Wetting and layering for Solid-on-Solid I: Identification of the wetting point and critical behavior

Abstract

We provide a complete description of the low temperature wetting transition for the two dimensional Solid-On-Solid model. More precisely we study the integer-valued field $(\varphi (x){)}_{x\in {\mathbb{Z}}^2}$, associated associated to the energy functional $$V(\varphi )=\beta \sum _{x\sim y}|\varphi (x)-\varphi (y)|-\sum _x\left(h{1}_{\{\varphi (x)=0\}}-\infty 1_{\{\varphi (x)<0\}}\right).$$ It is known since the pioneering work of Chalker (J. Phys. A {\bf 15} (1982) 481-485) that for every $\beta$, there exists $h_w(\beta )>0$ delimiting a transition between a delocalized phase ($h<h_w(\beta )$) where the proportion of points at level zero vanishes, and a localized phase ($h>h_w(\beta )$) where this proportion is positive. We prove in the present paper that for $\beta$ sufficiently large we have $$h_w(\beta )=\log \left(\frac{e^{4\beta }}{e^{4\beta }-1}\right).$$ Furthermore we provide a sharp asymptotic for the free energy at the vicinity of the critical point: We show that close to $h_w(\beta )$, the free energy is approximately piecewise affine and that the points of discontinuity for the derivative of the affine approximation forms a geometric sequence accumulating on the right of $h_w(\beta )$. This asymptotic behavior provides a strong evidence for the conjectured existence of countably many "layering transitions" at the vicinity of the critical point, corresponding to jumps for the typical height of the field.

39 pages, 4 Figures

Cited by in corpus (2)

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