Abstract. We study some spectral properties of random walks on infinite countable amenable groups with an emphasis on locally finite groups, e.g. the infinite symmetric group S ∞ . On locally finite groups, the random walks under consideration are driven by infinite divisible distributions. This allows us to embed our random walks into continuous time Lévy processes whose heat kernels have shapes similar to the ones of α-stable processes. We obtain examples of fast/slow decays of return probabilities, a recurrence criterion, exact values and estimates of isospectral profiles and spectral distributions, formulae and estimates for the escape rates and for heat kernels.

Let π : G Ñ U pHq be a unitary representation of a locally compact group. The braiding operator F : H b H Ñ H b H, which flips the components of the Hilbert tensor product F pv b wq " w b v, belongs to the von Neumann algebra W ˚ppπ b πqpG ˆGqq if and only if π is irreducible. Suppose G is semisimple over a local field. If G is non-compact with finite center, P ă G is a minimal parabolic, π : G Ñ U pL 2 pG{P qq is the quasi-regular representation, then lim nÑ8 1 ş Bn Ξpgq 2 dg ż Bn πpgq b πpg ´1qdg " F, in the weak operator topology, where Ξ is the Harish-Chandra function of G and B n is the ball of radius n around the identity defined by a natural length function on G.

Let G be a locally compact non-compact metric group. Assuming that G is abelian we construct symmetric aperiodic random walks on G with probabilities n → P(S2n ∈ V) 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 G, the decay of the function n → P(S2n ∈ V) can be made as slow as possible by choosing appropriate aperiodic random walks Sn on G.