Nonconventional averages along arithmetic progressions and lattice spin systems

Details Nonconventional averages along arithmetic progressions and lattice spin systems Nonconventional averages along arithmetic progressions and lattice spin systems

2011-10-11

arxiv.org/abs/1110.2354v2

Mathematics

Probability

18 pages, 3 figures. A new section was added on arithmetic progressions of length larger than 2 and statistical mechanics mode

Scientific paper

We consider nonconventional averages in the context of lattice spin systems, or equivalently random colourings of the integers. For iid colouring, we prove a large deviation principle for the number of monochromatic arithmetic progressions of size two in the box $[1,n]\cap \Z$ with an explicit rate function. For more general colourings, we prove simple bounds for the number of monochromatic arithmetic progressions of arbitrary size, as well as for the maximal progression inside the box $[1,n]\cap\Z$.

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