Power Lower Bounds for the Central Moments of the Last Passage Time for Directed Percolation in a Thin Rectangle

In directed last passage site percolation with i.i.d.~random weights with finite support over a $n\times\lfloor n^{\alpha}\rfloor$ grid, we prove that for $n$ large enough, the order of the $r$-th central moment, $1\le r<+\infty$, of the last passage time is lower bounded by $n^{r(1-\alpha)/2}$, $0<\alpha<1/3$.