On the Free Energy of a Gaussian Membrane Model with External Potentials

Abstract

We consider a class of d-dimensional Gaussian lattice field which is known as a model of semi-flexible membrane. We study the free energy of the model with external potentials and show the following:

(1) We consider the model with δ-pinning and prove that the field is always localized when d≥4.

(2) Consider the model confined between two hard walls. We give asymptotics of the free energy as the height of the wall goes to infinity.

Appendix

We consider the Hamiltonian (2). More precisely, consider

$$ \widetilde{H}(\phi) = \kappa_1 \sum _{\substack{ x, y \in\mathbb{Z}^d\\ |x-y|=1}} |\phi_x-\phi_y|^2 + \kappa_2 \sum _{x\in\mathbb{Z}^d}|\varDelta \phi_x |^2.$$

(16)

For simplicity we may assume that $\kappa_{1}=\frac{\kappa}{8d}$ and $\kappa_{2}=\frac{1}{2}, \kappa>0$. By the same computation as in introduction, we have

Hence the corresponding Gibbs measure on Λ _N coincides with the law of a centered Gaussian field on $\mathbb{R}^{\varLambda_{N}}$ with the covariance matrix $(-\kappa \varDelta _{N}+ \varDelta ^{2}_{N})^{-1}$. By Brascamp-Lieb inequality,

Also, since the matrices −Δ _N and κ−Δ _N commute, we can compute that

Therefore the asymptotics of the variance for the Gibbs measure corresponding to Hamiltonian (16) is similar to that of the ∇ϕ model and the similar result for the ∇ϕ model would hold in this case (though some extra work might be needed).