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We study and classify proper q-colourings of the ℤd lattice, identifying three regimes where different combinatorial behaviour holds. (1) When
, there exist frozen colourings, that is, proper q-colourings of ℤd which cannot be modified on any finite subset. (2) We prove a strong list-colouring property which implies that, when
, any proper q-colouring of the boundary of a box of side length
$n \ge d+2$
can be extended to a proper q-colouring of the entire box. (3) When
, the latter holds for any
$n \ge 1$
. Consequently, we classify the space of proper q-colourings of the ℤd lattice by their mixing properties.
Consider an ergodic Markov chain on a countable state space for which the return times have exponential tails. We show that the stationary version of any such chain is a finitary factor of an independent and identically distributed (i.i.d.) process. A key step is to show that any stationary renewal process whose jump distribution has exponential tails and is not supported on a proper subgroup of ℤ is a finitary factor of an i.i.d. process.
A cutset is a non-empty finite subset of ℤd which is both connected and co-connected. A cutset is odd if its vertex boundary lies in the odd bipartition class of ℤd. Peled  suggested that the number of odd cutsets which contain the origin and have n boundary edges may be of order eΘ(n/d) as d → ∞, much smaller than the number of general cutsets, which was shown by Lebowitz and Mazel  to be of order dΘ(n/d). In this paper, we verify this by showing that the number of such odd cutsets is (2+o(1))n/2d.
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