Upper tail large deviation for the first passage time in the frog model on $Z^d$ with $d\ge 2$

We study upper tail large deviations for the first passage time in the frog model on ${\mathbb{Z}}^d$, $d\ge 2$. Let $\mathrm{T}(0,x)$ denote the first time at which the infection, started from the origin, reaches $x$, and let $\mu$ be the associated time constant. We prove that, for any Euclidean unit vector $v\in {\mathbb{R}}^d$ and $\xi >0$, $$\underset{n\to \mathrm{\infty }}{\lim }-\frac{1}{{\mathfrak{s}}_d(n)}\log \mathbb{P}(\mathrm{T}(0,\lfloor nv\rfloor )\ge (\mu (v)+\xi )n)={\mathfrak{r}}_d\xi ,$$ where ${\mathfrak{s}}_2(n)=n\mathrm{/}\log n$ and ${\mathfrak{r}}_2=\pi$ for $d=2$; ${\mathfrak{s}}_d(n)=n$ and ${\mathfrak{r}}_d$ is the escape probability of simple random walk for $d\ge 3$. Moreover, on the upper tail large deviation event, we show that a certain slowdown mechanism occurs near the endpoint. To rule out other mechanisms, we analyze the growth of the infection near the starting point using a modified exploration process, and investigate its propagation through the intermediate region using the renormalization and covering arguments.