Abstract
We establish conditions on sequences of graphs which ensure that the mixing times of the random walks on the graphs in the sequence converge. The main assumption is that the graphs, associated measures and heat kernels converge in a suitable Gromov-Hausdorff sense. With this result we are able to establish the convergence of the mixing times on the largest component of the Erdős-Renyi random graph in the critical window, sharpening previous results for this random graph model. Our results also enable us to establish convergence in a number of other examples, such as finitely ramified fractal graphs, Galton-Watson trees and the range of a high-dimensional random walk.
Highlights
The geometric and analytic properties of random graphs have been the subject of much recent research
We establish conditions on sequences of graphs which ensure that the mixing times of the random walks on the graphs in the sequence converge
The main assumption is that the graphs, associated measures and heat kernels converge in a suitable Gromov-Hausdorff sense
Summary
The geometric and analytic properties of random graphs have been the subject of much recent research. We are ready to state the assumption under which we are able to prove the convergence of mixing times for the random walks on a sequence of graphs This captures the idea that, when suitably rescaled, the discrete state spaces, invariant measures and transition densities of a sequence of graphs converge to (F, dF ), π and (qt(x, y))x,y∈F,t>0, respectively. We use these ideas to derive tail estimates of mixing times on random graphs in the case of the continuum random tree and the Erdos-Renyi random graph. The proofs of these results can be found in the Appendix
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