## 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 Λ_{N} be a N×N box of Z^{2}, and we consider a nonnegative integer-valued field (ϕ(x))_{x∈ΛN } with zero boundary conditions (i.e. ϕ|Λ_{N}^{∁}=0) associated with the energy functional V(ϕ)=β∑x∼y|ϕ(x)−ϕ(y)|−∑xh1_{{ϕ(x)=0}}, where β>0 is the inverse temperature and h≥0 is the pinning parameter. Lacoin has shown that for sufficiently large β, there is a phase transition between delocalization and localization at the critical point h_{w}(β)=log [Formula presented]. In this paper we show that for β≥1 and h∈(0,h_{w}), the values of ϕ concentrate at the height H=⌊(4β)^{−1}logN⌋ with constant order fluctuations. Moreover, at criticality h=h_{w}, we provide evidence for the conjectured typical height H_{w}=⌊(6β)^{−1}logN⌋.

Original language | English |
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Pages (from-to) | 168-182 |

Number of pages | 15 |

Journal | Stochastic Processes and their Applications |

Volume | 165 |

DOIs | |

State | Published - Nov 2023 |

### Bibliographical note

Publisher Copyright:© 2023 Elsevier B.V.

## Keywords

- Delocalization behavior
- Random surface
- Solid-On-Solid
- Typical height
- Wetting