Upper tail large deviations for a class of distributions in First-passage percolation

In this paper we consider the first passage percolation with identical and independent exponentially distributions, called the Eden growth model, and we study the upper tail large deviations for the first passage time ${\rm T}$. Our main results prove that for any $\xi>0$ and $x\neq 0$, $\mathbb{P}({\rm T}(0,nx)>n(\mu+\xi))$ decays as $\exp{(-(2d\xi +o(1))n)}$ with a time constant $\mu$ and a dimension $d$. Moreover, we extend the result to stretched exponential distributions. On the contrary, we construct a continuous distribution with a finite exponential moment where the rate function does not exist.