So I define a random graph from $\mathbb{Z}^d[\math] (edges between vertices that are next to each other) by opening each edges independently with probability <script type="math/tex"> p[\math]. In dimension one the graphs are isomorphic to one another with probability one. The same argument (based on connected compnents and Borel-Cantelli) also works for <script type="math/tex"> d &gt; 1 [\math] for <script type="math/tex">p \<script type="math/tex">below the percolation threshold. Is the result true in all dimension and for all <script type="math/tex"> p [\math] ?$