We prove that the spectral gap of a finite planar graph $X$ is bounded by $lambda_1(X)le C(frac{log(diam X)}{diam X})^2$ where $C$ depends only on the degree of $X$. We then give a sequence of such graphs showing the the above estimate cannot be improved. This yields a negative answer to a question of Benjamini and Curien on the mixing times of the simple random walk on planar graphs.
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