A Central Limit Theorem for the Simple Random Walk on a Crystal Lattice

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Consider a lattice graph X realized in a k-dimensional vector space V (we shall use the same symbol X also for the set of vertices by abuse of notations). What we should have here in mind as a lattice graph is a generalization of classical lattices graphs such as the hyper-cubic lattice in ℝ k , the triangular lattice and the hexagonal lattice in ℝ2. We shall show, by using our previous result [1], that, as the mesh of X goes to zero, the simple (isotropic) random walk on X “converges” to the Brownian motion on V with a suitable Euclidean structure.