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Consider the first neighbors Ising model in $\mathbb Z^2$, with the Hamiltonian in the finite volume $\Lambda\subset\mathbb{Z}^2$ given by $$ H_{\Lambda}(\sigma|\omega)=-J\sum_{i,j\in\Lambda\atop{\|i …

Hi Gowers, In this paper: Sidoravicius, V. and Kesten, H . A shape theorem for the spread of an infection. Annals of Mathematics, v. 167, p. 1-63, 2008, the authors consider a different model, but …

Hi Marcin, recently writing a paper about phase transition on the Ising model with positive non-uniform magnetic field in infinite graphs I discovered that joining some results in the literature we …

I want to consider one-dimensional system on the lattice $\mathbb{L}=\mathbb{N}$. Let be $A:(\mathbb{S}^1)^{\mathbb{L}}\to\mathbb{R}$ a lipschtiz potential. Consider the Ruelle operator $$ \mathcal{L …

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 …

Hi all, I am doing a work in collaboration with other mathematicians about phase transition in the Ising model and we need to know if exponential upper bounds exist for the number of lattice animals …

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 $ …

This answer is just an expanding version of Kconrad answer's. I am posting it here because this argument support the variational method for finite state space and also touch in the observation made by …