Nonconventional averages along arithmetic progressions and lattice spin systems

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