Nonconventional averages along arithmetic progressions and lattice spin systems

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 ] ∩ N, as N → ∞, 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 ] ∩ N. Finally, we relate nonconventional sums along arithmetic progressions of size greater than two to statistical mechanics models in dimension larger than one.

Fluctuations of Observables in Dynamical Systems: From Limit Theorems to Concentration Inequalities

“…This theorem follows at once from (12) and the fact that the function K defined above has all its Lipschitz constants bounded by 1/n. A natural step further is to try to get an upper bound for dist K (E n (·), µ)dµ.…”

“…The previous procedure is usually called the 'Chernoff bounding trick'. Of course, we can apply this inequality to −K and deduce at once (12). Inequality (13) follows immediately from Markov's inequality.…”

“…Large deviations seem much more difficult to analyse and turn out to be nontrivial even for i.i.d. processes (see [12]).…”

Large Deviation Principle of Multidimensional Multiple Averages on $\mathbb{N}^d$

“…Denote by P r the product of Bernoulli with the parameter r on A. For σ ∈ A Z , the authors [6] study the thermodynamic limit of the free energy function associated to the sum…”

“…Note that the Hamiltonian ( 6) is longrange, non-translation invariant interaction and much more difficult to treat. In [6], the authors prove that the sequence of multiple average S N N satisfies a LDP with the rate function…”