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The scaling limits of several families of random graphs are shown to exist by using the idea of Gromov-Hausdorff convergence to certain random metric spaces.

For instance, uniformly chosen triangulations of the sphere with $n$ faces endowed with the graph distance have been proved to converge (in the Gromov-Hausdorff sense) after rescaling distances by $n^{-1/4}$ to a particular random metric space called the Brownian map. See the references in this earlier answer of minethis earlier answer of mine.

The scaling limits of several families of random graphs are shown to exist by using the idea of Gromov-Hausdorff convergence to certain random metric spaces.

For instance, uniformly chosen triangulations of the sphere with $n$ faces endowed with the graph distance have been proved to converge (in the Gromov-Hausdorff sense) after rescaling distances by $n^{-1/4}$ to a particular random metric space called the Brownian map. See the references in this earlier answer of mine.

The scaling limits of several families of random graphs are shown to exist by using the idea of Gromov-Hausdorff convergence to certain random metric spaces.

For instance, uniformly chosen triangulations of the sphere with $n$ faces endowed with the graph distance have been proved to converge (in the Gromov-Hausdorff sense) after rescaling distances by $n^{-1/4}$ to a particular random metric space called the Brownian map. See the references in this earlier answer of mine.

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The scaling limits of several families of random graphs are shown to exist by using the idea of Gromov-Hausdorff convergence to certain random metric spaces.

For instance, uniformly chosen triangulations of the sphere with $n$ faces endowed with the graph distance have been proved to converge (in the Gromov-Hausdorff sense) after rescaling distances by $n^{-1/4}$ to a particular random metric space called the Brownian map. See the references in this earlier answer of mine.