3 typos; "dimension" <--> "size";

The Ising model on $\mathbb{Z} / (2d+1)\mathbb{Z}$ 2d\mathbb{Z}$gives to the configuration$x=(x_0, \ldots, x_{2d+1}x_{2d-1}) \in {-1,+1}^{2d+1}$\{-1,+1\}^{2d}$ a probability proportional to $\exp(\beta \exp\big(\beta \sum_i x_ix_{i+1})$. x_ix_{i+1} \big)$. The Gibbs sampler with block updates is a Markov chain$X_k$that evolves on the set of such configurations and updates the odd (resp. even) indices conditionally on the even (resp. odd) indices with probability a half. It seems like a relatively straightforward application of the path coupling [1] approach (two configurations are neighbours if they agree on all odd or all even coordinates; distance between two neighbours is$1+H(x,y)/d$where$H$is the Hamming distance) shows that the mixing time of the Gibbs sampler does not depend on stays bounded as the dimension size$d$, d$ of the system goes to infinity, which looks rather surprising. Any intuition behind that? If this is already written somewhere, any reference concerning this (or similar) result?

• [1] Chapter 14 of Markov Chains and Mixing Times by D. Levin, Y. Peres and E. Wilmer
2 add reference for path coupling

The Ising model on $\mathbb{Z} / (2d+1)\mathbb{Z}$ gives to the configuration $x=(x_0, \ldots, x_{2d+1}) \in {-1,+1}^{2d+1}$ a probability proportional to $\exp(\beta \sum_i x_ix_{i+1})$. The Gibbs sampler with block updates is a Markov chain $X_k$ that evolves on the set of such configurations and updates the odd (resp. even) indices conditionally on the even (resp. odd) indices with probability a half.

It seems like a relatively straightforward application of the path coupling [1] approach (two configurations are neighbours if they agree on all odd or all even coordinates; distance between two neighbours is $1+H(x,y)/d$ where $H$ is the Hamming distance) shows that the mixing time of the Gibbs sampler does not depend on the dimension $d$, which looks rather surprising. Any intuition behind that? If this is already written somewhere, any reference concerning this (or similar) result?

• [1] Chapter 14 of Markov Chains and Mixing Times by D. Levin, Y. Peres and E. Wilmer
1

# Ising model on a cycle

The Ising model on $\mathbb{Z} / (2d+1)\mathbb{Z}$ gives to the configuration $x=(x_0, \ldots, x_{2d+1}) \in {-1,+1}^{2d+1}$ a probability proportional to $\exp(\beta \sum_i x_ix_{i+1})$. The Gibbs sampler with block updates is a Markov chain $X_k$ that evolves on the set of such configurations and updates the odd (resp. even) indices conditionally on the even (resp. odd) indices with probability a half.

It seems like a relatively straightforward application of the path coupling approach (two configurations are neighbours if they agree on all odd or all even coordinates; distance between two neighbours is $1+H(x,y)/d$ where $H$ is the Hamming distance) shows that the mixing time of the Gibbs sampler does not depend on the dimension $d$, which looks rather surprising. Any intuition behind that? If this is already written somewhere, any reference concerning this (or similar) result?