bound. This thesis gives an exposition of the arguments used to deduce this result.