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.
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Acknowledgements
The author would like to thank the referee for his very useful comments. This work was partially supported by the Ministry of Education, Culture, Sports, Science and Technology, Grant-in-Aid for Young Scientists (B), No. 23740086.
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Appendix
Appendix
We consider the Hamiltonian (2). More precisely, consider
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).
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Sakagawa, H. On the Free Energy of a Gaussian Membrane Model with External Potentials. J Stat Phys 147, 18–34 (2012). https://doi.org/10.1007/s10955-012-0475-0
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DOI: https://doi.org/10.1007/s10955-012-0475-0