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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 …

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 …

Let be $G=(V,E)$, where $V=\{1,\ldots,n\}$ and $E=\{\{i,j\}\subset V;|i-j|\leq k\}$ and $k<n$. For which values of $k\geq 2$, can we count explicitly the number of Hamiltonian paths in $G$ ?

(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 all, Let be $G_n=(V_n,E_n)$ a finite graph, where $V_n= \{0,1,\ldots, n\} \times\{0,1,\ldots,n\}$ and $E_n\subset V_n^{(2)}$ is the edge set of the nearest neighbors in the $\ell^1$ norm, that …