What is the exact definition of a sharp transition? In "Sharp threshold phenomena in statistical physics", H. Duminil-Copin, Japanese J. of Math. 14, 2019, a sharp transition of a boolean function is defined as follows:

A sequence of increasing boolean functions $(\mathbf{f_n})$ undergoes a sharp threshold
at $(p_n)$ if there exists a $(\delta_n)$ tending to 0 such that
$f_n(p_n - \delta_n) \to 0$ and  $f_n(p_n + \delta_n) \to 1$.

An example is $p_n = 1/n$, for the indicator $f=\mathbb{1}_{\text{giant component}}$ of the random graph.
What if I write $p_n = n^{-c}$, for $c<1$ and $c>1$, and consider this the threshold point for $\mathbb{1}_{\text{giant component}}$? Have I followed the correct definition of a sharp threshold above? That is, if I show that the limit of the indicator $\mathbb{1}_{\text{giant component}}$ is either 0 or 1 for $c<1$ or $c>1$, have completed an equivalent task to the above definition?
How are these two formulations (the adding $\delta_n$ method and the use of a power $c$) of the threshold related? Perhaps, I can write $f(n^{-c}) = f(n^{-1} ± \Delta_n(c))$, and find the corresponding sequence $(\Delta_n(c))$, and therefore show the two limits above are 0 or 1?
 A: The parameters $p_n$ are not arbitrary numerical parameters. They represent the expectation of one binary variable in a product space. Changing them additively is very different from changing the power  in the expression $n^{-c}$.
I believe definition 1.1 in the cited paper has a typo, and for $p_n$ tending to 0, we should require $\delta_n/p_n \to 0$ for a sharp threshold, and not just $\delta_n \to 0$. That is the requirement in the key papers that consider $p_n \to 0$, e.g. [1].
For the Erdos-Renyi graph $G(n,p)$ and a fixed vertex $v$, consider the property $$A_n:=\{\,   \text{deg} (v) \ge 1\,\}.$$
Then $A_n$ is an increasing property, and $$f_n(p)=P_p(A_n)=1-(1-p)^{n-1}$$ satisfies
$$\lim_n  f_n(c/n) =1-e^{-c}$$ so it definitely does not satisfy the definition of sharp threshold that requires $\delta_n/p_n \to 0$ (i.e. given that definition of "sharp", since small changes in $c$ don't have the required effect). However,
$\lim_n f_n(n^{-c})=0$   for $c>1$ and $\lim_n f_n(n^{-c})=1$ for $c<1$.
[1] Friedgut, Ehud, and Jean Bourgain. "Sharp thresholds of graph properties, and the -sat problem." Journal of the American mathematical Society 12, no. 4 (1999): 1017-1054.
