Exponential Rates of Convergence in the Ergodic Theorem: A Constructive Approach

Bosco, G.; Machado, Fábio Prates; Ritchie, Thomas Logan

doi:10.1007/s10955-010-9945-4

Cited by 4 publications

(15 citation statements)

References 8 publications

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“…Recall that for v ∈ L, ∆h v = 1 Note that Z n and Z n are measurable with respect to σ| Λ n+1 and σ | Λ n+1 , respectively. As both ∆h and ∆h are (Z 2 ) even -ffiid (by our assumption), we can deduce that the convergence in the ergodic theorem occurs at an exponential rate [9] (the statement there is for the entire group of translations, but the argument does not require this). Hence, P(Z n ≤ 0) + P(Z n ≥ 0) ≤ Be −bn 2 for some B, b > 0 and for all n ≥ 1.…”

Section: Recall the Definition Of The Diagonal Gradientmentioning

confidence: 75%

“…Let us mention a simple consequence of Theorem 1.1 and the previously stated fact that the Ising model is ffiid for all β ≤ β c (d). It is a general fact that any ffiid random field satisfies the ergodic theorem with an exponential rate of convergence [9]. Applied to the gradient of the Ising, this yields a volume-order large deviation estimate for the energy H V (σ) defined in (1.1).…”

Section: The Ising and Potts Models At Low Temperaturementioning

confidence: 98%

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

Ray,

Spinka

2019

Preprint

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It is known that the Ising model on Z d at a given temperature is a finitary factor of an i.i.d. process if and only if the temperature is at least the critical temperature. Below the critical temperature, the plus and minus states of the Ising model are distinct and differ from one another by a global flip of the spins. We show that it is only this global information which poses an obstruction for being finitary by showing that the gradient of the Ising model is a finitary factor of i.i.d. at all temperatures. As a consequence, we deduce a volume-order large deviation estimate for the energy. A similar result is shown for the Potts model.A result in the same spirit is also shown for the six-vertex model, which is itself the gradient of a height function, with parameter c 6.4. We show that the gradient of the height function is not a finitary factor of an i.i.d. process, but that its "Laplacian" is. For this, we introduce a coupling between the six-vertex model with c ≥ 2 and a new graphical representation of it, reminiscent of the Edwards-Sokal coupling between the Potts and random-cluster models. We believe that this graphical representation may be of independent interest and could serve as a tool in further understanding of the six-vertex model.To provide further support for the ubiquity of this type of phenomenon, we also prove an analogous result for the so-called beach model.The tools and techniques used in this paper are probabilistic in nature. The heart of the argument is to devise a suitable tree structure on the clusters of the underlying percolation process (associated to the graphical representation of the given model), which can be revealed piece-by-piece via exploration.

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Section: Recall the Definition Of The Diagonal Gradientmentioning

confidence: 75%

Section: The Ising and Potts Models At Low Temperaturementioning

confidence: 98%

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

Ray,

Spinka

2019

Preprint

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“…Define

Z_{n} : = \frac{1}{| Λ_{n} \cap 𝕃 |} \sum_{v \in Λ_{n} \cap 𝕃} Δ h_{v} and Z_{n}^{'} : = \frac{1}{| Λ_{n} \cap 𝕃 |} \sum_{v \in Λ_{n} \cap 𝕃} Δ h_{v}^{'} .

Note that

Z_{n}

and

Z_{n}^{'}

are measurable with respect to

σ |_{Λ_{n + 1}}

and

σ^{'} |_{Λ_{n + 1}}

, respectively. As both

Δ h

and

Δ h^{'}

are

{(ℤ^{2})}_{even}

‐ffiid (by our assumption), we can deduce that the convergence in the ergodic theorem occurs at an exponential rate [11] (the statement there is for the entire group of translations, but the argument does not require this). Hence,

ℙ (Z_{n} \leq 0) + ℙ (Z_{n}^{'} \geq 0) \leq B e^{- b n^{2}}

…”

Section: The Six‐vertex Modelmentioning

confidence: 78%

“…Let us mention a simple consequence of Theorem 5 and the previously stated fact that the Ising model is ffiid for all 𝛽 ≤ 𝛽 c (𝑑). It is a general fact that any ffiid random field satisfies the ergodic theorem with an exponential rate of convergence [11]. Applied to the gradient of the Ising, this yields a volume-order large deviation estimate for the energy H V (𝜎) defined in (6).…”

Section: The Ising and Potts Modelsmentioning

confidence: 98%

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

Ray

Spinka

2021

Random Struct Algorithms

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It is known that the Ising model on Z 𝑑 at a given temperature is a finitary factor of an i.i.d. process if and only if the temperature is at least the critical temperature. Below the critical temperature, the plus and minus states of the Ising model are distinct and differ from one another by a global flip of the spins. We show that it is only this global information which poses an obstruction to being finitary by showing that the gradient of the Ising model is a finitary factor of i.i.d. at all temperatures. As a consequence, we deduce a volume-order large deviation estimate for the energy. Results in the same spirit are shown for the Potts model, the so-called beach model, and the six-vertex model. We also introduce a coupling between the six-vertex model with c ≥ 2 and a new Edwards-Sokal type graphical representation of it, which we believe is of independent interest.

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Finitary codings for spatial mixing Markov random fields

Spinka¹

2020

Ann. Probab.

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Exponential Rates of Convergence in the Ergodic Theorem: A Constructive Approach

Cited by 4 publications

References 8 publications

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

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

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

Finitary codings for spatial mixing Markov random fields

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