Barbara Dembin, Shuta Nakajima: Upper tail large deviation rate function for chemical distance in supercritical percolation:


We consider supercritical bond percolation on Z^d and study the chemical distance, i.e., the graph distance on the infinite cluster. It is well-known that there exists a deterministic constant μ(x) such that the chemical distance D(0,nx) between two connected points 0 and nx grows like nμ(x). We prove the existence of the rate function for the upper tail large deviation event {D(0,nx)>nμ(x)(1+ϵ),0↔nx} for d>=3. To achieve this, we introduce a new notion that relates to the presence of a cut-point, which is responsible for the occurence of upper tail large deviations.

In the first part, we present our results and the main ideas of the proofs. In the second part, we will give a more detailed sketch of the proofs.

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