Random walks on unimodular p-adic groups

In a recent paper Pittet and Saloff-Coste established the lower bound p 2 n ( e , e ) ⩾ c exp ( - Cn 1 / 3 ) , n = 1 , 2 , … for the large times asymptotic behaviours of the probabilities p 2 n ( e , e ) of return to the origin at even times 2 n , for random walks associated with finite symmetric generating sets of solvable groups of finite Prüfer rank and asked if a similar lower bound is available in the case of the semi-direct product Sol ( Q p ) = Q p 2 ⋉ Q p * . In this paper, we give an answer to this problem.