Nonconventional averages along arithmetic progressions and lattice spin systems

TL;DR

This paper investigates nonconventional averages in lattice spin systems and random colorings, establishing large deviation principles and bounds, and connecting these sums to higher-dimensional statistical mechanics models.

Contribution

It introduces large deviation results for monochromatic arithmetic progressions in i.i.d. colorings and extends bounds to general colorings, linking nonconventional sums to higher-dimensional models.

Findings

Large deviation principle for monochromatic pairs with explicit rate function

Bounds for the number of monochromatic progressions of arbitrary size

Connection of nonconventional sums to higher-dimensional statistical mechanics

Abstract

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.