In "[Sharp threshold phenomena in statistical physics](https://link.springer.com/article/10.1007/s11537-018-1726-x)", *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?



What is the exact definition of a sharp transition?
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.


