  The Bethe equations, arising in description of the spectrum of the dilatation
operator for the su(2) sector of the N=4 supersymmetric Yang-Mills theory, are
considered in the anti-ferromagnetic regime. These equations are deformation of
those for the Heisenberg XXX magnet. It is proven that in the thermodynamic
limit roots of the deformed equations group into strings. It is proven that the
corresponding Yang's action is convex, which implies uniqueness of solution for
centers of the strings. The state formed of strings of length (2n+1) is
considered and the density of their distribution is found. It is shown that the
energy of such a state decreases as n grows. It is observed that
non-analyticity of the left hand side of the Bethe equations leads to an
additional contribution to the density and energy of strings of even length.
Whence it is concluded that the structure of the anti-ferromagnetic vacuum is
determined by the behaviour of exponential corrections to string solutions in
the thermodynamic limit and possibly involves strings of length 2.
