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

TL;DR

This paper investigates the typical height of a (2+1)-D solid-on-solid surface with pinning above a wall in the delocalized phase, revealing how the height scales logarithmically with system size and providing evidence for critical behavior.

Contribution

It establishes the logarithmic scaling of the typical height in the delocalized phase and offers conjectural insights into the critical height at phase transition.

Findings

Height concentrates at loor((4eta)^{-1} \u221d ig(log N) for h < h_w

Provides evidence for the conjectured critical height loor((6eta)^{-1} ig(log N) at h = h_w

Shows constant order fluctuations around the typical height

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 $(\phi(x))_{x \in \Lambda_N}$ with zero boundary conditions (i.e. $\phi|_{\Lambda_N^{\complement}}=0 $) associated with the energy functional $$ \mathcal{V} (\phi)= \beta \sum_{x \sim y} \vert \phi(x)-\phi(y) \vert- \sum_{x} h \mathbf{1}_{\{ \phi(x)=0\}},$$ 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 $\phi$ 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 $H_w= \lfloor (6 \beta)^{-1} \log N \rfloor$.