During my study of percolation I came across exponential decay and there are some parts I do not understand about this.
The definition of exponential decay is as follows:
$f(t)$ decays exponentially means that there are parameters $c,\lambda$ such that $$ f(t) = c\,e^{\lambda t} $$
Now in percolation theory they say that the tail of $C$ decays exponentially when $$ P(|C|\geq n)\leq e^{-cn}$$ Why?
Furthermore they say that there is no exponential decay when $$P(n\leq C<\infty)\geq e^{-cn^{(d-1)/d}}$$
Why?
Any help is appreciated.