Localisation in the Bouchaud–Anderson model

Abstract

Abstract. It is well-known that both random branching and trapping mechanisms can induce localisation phenomena in random walks; the prototypical examples being the parabolic Anderson and Bouchaud trap models respectively. Our aim is to investigate how these localisation phenomena interact in a hybrid model combining the dynamics of the parabolic Anderson and Bouchaud trap models. Under certain natural assumptions, we show that 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 (the 'fit and stable' hypothesis of population dynamics). Minor revision to published version. This is an updated version of [23] containing the following minor revisions:• A typo in the statement of Proposition 3.14 has been corrected;• The coupling used in Section 5 has been slightly modified to fix a gap in its original statement; we thank Renato Soares dos Santos for pointing this out to us; • A slight correction has been made to the proof of Proposition 4.11; and • The bibliography has been updated.

Delocalising the parabolic Anderson model through partial duplication of the potential

“…More recently, it has been shown that for sufficiently heavy-tailed potentials (Pareto [11], exponential [12], Weibull [5,17]), the solution exhibits the strongest possible form of localisation: complete localisation. In [14] this was shown to also be the case for a model that replaced the Laplacian with the generator of a trapped random walk. By contrast, in very recent work [3] it has been shown that in the double-exponential case the PAM localises on a single connected island, rather than on a single site.…”

The Parabolic Anderson Model

“…In the sense of convergence in probability, concentration with n t D 1 was proved for some related potential distributions in [LacMör12,SidTwa14,FioMui14] (even in random environment, see [MuiPym14] and Sect. 7.9.2).…”

“…Recently [MuiPym14], the PAM was studied in another class of random environment, more precisely, the underlying random walk was replaced by the Bouchaud random walk, better known under the name Bouchaud trap model, where the randomness of the holding times does not sit in the bonds, but in the sites, and is chosen very heavy-tailed. More explicitly, the generator is given by…”

“…Indeed, in [MuiPym14], the potential distribution is taken as Weibull-distributed, in which case these phenomena for the PAM are known from [FioMui14] and [SidTwa14]. 6.4.2.…”

“…The main result of [MuiPym14] is the complete localisation of the solution u.t; / in one random, t-dependent site Z t 2 Z d in the sense that u.t; Z t /=U.t/ converges to one in probability. Furthermore, depending on the details of the distribution of potential, the potential values in the neighbouring sites of the concentration site Z t are characterised; they indeed show an interesting limiting behaviour, and it can be seen in some sense that the one-site island slowly starts to emerge an interesting shape as the potential distribution gets less heavy-distributed.…”