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A: mixed
B: mixed
+J: maximum
-J: minimum
-J: minimum
+J: maximum
3
3
1
2
1
2
A: mixed
B: happy
A: happy
B: mixed
-J: minimum
-J: minimum
Initial configuration
Unstable
First possibility
for next iteration
A: happy
B: unhappy
-J: minimum
-J: minimum
+J: maximum
+J: maximum
3
3
1
2
1
2
B: mixed
B: mixed
B: unhappy
B: unhappy
+J: maximum
+J: maximum
Second possibility
for next iteration
Third possibility
for next iteration
Fig.1.
Top left shows one possible configuration of alliances with countries 1 and 2
in A and country 3 in B. From it, countries 1 and 2 being in a mixed situation with
respect to optimzing their respective bilateral interations, three possible and equiprob-
able distributions are possible. In the first possible following configuration (top right),
country 1 has shifted alliance from A to B. However its move keeps it in its mixed
situation while making country 2 happy and country 3 mixed. Instead it could have
been country 2 which had shifted alliance (low right) making 1 happy and 3 mixed.
Last possibility (low left) is both 1 and 2 shifting simultaneously. It is the worse since
each country is unhappy.
to the Hamiltonian
H
=
1
2
P
N
i
=1
E
i
. Eq. (6) is minimum for
η
i
and
h
i
having
the same sign. For a given
h
i
there exists always a well defined coalition choice
except for
h
i
= 0. In this case site
i
is unstable. Then both coalitions are identical
with respect to its local energy which stays equal to zero. An unstable site flips
continuously with probability
1
2
(see Fig. (1)).
