Upper bounds on the percolation correlation length

Friday, 8 February, 2019

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We study the size of the near-critical window for Bernoulli percolation on \mathbb Z^d. More precisely, we use a quantitative Grimmett-Marstrand theorem to prove that the correlation length, both below and above criticality, is bounded from above by \exp(C/|p-p_c|^2). Improving on this bound would be a further step towards the conjecture that there is no infinite cluster at criticality on \mathbb Z^d for every d\ge2.


Hugo Duminil-Copin
Gady Kozma
Vincent Tassion