  We study the so-called nonconventional averages in the context of lattice
spin systems, or equivalently random colourings of the integers. For i.i.d.
colourings, we prove a large deviation principle for the number of
monochromatic arithmetic progressions of size two in the box $[1,N]\cap \N$, as
$N\to\infty$, with an explicit rate function related to the one-dimensional
Ising model. For more general colourings, we prove some bounds for the number
of monochromatic arithmetic progressions of arbitrary size, as well as for the
maximal progression inside the box $[1,N]\cap \N$. Finally, we relate
nonconventional sums along arithmetic progressions of size greater than two to
statistical mechanics models in dimension larger than one.
