Large deviations for the Ginzburg–Landau ∇φ interface model

Abstract

Hydrodynamic large scale limit for the Ginzburg-Landau ∇φ interface model was established in [6]. As its next stage this paper studies the corresponding large deviation problem. The dynamic rate functional is given byis the surface tension for mean tilt u ∈ R d . Our main tool is H −1 -method exploited by Landim and Yau [9]. The relationship to the rate functional obtained under the static situation by Deuschel et al.[3] is also discussed.

Hydrodynamic limit for ∇φ interface model on a wall

“…In a similar manner to what we did for I 4,f , we have I 4,g ≤ 0 since g − (r) ≤ 0. These estimates taking β sufficiently small combined with Corollary 3.2 and Proposition 3.3 of [7] prove (6.4); see the proof of (6.4) in [7]. Remark 6.1.…”

“…In fact, for some sufficiently small positive constant λ, the right hand side of (2.1) might be replaced with C + λh and (2.2) by −C ≤ ∂f/∂h ≤ λ. The constant λ = λ V is determined by Corollary 3.2 and Proposition 3.3 of [7] as we saw in the proof.…”

“…The formula (6.3) is an application of Girsanov's theorem and then Itô's formula for dG − (t, x, φ t (x)) under P N ; see (6.5) of [7] for a similar expression. To prove (6.4), one has from (6.3)…”

“…In fact, the hydrodynamic limit (law of large numbers), equilibrium fluctuation (central limit theorem) and large deviations at static or dynamic levels were investigated by recent papers [9], [10], [5] and [7], respectively. The paper [6] reviews these results.…”

The Random Walk Representation for Interacting Diffusion Processes

“…Also the hydrodynamic limes of such a system has been derived in several papers including study of the corresponding large deviations and fluctuations, cf. [17], [21]. In this case the diffusion coefficient is constant:…”