Peierls's argument for the Ising model I was reading about Peierls's argument for the Ising model at this link and got a question.
The Hamiltonian is $$ H = -J \sum_{(i,j)} \sigma_i \sigma_j - \sum_i h_i \sigma_i $$ with the first summation extending over nearest neighbours, $J > 0$.
They talk about both $1$ and $2$D and I have the same question for both cases, although for the one-dimensional case it does not affect the final conclusion.
The competition between energy $U$ and entropy $S$ in the free energy $F = U - TS $ is considered.
For the $2$D case it is said that the energy difference between a completely ordered lattice with spins pointing up and a configuration with an island of spins pointing downwards, of perimeter $L$ equals
$$\Delta U = 2JL$$
To estimate the entropy contribution it is therein said " the number of ways to build an island which includes a fixed spin X " (bold mine) is roughly $$\mu^L$$, where $\mu$ is a number less than $3$.
The change in free energy is hence
$$ \Delta F = 2JL -kT L \log \mu$$ and contrary to the $1$D case both terms scale with $L$ and they can effectively compete to the point that a magnetisation different from $0$ appears at a sufficiently low temperature.
I do not understand why entropy is estimated with the condition of imposing the "flipped island" has to contain a fixed spin $X$.
The change in energy is independent of the location of the island. Wherever the flipped island appears, the energy change is the same. But the flipped island can appear anywhere on the lattice, why is this not accounted for in the entropy estimate?
Should not entropy be computed, including all the islands, with fixed perimeter $L$?
It seems at first sight entropy is severely underestimated if this fixed spin constraint is exerted.
 A: Peierls argument is to show that it is favorable to create a domain here (wherever the experimenter defines as "here"). In the thermodynamic limit, if it's favorable to create a domain "here", then it's favorable to create a domain anywhere else, and the whole system becomes disordered (i.e. the ordered state is not stable).
It's not enough to argue that it's favorable to create a domain wall anywhere. In the thermodynamic limit, it's always favorable, since as you pointed out, that entropic contribution scales with the system size. But that doesn't tell you anything about the phase: just because there's a domain somewhere, doesn't mean the system has become disordered. You need to show that the domain can be created "here" (and by symmetry, that means it's favorable to create domains everywhere, and put the system in disorder).
EDIT: more details to follow.
I believe your qualm is with the style of the argument, which I agree is a little unintuitive. Here's how I understand it.
Imagine we flip a single island of size $L$, some very large distance away. Compute the magnetization now. It's still 1 (or -1) because in the thermodynamic limit, the contribution of a single island is negligible. So, this (by itself) doesn't tell us anything about the emergence of the disordered phase.
Ok, but of course we are interested in the case where islands can spontaneously pop up anywhere and everywhere, thus truly changing the magnetization even in the thermodynamic limit. As you pointed out in the original question, the "perfect" calculation will involve computing the energy and entropy changes of any number of islands of any size. These calculations (which directly account for fluctuations) are extremely hard to do.
As a proxy for the "perfect" calculation, we use Peierls argument: compute the $\Delta F$ of a single island, and then extrapolate it to the multiple-island case. But in order for this extrapolation to make sense, we have to only consider local fluctuations (i.e. islands that appear at a particular place). The reason is because being able to flip one island at an arbitrary location doesn't imply that it's favorable to flip islands everywhere; as you add more islands, the premise of the original calculation breaks down. On the other hand, if a local fluctuation is favorable, then it does follow that fluctuations everywhere are favorable.
