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.

The parabolic Anderson model on Z d with i.i.d. potential is known to completely localise if the distribution of the potential is sufficiently heavy-tailed at infinity. In this paper we investigate a modification of the model in which the potential is partially duplicated in a symmetric way across a plane through the origin. In the case of potential distribution with polynomial tail decay, we exhibit a surprising phase transition in the model as the decay exponent varies. For large values of the exponent the model completely localises as in the i.i.d. case. By contrast, for small values of the exponent we show that the model may delocalise. More precisely, we show that there is an event of non-negligible probability on which the solution has non-negligible mass on two sites.

Oliveira conjectured that the order of the mixing time of the exclusion process with k-particles on an arbitrary n-vertex graph is at most that of the mixing-time of k independent particles. We verify this up to a constant factor for d-regular graphs when each edge rings at rate 1/d in various cases:(1) when d = (log n/k n),(2) when gap := the spectral-gap of a single walk is O(1/ log 4 n) and k ≥ n (1) ,(3) when k n a for some constant 0 < a < 1. In these cases, our analysis yields a probabilistic proof of a weaker version of Aldous' famous spectral-gap conjecture (resolved by Caputo et al.). We also prove a general bound of O(log n log log n/gap), which is within a log log n factor from Oliveira's conjecture when k ≥ n (1) . As applications, we get new mixing bounds:(a) O(log n log log n) for expanders, (b) order d log(dk) for the hypercube {0, 1} d , (c) order (Diameter) 2 log k for vertex-transitive graphs of moderate growth and for supercritical percolation on a fixed dimensional torus.

Oliveira conjectured that the order of the mixing time of the exclusion process with kparticles on an arbitrary n-vertex graph is at most that of the mixing-time of k independent particles. We verify this up to a constant factor for d-regular graphs when each edge rings at rate 1/d in various cases: (1) when d = Ω(log n/k n) or (2) when gap := the spectral-gap of a single walk is O(1/ log 4 n) and k n Ω(1) or (3) when k ≍ n a for some constant 0 < a < 1. In these cases our analysis yields a probabilistic proof of a weaker version of Aldous' famous spectral-gap conjecture (resolved by Caputo et al.). We also prove a general bound which is at worst O(log n log log n/gap), which is within a log log n factor from Oliveira's conjecture when k n Ω(1) . As applications we get new mixing bounds: (a) O(log n log log n) for expanders, (b) order d log(dk) for the hypercube {0, 1} d and (c) order (Diameter) 2 log k for vertex-transitive graphs of moderate growth and for supercritical percolation on a fixed dimensional torus.