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The boundary of a square tiling of a graph coincides with the Poisson boundary

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Figure

Figure 1: The infinite binary tree and its square tiling.
Figure 2: The local situation at a vertexM x whose span T (x) is dissected by a meridian in the tiling and the graph.
Figure 3: The path P c�.
Figure 4: A counterexample to Conjecture 8.2. The righmost and leftmost geodesicsare to be thought as being identified

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