I have some questions regarding the Ising model in the presence of a magnetic field which is non-uniform.
Let us define the Ising Hamiltonian on a $d-$dimensional lattice,
$$ H = -\frac{1}{2} \sum_{i,j}J \sigma_i \sigma_j - \sum_i h_i \sigma_i,$$
where $J$ is the interaction parameter, $h_i$ the magnetic field on site $i$, and $\sigma_{i,j}$ the spin variable on sites $(i,j)$ ($i.e.$, $\sigma_{i,j} = +1$ or $-1$).
I have two questions:
(1) When $h$ is uniform, according to the Lee-Yang theorem, there is no phase transition in the thermodynamic limit with respect to temperature (I guess, $\forall J$ positive or negative). Does this result apply for any dimension $d$ and in particular, in the infinite dimensional lattice (= mean-field) case? Also, does it depend on the lattice structure ?
(2) When $h$ is non-uniform, is there any phase transition with respect to temperature ? In particular, I am interested in the mean-field case. I found some solutions in this paper. It said that "for any positive (resp. negative) and bounded magnetic field, the model does not present the phase transition phenomenon whenever lim inf $h_i > 0$", but I am not sure to understand. Does that mean that if the minimum value of the magnetic field over all the lattice is positive and not infinite, then there is no phase transition ?