Muirhead, SJ; (2016) Intermittency and Localisation Phenomena in the Parabolic Anderson and Bouchaud Trap Models. Doctoral thesis , UCL (University College London).

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Abstract

This thesis studies intermittency and localisation phenomena in the parabolic Anderson model (PAM) and the Bouchaud trap model (BTM), models for random walks in a random branching environment and a random trapping landscape respectively. In the PAM, we study the phenomenon of complete localisation, which describes the eventual concentration of the (renormalised) mass function on a single site with overwhelming probability. Our main result is that complete localisation holds for potential distributions with (i) Weibull tail decay, and (ii) fractional-double-exponential tail decay. Since complete localisation is strongly conjectured to break down for potentials with double-exponential tail decay, in a sense our work completes the program of establishing complete localisation in the PAM. In the Weibull case, we further give a detailed geometric description of the complete localisation behaviour. In the BTM, we study the regime of slowly varying traps, that is, when the survival function of the trap distribution has a slowly varying tail at infinity. Our main result is that the BTM on the integers exhibits extremely strong localisation behaviour that is qualitatively different to the known localisation behaviour in the regularly varying case. More precisely, we demonstrate that (i) the mass function of the BTM concentrates on two-sites with overwhelming probability, and (ii) the rescaled BTM converges to a highly-singular process we call the extremal FIN process. Finally, we explore the interaction between the localisation phenomena due to random branching and trapping mechanisms by studying a hybrid model which combines these mechanisms. Our main result is that, under certain natural assumptions, the localisation effects due to random branching and trapping mechanisms tend to (i) mutually reinforce, and (ii) induce a local correlation in the random fields (c.f. the `fit and stable' hypothesis of population dynamics).

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