# What is the lower critical dimension (LCD) of the bond-diluted Ising model?

It is known that the lower critical dimension (LCD) $$d_l$$ of the Ising model is $$d_l=1$$, that the LCD of the Edwards-Anderson model is $$d_l=5/2$$ (source) and that the LCD of the random field Ising model is $$d_l=2$$ (source).

My question is: what is the LCD of the bond-diluted Ising model?

The bond-diluted Ising model is defined by the hamiltonian

$$H = - \sum_{\langle ij \rangle} J_{ij} \sigma_i \sigma_j$$

where the sum runs over nearest neighbors, $$\sigma_i=\pm 1$$ are the usual spin variables and $$J_{ij}$$ is $$1$$ with probability $$p$$ and $$0$$ with probability $$1-p$$.

Clarification: I know that there can be no phase transition in $$d=1$$ because the one-dimensional bond percolation threshold is $$p_c(1)=1$$ and a phase transition is possible only if $$p>p_c(d)$$ (see Yvan Velenkis's answer and following comments), but I was wondering whether more general results including non-integer dimensions are available (see this paper and this paper for examples of discussions where the dimension is treated as a real variable).

If the bonds of the two-dimensional square lattice are removed independently with probability $p$, then there is a phase transition for the Ising model on the resulting graph, at large enough $\beta$, for any $p>p_{\rm c}(2)$, where $p_{\rm c}(2)=\tfrac12$ is the critical percolation threshold for bond percolation on $\mathbb{Z}^2$.
• Thank you for your answer. I am aware of this result, as far as I know can be generalized to $\mathbf Z^n$ if $p_c(n)$ is known. I was actually wondering what the lower critical dimension for this model is, i.e. what is the dimension $d_l$ such that there is no phase transition for $d\leq d_l$, regardless of $p$ and $\beta$. Oct 9, 2016 at 18:29
• There is no phase transition in $d=1$, because $p_c(1)=1$, but I was wondering wether more general results considering also non-integer dimension exist. Oct 9, 2016 at 18:45
• Actually, the following paper seems to address the Ising model on a large class of graphs with dimension between $1$ and $2$: the paper. I don't know whether their criteria are robust enough to give you the result you want. I also haven't read this paper, so I cannot guarantee that the result is rigorous. Oct 9, 2016 at 19:12
• Thank you, I will take a look at the paper. Anyway, the idea is usually just to treat $d$ as a real variable, like they do in this paper, where they estimate the free energy cost of the creation of a domain wall between the two phases and use this estimate to infer that $d_l=5/2$ (Edwards-Anderson model). They don't actually give details about the meaning of this non-integer dimension, but I guess that with some effort the definition could me made more rigorous (some kind of fractal dimension, probably). Oct 9, 2016 at 19:29