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, $1\le r<+\mathrm{\infty }$, of the last passage time is lower bounded by $n^{r(1-\alpha )\mathrm{/}2}$, $0<\alpha <1\mathrm{/}3$.