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We consider the parabolic Anderson model with Weibull potential field, forall values of the Weibull parameter. We prove that the solution is eventuallylocalised at a single site with overwhelming probability (completelocalisation) and, moreover, that the solution has exponential shape around thelocalisation site. We determine the localisation site explicitly, and derivelimit formulae for its distance, the profile of the nearby potential field andits ageing behaviour. We also prove that the localisation site is determinedlocally, that is, by maximising a certain time-dependent functional thatdepends only on: (i) the value of the potential field in a neighbourhood offixed radius around a site; and (ii) the distance of that site to the origin.

Complete Localisation and Exponential Shape of the Parabolic Anderson Model with Weibull Potential Field