Question about Peierls Droplets argument I am trying to understand the logic of considering Peierls droplets. The basic idea is that the entropy and energy of the loop are proportional to its length L (see Tong's lectures, p.161). As a result, in the thermodynamic limit, we find that a low-temperature (ordered phase) exists up to a certain critical temperature (order parameter = constant) $T_c$, and at $T > T_c$ the ordered phase begins to smoothly disorder (the order parameter smoothly changes with temperature). But in reality we have a different situation, in a sense, the opposite: A completely disordered phase with zero average energy per spin exists above a certain critical temperature, and begins to smoothly order with decreasing temperature. Is there any way to explain the discrepancy of this model?
 A: I am not sure I understand your question, as it phrases Peierls' argument as a general, but rough (basically heuristic) thermodynamic type argument (which it is not, even though it is often presented as one). In any case, I'll try to address some issues that might help.

My main point is that Peierls' contour argument does not tell you anything about the phase transition from the paramagnetic phase to the ferromagnetic phase, only that the latter exists at sufficiently low temperatures.
This can be seen by observing that in dimensions $3$ and more, the $-$ spins percolate in the $+$ phase for temperatures $T$ satisfying $T_0<T<T_c$ for some $T_0$ depending on the dimension. So, the probability of having an infinitely large "droplet" surrounding a given point is strictly positive when $T$ satisfies $T_0<T<T_c$. This means that Peierls' argument cannot be applied at a temperature above $T_0$ and thus cannot detect the transition occurring at $T_c$.
Of course, one might try to resort to more sophisticated variants. Say, by introducing block-spins of a suitable size $R(T)$ (partition the system in boxes of size $R(T)$ and declare a box to be of type $+$ if you see something typical of the $+$ phase inside, of type $-$ if you see something typical of the $-$ phase inside, and $0$ otherwise). Peierls contours defined in terms of such block spins would be fine. Notice, however, that $R(T)$ will have to diverge as $T$ gets close to $T_c$, so that, again, Peierls' argument will not tell you anything about the transition at $T_c$.

Even in dimension $2$, where $T_0=T_c$, I don't get your question. The typical diameter of a droplet surrounding a given point diverges as $T$ approaches $T_c$ in this case. So Peierls' argument certainly does not imply that the order parameter is constant for $T<T_c$, nor that it smoothly changes for $T>T_c$. Indeed, the opposite is true: the order parameter smoothly decreases to $0$ as $T$ increases towards $T_c$, and remains identically $0$ at and above $T_c$. 
