We discuss the long term behaviour of trap models on the integers with asymptotically vanishing drift, providing scaling limit theorems and ageing results. Depending on the tail behaviour of the traps and the strength of the drift, we identify three
different regimes, one of which features a previously unobserved limit process.
Consider a discrete-time one-dimensional supercritical branching random walk. We study the probability that there exists an infinite ray in the branching random walk that always lies above the line of slope $gamma-epsilon$, where $gamma$ denotes the
asymptotic speed of the right-most position in the branching random walk. Under mild general assumptions upon the distribution of the branching random walk, we prove that when $epsilonto 0$, the probability in question decays like $exp{- {beta + o(1)over epsilon^{1/2}}}$, where $beta$ is a positive constant depending on the distribution of the branching random walk. In the special case of i.i.d. Bernoulli$(p)$ random variables (with $0<p<{1over 2}$) assigned on a rooted binary tree, this answers an open question of Robin Pemantle.
It is well known that the distribution of simple random walks on $bf{Z}$ conditioned on returning to the origin after $2n$ steps does not depend on $p= P(S_1 = 1)$, the probability of moving to the right. Moreover, conditioned on ${S_{2n}=0}$ the max
imal displacement $max_{kleq 2n} |S_k|$ converges in distribution when scaled by $sqrt{n}$ (diffusive scaling). We consider the analogous problem for transient random walks in random environments on $bf{Z}$. We show that under the quenched law $P_omega$ (conditioned on the environment $omega$), the maximal displacement of the random walk when conditioned to return to the origin at time $2n$ is no longer necessarily of the order $sqrt{n}$. If the environment is nestling (both positive and negative local drifts exist) then the maximal displacement conditioned on returning to the origin at time $2n$ is of order $n^{kappa/(kappa+1)}$, where the constant $kappa>0$ depends on the law on environment. On the other hand, if the environment is marginally nestling or non-nestling (only non-negative local drifts) then the maximal displacement conditioned on returning to the origin at time $2n$ is at least $n^{1-varepsilon}$ and at most $n/(ln n)^{2-varepsilon}$ for any $varepsilon>0$. As a consequence of our proofs, we obtain precise rates of decay for $P_omega(X_{2n}=0)$. In particular, for certain non-nestling environments we show that $P_omega(X_{2n}=0) = exp{-Cn -Cn/(ln n)^2 + o(n/(ln n)^2) }$ with explicit constants $C,C>0$.
We consider a one-dimensional recurrent random walk in random environment (RWRE). We show that the - suitably centered - empirical distributions of the RWRE converge weakly to a certain limit law which describes the stationary distribution of a rando
m walk in an infinite valley. The construction of the infinite valley goes back to Golosov. As a consequence, we show weak convergence for both the maximal local time and the self-intersection local time of the RWRE and also determine the exact constant in the almost sure upper limit of the maximal local time.
