The Nonbacktracking Lace Expansion
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In the previous chapters, we have concluded the argument that percolation has mean-field behavior in sufficiently high dimensions. The analysis, however, is not very explicit about what dimension suffices in order to make the analysis work. The main result of this chapter is \(d \ge 11\) suffices, as recently proved by Fitzner and van der Hofstad. To this end, we introduce non-backtracking random walk and explain that this yields a better approximation to the percolation twopoint function than the random walk Green function. We explain the non-backtracking lace expansion, which makes this perturbation statement precise. Finally, we show how the bootstrap argument can be performed in this new setting. The proof that is highlighted in this chapter is computer assisted, and we finish with a discussion of the numerical aspects of the analysis.