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# Publications of the Research Institute for Mathematical Sciences

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**Volume 48, Issue 2, 2012, pp. 279–338**

**DOI: 10.2977/PRIMS/70**

Published online: 2012-06-03

Scaling Limit for the Random Walk on the Largest Connected Component of the Critical Random Graph

David A. Croydon^{[1]}(1) Warwick University, Coventry, UK

A scaling limit for the simple random walk on the largest connected component of the Erd\H{o}s-R\'{e}nyi random graph $G(n,p)$ in the critical window, $p=n^{-1}+\lambda n^{-4/3}$, is deduced. The limiting diffusion is constructed using resistance form techniques, and is shown to satisfy the same quenched short-time heat kernel asymptotics as the Brownian motion on the continuum random tree.

*Keywords: *random graphs, random walk in random environment, scaling limit, continuum random tree

Croydon David: Scaling Limit for the Random Walk on the Largest Connected Component of the Critical Random Graph. *Publ. Res. Inst. Math. Sci.* 48 (2012), 279-338. doi: 10.2977/PRIMS/70