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

Source abstract

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 $μ$ be the associated time constant. We prove that, for any Euclidean unit vector $v\in\mathbb R^d$ and $ξ>0$, $$\lim_{n\to\infty} -\frac{1}{\mathfrak{s}_d(n)} \log \mathbb{P}\bigl(\mathrm{T}(0,\lfloor nv\rfloor) \ge (μ(v)+ξ)n\bigr) =\mathfrak{r}_dξ,$$ where $\mathfrak s_2(n)=n/\log n$ and $\mathfrak r_2=π$ for $d=2$; $\mathfrak s_d(n)=n$ and $\mathfrak r_d$ is the escape probability of simple random walk for $d\ge3$. 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.