Finitary codings for spatial mixing Markov random fields

Finitary codings for gradient models and a new graphical representation for the six‐vertex model

“…Let us now outline some of the ideas and ingredients that go into the proof Theorem 8. The part of the result concerning the nonexistence of a finitary coding follows a similar argument as the one in [45] for general Markov random fields (though the argument here is slightly complicated by the existence of hard constraints) and we do not expand on it here. Let us explain the second part of the result, namely, that the absolute value of the diagonal gradient is ffiid.…”

“…It has been shown that the two‐type beach model (more precisely, any constant‐type Gibbs measure) is ffiid if and only if there is a unique Gibbs measure (see [45, Corollary 1.7]; the statement there refers to whether is above or below , but the proof only relies on whether the Gibbs measure is unique or not), and the beach‐random‐cluster representation allows to extend this to the multitype model. We are therefore led to consider a gradient of the model.…”

“…It is well known [13, 33] that when is sufficiently large (as a function of d ), there are two distinct extremal Gibbs measures, which we call here the plus and minus Gibbs measures, related to each other by a global flip of the spins. In this case, these measures are not ffiid (see [45, Section 1.1.4]), and the question of whether a gradient of theirs is arises. The gradient we consider here is simply the pointwise absolute value (a more informative gradient would be the random field defined for all pairs of vertices, not just nearest neighbors; to keep things simple, we do not consider this here).…”

Perfect Sampling in Infinite Spin Systems via Strong Spatial Mixing

“…Note that this task was first achieved by van den Berg and Steif [BS99] using a more complicated algorithm inspired by Propp and Wilson's Coupling From The Past (CFTP). Spinka [Spi20] also covers this situation in the context of a more general treatment.…”

Ising model on trees and factors of IID

“…For a measure ν on a measurable space Ω T 1 , ν is called a factor of IID if there exists a measurable space Ω 0 , a measure µ 0 on Ω 0 , and a T -factor φ : Ω T 0 → Ω T 1 such that ν is the φ-push-forward of the product measure µ ⊗T 0 . Besides the classical works from ergodic theory [19][20][21]4], study of factor of IIDs has drawn extensive interests in probability theory, e.g., [10,9,14,28,1,2,23,25,26]. In particular, a factor of IID on T can be interpreted as an infinite analogue of local algorithms on the random d-regular graph (whose limiting local structure is T ) [5,8,22].…”