A remark on the norm of a random walk on surface groups
We show that the norm of the random walk operator on the Cayley graph of the surface group in the standard presentation is bounded by 1/√g where g is the genus of the surface.
On the norms of the random walks on planar graphs
We consider the nearest neighbor random walk on planar graphs. For certain families of these graphs, we give explicit upper bounds on the norm of the random walk operator in terms of the minimal number of edges at each vertex. We show that for a wide range of planar graphs the spectral radius of the random walk is less than one.
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