A Sketch of Menshikov’s Theorem

Let −→ Λ be an infinite, locally finite oriented multi-graph with C− → Λ finite and strongly connected, and let p < pH( −→ Λ ). Menshikov’s theorem states that there exists an α > 0 s.t. Pp(x n →) ≤ exp(−αn/(logn)) for all sites x and n ≥ 2. In this paper, we begin backwards, explaining the importance of this theorem and then giving a sketch of the proof, emphasizing the intuition and motivations behind the methods used. Unfortunately, due to constraints on space, we’ll sometimes simply summarize a statement and defer its rigorous proof to the text, when the proof is relatively short or simple or when the proof is too technical for an article.