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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.
In the context of stationary
nearest-neighbour Gibbs measures
satisfying strong spatial mixing, we present a new combinatorial condition (the topological strong spatial mixing property) on the support of
that is sufficient for having an efficient approximation algorithm for topological pressure. We establish many useful properties of topological strong spatial mixing for studying strong spatial mixing on systems with hard constraints. We also show that topological strong spatial mixing is, in fact, necessary for strong spatial mixing to hold at high rate. Part of this work is an extension of results obtained by Gamarnik and Katz [Sequential cavity method for computing free energy and surface pressure. J. Stat. Phys.137(2) (2009), 205–232], and Marcus and Pavlov [An integral representation for topological pressure in terms of conditional probabilities. Israel J. Math.207(1) (2015), 395–433], who gave a special representation of topological pressure in terms of conditional probabilities.
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