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).


1 Answer 1


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 $\mathbb{Z}^2$.

The first (rigorous) proof, as far as I know, can be found in H.-O. Georgii, Spontaneous magnetization of randomly dilute ferromagnets, J. Statist. Phys. 25 (1981), no. 3, 369-396.

  • $\begingroup$ 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 $\beta$. $\endgroup$
    – valerio
    Commented Oct 9, 2016 at 18:29
  • $\begingroup$ 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. $\endgroup$
    – valerio
    Commented Oct 9, 2016 at 18:45
  • $\begingroup$ I am unfortunately not familiar with the literature on the Ising model on graphs of fractional dimension. It is not even obvious to me why the dimension of the graph would be the only relevant property. What class of graphs do you have in mind? Which notion of fractal dimension? $\endgroup$ Commented Oct 9, 2016 at 19:10
  • $\begingroup$ 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. $\endgroup$ Commented Oct 9, 2016 at 19:12
  • $\begingroup$ 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). $\endgroup$
    – valerio
    Commented Oct 9, 2016 at 19:29

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