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.
Our result can be understood from various perspectives. One of them is the so-called entropic repulsion. This question was asked in @cite_0 in the context of the Gaussian free field for @math . Namely, the authors studied the repulsive effect on the interface of the wall which is a fixed realization of an i.i.d. field @math . They observe that the tail of @math plays a fundamental role. When it is subgaussian than the effect of the wall is essentially equivalent to the wall given by @math , while when the tail is heavier than Gaussian the interface is pushed much more upwards. It would be interesting to ask an analogous question in our case. By Fact we know already that the disorder has a negligible effect when @math for @math . We expect that when @math the repulsion becomes much stronger.
@cite_0
1480598051 2592044494 2059566060 1572521168
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 study the tail behavior for the maximum of discrete Gaussian free field on a 2D box with Dirichlet boundary condition after centering by its expectation. We show that it exhibits an exponential decay for the right tail and a double exponential decay for the left tail. In particular, our result implies that the variance of the maximum is of order 1, improving an (o( n) ) bound by Chatterjee (Chaos, concentration, and multiple valleys, 2008) and confirming a folklore conjecture. An important ingredient for our proof is a result of Bramson and Zeitouni (Commun. Pure Appl. Math, 2010), who proved the tightness of the centered maximum together with an evaluation of the expectation up to an additive constant. We consider the lattice version of the free field in two dimensions and study the fractal structure of the sets where the field is unusually high (or low). We then extend some of our computations to the case of the free field conditioned on being everywhere nonnegative. For example, we compute the width of the largest downward spike of a given length. Through the prism of these results, we find that the extrema of the free field under entropic repulsion (minus its mean) and those of the unconditioned free field are identical. Finally, when compared to previous results these findings reveal a suggestive analogy between the square of the free field and the two-dimensional simple random walk on the discrete torus. We consider the lattice version of the free eld in two dimensions (also called harmonic crystal). The main aim of the paper is to discuss quantitatively the entropic repulsion of the random surface in the presence of a hard wall. The basic ingredient of the proof is the analysis of the maximum of the eld which requires a multiscale analysis reducing the problem essentially to a problem on a eld with a tree structure. 2000 MSC: 60K35, 60G15, 82B41