Nonconventional averages along arithmetic progressions and lattice spin systems

作者：Carinci Gioia; Chazottes Jean Rene; Giardina Cristian^*; Redig Frank

来源：Indagationes Mathematicae, 2012, 23(3): 589-602.

DOI：10.1016/j.indag.2012.05.010

• Large deviations
• Nonconventional averages
• Concentration inequalities
• Ising model

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] boolean AND N, as N -%26gt; infinity, with an explicit rate function related to the one-dimensional Ising model. %26lt;br%26gt;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] boolean AND N. %26lt;br%26gt;Finally, we relate nonconventional sums along arithmetic progressions of size greater than two to statistical mechanics models in dimension larger than one.

• 出版日期2012-9