The scaling limit of the $(\nabla+Δ)$-model

Scaling limit of semiflexible polymers: a phase transition

“…In this section we prove Theorem 2.1 by showing finite dimensional convergence and tightness. The proof is similar to the proofs of the lower dimensional results in Cipriani et al (2018b) and Cipriani et al (2018a). First we will show tightness and the finite dimensional convergence is similar to the proof of Theorem 2.8.…”

“…Here also we briefly give the definition of the Sobolev space H −s −∆ (D) and the Gaussian free field. For a detail discussion see Cipriani et al (2018a). By the spectral theorem for compact self-adjoint operators and elliptic regularity we know that there exist smooth eigenfunctions…”

“…We refer to Caravenna and Deuschel (2009), Cipriani et al (2018b), Hryniv and Velenik (2009) for the scaling limit of the membrane model in d ≥ 1. The literature on the case when κ 1 > 0, κ 2 > 0 is limited and has been considered in the works of Borecki (2010), Borecki and Caravenna (2010), Cipriani et al (2018a), Sakagawa (2018). Borecki (2010) and Borecki and Caravenna (2010) introduced this model as the (∇ + ∆)model (we will also refer to it as "mixed model") with constant κ 1 , κ 2 .…”

“…The results were extended to higher dimensions, together with further properties of the free energy, in Sakagawa (2018). In Cipriani et al (2018a) the scaling limit of the (∇ + ∆)-model is studied. There it is shown that if one takes the lattice size to go to zero, under a suitable scaling the Laplacian term is dominated by the gradient and the limit becomes the Gaussian free field.…”

“…Using Sobolev estimates it can be shown that the closeness of the solutions is related to the approximation of the discrete elliptic operator to the continuum one. This idea has been already employed in Cipriani et al (2018b) and Cipriani et al (2018a). But in the present scenario, the discrete elliptic operators have coefficients which depend on N and hence the estimates of Thomée (1964) are not applicable directly.…”