We study the persistence exponent for the first passage time of a random walk below the trajectory of another random walk. More precisely, let @math and @math be two centered, weakly dependent random walks. We establish that @math for a non-random @math . In the classical setting, @math , it is well-known that @math . We prove that for any non-trivial @math one has @math and the exponent @math depends only on @math . Our result holds also in the continuous setting, when @math and @math are independent and possibly perturbed Brownian motions or Ornstein-Uhlenbeck processes. In the latter case the probability decays at exponential rate.
We consider the harmonic crystal, or massless free field, ( ), ( ), that is the centered Gaussian field with covariance given by the Green function of the simple random walk on ℤ d . Our main aim is to obtain quantitative information on the repulsion phenomenon that arises when we condition ( ) to be larger than ( ), ( ) is an IID field (which is also independent of ϕ), for every x in a large region ( ), with N a positive integer and D a bounded subset of ℝ d . We are mostly motivated by results for given typical realizations of σ (quenched set–up), since the conditioned harmonic crystal may be seen as a model for an equilibrium interface, living in a (d+1)–dimensional space, constrained not to go below an inhomogeneous substrate that acts as a hard wall. We consider various types of substrate and we observe that the interface is pushed away from the wall much more than in the case of a flat wall as soon as the upward tail of σ 0 is heavier than Gaussian, while essentially no effect is observed if the tail is sub–Gaussian. In the critical case, that is the one of approximately Gaussian tail, the interplay of the two sources of randomness, ϕ and σ, leads to an enhanced repulsion effect of additive type. This generalizes work done in the case of a flat wall and also in our case the crucial estimates are optimal Large Deviation type asymptotics as ( ) of the probability that ϕ lies above σ in D N .