# Weird fluctuations in Ising model

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Not 100% sure this thread belongs in this section, sorry if it's out of place.

I was trying to run a simulation of a 2D Ising model; i thought everything was going fine until I started to look at the numerical results and I noticed that I get some wild fluctuations in my results; as an example, here are the values i get for final energy and magnetization (per site) after 10 different 10000-step runs on a 128x128 lattice at temperature T = 0.1 K(critical temperature is at 2.27 K), where the initial spin configuration is chosen randomly(+1 or -1 is assigned to each node with equal probability):
$$(E, |M|) = (-2, 1); (-1.95, 0.52); (-2, 1); (-1.95, 0.21); (-2, 1); (-1.95, 0.02); (-2, 1); (-1.95; 0.24); (-2, 1); (-2; 1)$$
With a temperature so low I would expect all $|M|$ to be exactly 1, or at least that fluctuations like these would be way less probable than 4/10, so I guess there must be some error in my code. Am I wrong in thinking this?
I would be very grateful if someone could take a look at my code and maybe point out where the error is, cause it seems like I can't find it.

Here is the function used to update the lattice(it's Metropolis algorithm):
Code:
void update(graph *pLattice, int *s, double *prob, int *pE, int *pM){
/* pLattice is the pointer to the graph/lattice; s is the pointer
* to the array containing the spin value for each site; prob is the pointer to
* the lookup table used to map differences in energy to their probability according
* to Boltzmann distribution; pE and pM point respectively to the energy and
* magnetization values of the lattice. */
int k, site, *pNeigh, siteConn, sum;
for(site = 0; site < pLattice->numNodes; site++){ // loop over all nodes
pNeigh = pLattice->neigh[site]; // = neigh[site]
siteConn = pLattice->conn[site]; // = 4
sum = 0;
for(k = 0; k < siteConn; k++, pNeigh++){
sum += s[*pNeigh];  // = s[neigh[site][k]]
}
sum *= s[site]; // this is now the energy of site
if(sum <= 0 || drand48() < prob[sum]){
/* this is legit because sum = half the variation of energy due to the flip of s[site]
* and 0 < prob[sum] = exp(-2.0 * sum / T) < 1; (T=temperature) */
s[site] = -s[site];
*pE += 2*sum;
*pM += 2*s[site];
}
}
return;
}
"graph" is basically an adjacency list, defined as
Code:
typedef struct {
/* conn points to an array of size numNodes, whose items are the number of
* connections of each node(i'm using periodic bc so conn[i] = 4 for every value of i).
* neigh points to a 2d array of dimensions numNodes*numLinks,
* where the neighbours of each node are stored. */
} graph;
The main function, besides declaration of variables and error checking, is basically just
Code:
  E = energy(pLattice, s);
M = magnetization(pLattice, s);
for(i=0;i<R;i++){
update(pLattice, s, probLUT, &E, &M);
}
fprintf(stderr, "E = %f; M = %f\n", (double)E/N, (double)M/N);
where N is the number of nodes.

The difference in energy value between "correct" and "weird" runs is small, so when I average over lot of runs for each T, and for many values of T, the plot EvsT
looks not too different from the expected curve; the plot of |M|vsT still shows the correct behaviour, but instead of getting constant |M|=1 for $T<T_c$ i get values oscillating around |M|=0.8. I tried running for different times, different lattice size, different initial conditions, with no different results.
I'll post full code, graphs or whatever if needed. Thanks.

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Updating just in case anyone happens to come by and help me out. I created some animated plots for the 'weird' results and I found that they correspond to situations where two connected clusters(one for each spin) arise and reach some sort of equilibrium configuration where they are just translated (always) in the $-x$ direction through the periodic boundary. Here a few pics of a 64x64 lattice taken at times 850,851,852,856:    Basically it evolves for some time, then stabilizes on a configuration like this which is mantained for the rest of the simulation. This makes no sense at all to me: why would something like this happen? and how? I also tried rewriting the program all over, without looking at the original one, but I get the exact same thing. I guess this must be related to the periodic boundaries, so I'll try to implement free boundaries and see what happens.

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