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Let be a locally compact non-compact metric group. Assuming that is abelian we construct symmetric aperiodic random walks on with probabilities $n\mapsto \mathbb{P}\left(S_{2n}\in V\right)$ of return to any neighborhood V of the neutral element decaying at infinity almost as fast as the exponential function n ↦ exp(-n). We also show that for some discrete groups , the decay of the function $n\mapsto \mathbb{P}\left(S_{2n}\in V\right)$ can be made as slow as possible by choosing appropriate aperiodic random walks Sₙ on .