Recent progress in high-dimensional percolation.
- Remco van der Hofstad (Eindhoven University of Technology, Netherlands)
It is now 25 years ago that Hara and Slade published their seminal work on the mean-field behavior of percolation in high-dimensions, showing that at criticality there is no percolation and identifying several percolation critical exponents. The main technique used is the lace expansion, a perturbation technique that allows us to compare percolation paths to random walks based on the idea that faraway pieces of percolation paths are almost independent in high dimensions.
In the past few years, a number of novel results have appeared for high-dimensional percolation. I intend to highlight the following topics:
- The recent computer-assisted proof, with Robert Fitzner, that identifies the critical behavior of nearest-neighbor percolation above 14 dimensions using the so-called Non-Backtracking Lace Expansion (NoBLE). While these results are expected to hold above 6 dimensions, the previous and unpublished proof by Hara and Slade only applied above 18 dimensions;
- The identification of arm exponents in high-dimensional percolation in two works by Asaf Nachmias and Gady Kozma, using a clever and novel difference inequality argument, and its implications for the incipient infinite cluster and random walks on them;
- The finite-size scaling for percolation on a high-dimensional torus, where the largest connected components share many features to the Erdos-Renyi random graph. In particular, substantial progress has been made concerning percolation on the hypercube, where joint work with Asaf Nachmias avoids the lace expansion altogether.
We assume no prior knowledge about percolation.