## Topics in percolation and sequence analysis

##### Abstract

This thesis studies three topics, two in percolation system and one in sequence analysis. In the first part, we prove that, for directed Bernoulli last passage percolation with i.i.d.~weights on vertices over a $n\times n$ grid and for $n$ large enough, the geodesics are shown to be concentrated in a cylinder, centered on the main diagonal and of width of order $n^{(2\kappa+2)/(2\kappa+3)}\sqrt{\ln n}$, where $1\le\kappa<\infty$ is the curvature power-index of the shape function at $(1,1)$. The methodology of proof is robust enough to also apply to directed Bernoulli first passage site percolation, and further to longest common subsequences in random words. In the second part, we prove that, in directed last passage site percolation over a $n\times\lfloor n^{\alpha}\rfloor$-grid and for i.i.d.~random weights having finite support, the order of the $r$-th central moment, $1\le r<+\infty$, of the last passage time is, for $n$ large enough,
lower bounded by $n^{r(1-\alpha)/2}$, $0<\alpha<1/3$. In the last part, we address a question and a conjecture on the expected length of the longest common subsequences of two i.i.d.$\ $random permutations of $[n]:=\{1,2,...,n\}$. The question is resolved by showing that the minimal expectation is not attained in the uniform case. The conjecture asserts that $\sqrt{n}$ is a lower bound on
this expectation, but we only obtain $\sqrt[3]{n}$ for it.