Typical height of the (2+1)-D Solid-on-Solid surface with pinning above a wall in the delocalized phase

Abstract

We study the typical height of the (2+1)-dimensional solid-on-solid surface with pinning interacting with an impenetrable wall in the delocalization phase. More precisely, let ${\Lambda }_N$ be a $N\times N$ box of ${\mathbb{Z}}^2$, and we consider a nonnegative integer-valued field $(\varphi (x){)}_{x\in {\Lambda }_N}$ with zero boundary conditions (i.e. $\varphi {|}_{{\Lambda }_N^{\complement }}=0$) associated with the energy functional $$\mathcal{V}(\varphi )=\beta \sum _{x\sim y}|\varphi (x)-\varphi (y)|-\sum _{x}h{1}_{\{\varphi (x)=0\}},$$ where $\beta >0$ is the inverse temperature and $h\ge 0$ is the pinning parameter. Lacoin has shown that for sufficiently large $\beta$, there is a phase transition between delocalization and localization at the critical point $$h_w(\beta )=\log \left(\frac{e^{4\beta }}{e^{4\beta }-1}\right).$$ In this paper we show that for $\beta \ge 1$ and $h\in (0,h_w)$, the values of $\varphi$ concentrate at the height $H=\lfloor (4\beta {)}^{-1}\log N\rfloor$ with constant order fluctuations. Moreover, at criticality $h=h_w$, we provide evidence for the conjectured typical height $H_w=\lfloor (6\beta {)}^{-1}\log N\rfloor$.

13 Pages, comments are welcome

References in corpus (1)

• Localization and delocalization of random interfaces