%0 Journal Article %1 addarioberry2012continuum %A Addario-Berry, L. %A Broutin, N. %A Goldschmidt, C. %D 2012 %I Springer-Verlag %J Probability Theory and Related Fields %K Gromov-Hausdorff continuum_random_tree phase_transition random_graphs stochastic_processes %N 3-4 %P 367-406 %R 10.1007/s00440-010-0325-4 %T The continuum limit of critical random graphs %U http://dx.doi.org/10.1007/s00440-010-0325-4 %V 152 %X We consider the Erdős–Rényi random graph G(n, p) inside the critical window, that is when p = 1/n + λn −4/3, for some fixed λ∈ℝ . We prove that the sequence of connected components of G(n, p), considered as metric spaces using the graph distance rescaled by n −1/3, converges towards a sequence of continuous compact metric spaces. The result relies on a bijection between graphs and certain marked random walks, and the theory of continuum random trees. Our result gives access to the answers to a great many questions about distances in critical random graphs. In particular, we deduce that the diameter of G(n, p) rescaled by n −1/3 converges in distribution to an absolutely continuous random variable with finite mean.

@article{addarioberry2012continuum, abstract = {We consider the Erdős–Rényi random graph G(n, p) inside the critical window, that is when p = 1/n + λn −4/3, for some fixed λ∈ℝ . We prove that the sequence of connected components of G(n, p), considered as metric spaces using the graph distance rescaled by n −1/3, converges towards a sequence of continuous compact metric spaces. The result relies on a bijection between graphs and certain marked random walks, and the theory of continuum random trees. Our result gives access to the answers to a great many questions about distances in critical random graphs. In particular, we deduce that the diameter of G(n, p) rescaled by n −1/3 converges in distribution to an absolutely continuous random variable with finite mean.}, added-at = {2015-05-19T21:37:32.000+0200}, author = {Addario-Berry, L. and Broutin, N. and Goldschmidt, C.}, biburl = {https://www.bibsonomy.org/bibtex/27a3a90a9da56db10d0cbd58c63e736da/peter.ralph}, doi = {10.1007/s00440-010-0325-4}, interhash = {4c1a531188a58825c70ea85a2d6640e6}, intrahash = {7a3a90a9da56db10d0cbd58c63e736da}, issn = {0178-8051}, journal = {Probability Theory and Related Fields}, keywords = {Gromov-Hausdorff continuum_random_tree phase_transition random_graphs stochastic_processes}, language = {English}, number = {3-4}, pages = {367-406}, publisher = {Springer-Verlag}, timestamp = {2015-05-19T21:37:32.000+0200}, title = {The continuum limit of critical random graphs}, url = {http://dx.doi.org/10.1007/s00440-010-0325-4}, volume = 152, year = 2012 }