Why does the Higgs field fall into the same ground state at all points across space? I'm struggling to understand the concept of spontaneous symmetry breaking. I understand that the sign of the coefficient $\mu^2 > 0$ in the Higgs potential:
$$ V(\phi) = \mu^2 \phi^{\dagger} \phi - \frac{\lambda}{4} (\phi^{\dagger}\phi)^2 $$
Leads to the minimum of the classical potential being nonzero and the Higgs developing a non-zero VEV $\langle \phi\rangle = \frac{\nu^2}{2}$
My troubles follow: am I correct in saying that the Higgs potential is 'replicated' at every point in space, and that the gauge symmetry of the SM is spontaneously broken when the Higgs field selects the same ground state across all points in 3-space? 
If so, why should the Higgs field at space point $x_1$ collapse into the same ground state as the Higgs field at point $x_2$? I understand that the circle of degenerate minima form a spherical shell $\phi^{\dagger}\phi = \frac{\nu^2}{2}$, so why does the Higgs field choose the same point on this shell across all space?
A similar question could be asked about minimization of free energy in the Ginzburg-Landau theory of superconductivity, in this case there is a $U(1)$ symmetry and the ground state picks a unique phase at every point across the system. But why?
 A: Important question!
When the Higgs symmetry breaks to give a particular vev at the first point $x_1$ then, as other answers point out, neighbouring points will break to the same value.
The domain will spread, presumably at lightspeed.
But in a large universe there may well be some point $x_2$, far from $x_1$,  which chooses a different vev, and will start its own spreading domain. In time the regions will meet and form a domain wall.  Like a ferromagnet.
(I have more than once sat at a circular table at a conference dinner where guests had to choose whether to use the glass on their left or on their right. Usually one brave person would make a choice, their neighbours would follow, and all would be well. Sometimes two brave people, some distance apart, made different choices so that one guest finally ended with two glasses and one with none: they were the domain walls.)
As the observed universe all has the same Higgs vev (as far as we can see) that means it must have been causally connected when the symmetry was broken. So this is a big argument for inflation. The Higgs symmetry was broken with different vevs in different regions and all sorts of interesting topological features, but an enormous expansion took a small homogenous patch and blew it up to the scale of our observed universe.
A: The OP asks:

I understand that the circle of degenerate minima form a spherical
  shell $\phi^{\dagger}\phi=\frac{\nu^2}{2}$, so why does the Higgs
  field choose the same point on this shell across all space?

The shape of the potential that gives rise to the degenerate minima, forms this spherical shell as a collection of continuously connected points that all give the same VEV. So, to say this differently, the value of the VEV would not have been different if another point on the shell has been chosen. So then why this particular point?
Actually, one should perhaps not assume that the point on the shell is the same throughout space. We already know that Goldstone bosons represent excitations of the motion along the valley, in other words, the motion of the point as it remains on the shell. So, in principle the point could be moving around on the shell as we move from point to point through space. However, as long as it remains on the shell, the VEV would be the same everywhere.
A: I like to answer this question after reading the comment of @Diracology.
Before the unification between the weak and e.m. force broke, very early in the history of the Universe [when the temperature was about $10^{15}(K)$], the field connected to the Mexican Hat potential (present at every point in space) was zero and corresponding to a maximum, constant energy throughout space.
When the temperature dropped the unification between the two forces broke (after which the weak and e.m. force became two distinguishable forces), because the zero field connected to the MH potential came to lie on a random point on the circle at the rim of the MH. Each of these points corresponded (corresponds) to the same vev. It is said in another answer:

If the Higgs field had different values at different points in space i.e., if it had a spacetime variation, then the gradient term would give a positive contribution to the Hamiltonian, and hence, the total energy will not be minimized. 

I don't think this is a real physical explanation, though. It just gives an explanation with the aid of math, which comes after the physics. I mean, the fact that the field doesn't fall at the same place on the rim of the hat everywhere is translated to in the "language" of math. The field doesn't "know" about the math and just falls in a configuration with least energy, which is one that has a non-random variation throughout space.
Just look at this picture:

Left one obviously sees the MH potential, while on the right one can see the cross-section of a cosmic string. The arrows represent the omnipresent Higgs field. The diameter is about the same as that of a proton [$1(fm)$]. Inside the string, the same conditions are present as the conditions in the early Universe, just before the electroweak symmetry breaking so the temperature inside the string is about $10^{15}(K)$. The configuration of the Higgs field has, in this case, a zero gradient (it may seem that the gradient isn't zero, as the electric classical field around an electron, but in this case the absolute value of the field, unlike its direction, has the same value everywhere), which means the total energy is minimized.
This one example shows that the Higgs field falls does not have to fall in the same ground state at all points in space. Though such a string (of which it is thought that there is one in each Hubble volume) has (probably) never been observed, there is a theoretical possibility that it exists. When these cosmic strings (not to be confused with the strings from string theory) exist and one passes the Earth closeby, I wouldn't be able anymore to write this down (let alone if one passes **through the Earth).
There were theories that postulated two-dimensional versions of the cosmic string (textures), but these were ruled out by experiment.
A: If I understand it correct, then in a nutshell, you are asking why is the VEV independent of spacetime. If the Higgs field had different values at different points in space i.e., if it had a spacetime variation, then the gradient term would give a positive contribution to the Hamiltonian, and hence, the total energy will not be minimized.
