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Random graph asymptotics on high-dimensional tori: volume, diameter and mixing time

van der Hofstad, R (Eindhoven)
Thursday 08 April 2010, 15:00-16:00

Seminar Room 1, Newton Institute


We study finite-size scaling in high-dimensional percolation, where edges are independently kept with probability p.

It is well known that percolation has a phase transition, i.e., there exists a critical value p_c such that for p>p_c there is a unique infinite connected component, while for p < p_c all connected components are finite.

In the last decades, substantial progress has been made in the understanding of percolation in high-dimensions. In particular, we know that there is no infinite cluster at the critical value, and various critical exponents have been identified as the ones for percolation on a tree. The reason for this is that the geometry trivializes, since far away pieces of connected components are close to independent.

In this talk we shall investigate the largest connected component for critical percolation on a high-dimensional torus. We shall prove that it shares many features with percolation on the complete graph, or the Erd\H{o}s-R\'enyi random graph (ERRG). We shall also discuss conjectures for the scaling limit, as well as for inhomogeneous spatial percolation models where the edges remain independent, but depend on certain vertex weights. For non-spatial versions, which Bollob\'as, Janson and Riordan call the rank-1 inhomogeneous random graph, we have a characterization when the scaling limit of the critical clusters ordered in size is equal to the one on the ERRG, and when it is different.

This is joint work with Shankar Bhamidi, Johan van Leeuwaarden, Markus Heydenreich, Gordon Slade, Christian Borgs, Jennifer Chayes and Joel Spencer.


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