A. Georgakopoulos & C. Panagiotis: The percolation density \mbox{\huge{\theta(p)}} is analytic.

Abstract: We prove that for Bernoulli bond percolation on \mathbb{Z}^d, d\geq 2, the percolation density \theta(p) (defined as the probability of the origin lying in an infinite cluster) is an analytic function of the parameter in the supercritical interval (p_c,1]. This answers a question of Kesten from 1981.

The proof involves a little bit of elementary complex analysis (Weierstrass M-test), a few well-known results from percolation theory (Aizenman-Barsky/Menshikov theorem), but above all combinatorial ideas. We used a new notion of contours, bounds on the number of partitions of an integer, and the inclusion-exclusion principle, to obtain a refinement of a classical argument of Peierls that settled the 2-dimensional case in 2018. More recently, we coupled these techniques with a renormalisation argument to handle all dimensions.


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