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Consider the infinite cartesian product $\Omega=\{0,1\}^{\mathbb{N}}$ as a measurable space endowed with the $\sigma$-algebra $\mathscr{F}$ generated by the cylinder sets and $\sigma:\Omega\to\Omega$ …

Consider the two-dimensional lattice $G=(\mathbb{Z}^2,\mathbb{E}^2)$, where edge set $\mathbb{E}^2$ is given by the pairs of nearest neighbors in the $\ell^1$ norm in $\mathbb{Z}^2$. Let $V\subset\ma …

Let be $\mathcal M(\partial\mathbb D)$ denote the set of all Borel complex probability measures on $\partial\mathbb D$ (unit circle in the complex plane). Define a mapping $\Phi:\mathcal M(\partial\ma …

I would like to point out that extremal Gibbs measures are interesting class of measures that can be defined over $2^{\omega}$ and satisfies a zero-one law, not in the whole $\sigma$-algebra generated …

(It is not an answer but I put it here because I am having problems to post it in the comments) Hi Gil, thinking about the question 3 comes in my mind the Gibbs measures. It does not maximize the en …

Hi Joseph, I think the literature about independent site percolation model is relevant for your questions. There is a nice book, that address yours questions in this post. Percolation by Geoffrey G …

Let $\mathbb Z^2$ denote the two-dimensional integer lattice with norm of $i=(i_1,i_2)$ given by $\|i\|=|i_1|+|i_2|$. For each $x\in\mathbb Z^2$, we assign a uniform random variable, $\sigma_x$ taki …

Let $d>0$ be an integer and consider the first neighbors independent bond percolation model in $\mathbb Z^d$, where each edge is open with probability $p\in[0,1]$. I would like to know, if can we gen …

$\ \ \ $ In Time Evolution of Infinite Anharmonic Systems, Lanford,Lebowitz and Lieb, roughly speaking, proved that for some families of functions $F_v$ $(v\in\mathbb Z^d)$ and a large set of initial …

Hi Yaroslav, With the additional hypothesis you are considering, I guess that the inequality can be proved following the text that you linked. Fix any probability vector $(p_1,\ldots,p_k)$ and consi …

Probably is not general as you want, but if you don't think before about that can be a begining... Proposition: If $\pi:Y\to X$ is bijective function such that $\pi^{-1}$ is continuous then $\mathbb …

Background: Let $\mathbb Z^d$ denote the $d$-dimensional integer lattice with norm $|x|=\sum_i|x_i|$. For each $x\in\mathbb Z^d$ we associate a spin variable, $\sigma_x$ taking values on the set $ …

if we assume that the index $t$ in your process is countable then what you are looking for is described in Georgii's book, Gibbs Measures and Phase Transition. In the language of mathematical Statisti …