Scaling limits of equilibrium wetting models in (1+1)–dimension

Abstract

We study the path properties for the δ-pinning wetting model in (1 + 1)-dimension. In other terms, we consider a random walk model with fairly general continuous increments conditioned to stay in the upper half plane and with a δ-measure reward for touching zero, that is the boundary of the forbidden region. It is well known that such a model displays a localization/delocalization transition, according to the size of the reward. Our focus is on getting a precise pathwise description of the system, in both the delocalized phase, that includes the critical case, and in the localized one. From this we extract the (Brownian) scaling limits of the model.

A Numerical Approach to Copolymers at Selective Interfaces

“…In the localized phase it is known [24,2] that the laws of S N under P λ,h N,ω r are tight, which means that the polymer is essentially at O(1) distance from the x-axis. The situation is completely different in the (interior of the) delocalized phase, where one expects that S N = O( √ N ): in fact the conjectured path behavior (motivated by the analogy with the known results for non disordered models, see in particular [21], [11] and [8]) should be weak convergence under diffusive scaling to the Brownian meander process (that is Brownian motion conditioned to stay positive on the interval [0, 1], see [23]). Therefore in the (interior of the) delocalized phase the law of S N / √ N under P λ,h N,ω r should converge weakly to the corresponding marginal of the Brownian meander, whose law has density x exp(−x 2 /2)1 (x≥0) .…”

“…However they give at the same time still a weak information on the paths, above all if compared to what is available for non disordered models, see e.g. [21], [11], [8] and references therein, namely Brownian scaling, which in turn is a consequence of the fact that all the visits in the unfavorable solvent happen very close to the boundary points, that is the origin, under the measure P λ,h N,ω . In non disordered models one can in fact prove that the polymer becomes transient and that it visits the unfavorable solvent, or any point below a fixed level, only a finite number of times.…”

Estimates on path delocalization for copolymers at selective interfaces

“…In this respect, based on what is known on non-disordered models, see e.g. [9] and [14], it is tempting to conjecture the following scenario: for h > h c (λ) C.1 there are only a finite number of visits to the unfavorable solvent, that is P( dω)-a.s. lim , with S distributed according to P f N,ω , converges weakly as N → ∞ to the law of the Brownian meander, that is the law of a standard Brownian process conditioned not to enter the lower half plane. The standard reference for the Brownian meander is [17].…”

“…The same theorem tells us that, if R n := max ℓ − k : k and ℓ even , 0 ≤ k < ℓ ≤ n, Notice that the longest atypical stretch, in the sense of (4.6), for n = τ N ranges from τ N − R τ N to τ N , so The second inequality holds for N sufficiently large. Now we set δ := sup 9) and observe that sup q<h (−2λq − Σ h (q)) is positive if and only if sup q∈R (−2λq − Σ h (q)) is positive. The latter expression is the Legendre transform of Σ h (·) and therefore it coincides with −2λh+log M(2λ), which is positive for h < h(λ).…”

The Random Walk Representation for Interacting Diffusion Processes

“…Only in the one dimensional case where both fluctuations and repulsion are of the same size, one can see that starting from equilibrium in a box of size N , the properly rescaled space-time process converges as N → ∞ to reflected partial stochastic differential equation of the Nualard-Pardoux type, cf. [18], [30], [12].…”