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.
The paper @cite_0 was followed by @cite_4 which could be seen as an analogue of our work. Namely, the topic of this paper is a Gaussian free field interface conditioned to be above the fixed realization of another Gaussian free field. The authors obtain the precise estimates for the probability of this event and the entropic repulsion induced by the conditioning.
@cite_0 @cite_4
1480598051 2002513116
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 . We analyze a model of an interface fluctuating above a rough substrate. It is based on harmonic crystals, or lattice free fields, indexed by ℤ d , d ≥ 3. The phenomenon for which we want to get precise quantitative estimates is the repulsion effect of the substrate on the interface: the substrate is itself a random field, but its randomness is quenched (this generalizes the widely considered case of a flat deterministic substrate). With respect to [2] in which the substrate has been taken to be an IID field, here the substrate is an harmonic crystal, as the interface, and as such it is strongly correlated. We obtain the leading asymptotic behavior of the model in the limit of a very extended substrate: we show in particular that, to leading order, the effect of an IID substrate cannot be distinguished from the effect of an harmonic crystal substrate. We observe however that, unlike in the IID substrate case, annealed and quenched models display sharply different features.