本文考虑$(2+1)$维广义固体-固体（SOS）模型，即具有$|\nablaϕ|^{p}$形式的梯度势的随机离散表面，其中$p\in [1,+\infty]$。我们证明了在低温下，对于具有侧边$L$的正方形区域$Λ$，无论是在无限体积测度下还是在$Λ$周围零边界条件下，$Λ$中表面非负的概率都表现出类似于$\exp(-4βτ_{p,β} L H_p(L) )$的行为，其中$β$是逆温度，$τ_{p,β}$是零倾斜时的表面张力或阶跃自由能，以及$H_p(L)$是熵排斥高度，即当施加正性约束时场的典型高度。这推广了最近在\cite{CMT}中获得的标准SOS模型（$p=1$）的结果。
英文摘要：
We consider the $(2+1)$-dimensional generalized solid-on-solid (SOS) model, that is the random discrete surface with a gradient potential of the form $|\nablaϕ|^{p}$, where $p\in [1,+\infty]$. We show that at low temperature, for a square region $Λ$ with side $L$, both under the infinite volume measure and under the measure with zero boundary conditions around $Λ$, the probability that the surface is nonnegative in $Λ$ behaves like $\exp(-4βτ_{p,β} L H_p(L) )$, where $β$ is the inverse temperature, $τ_{p,β}$ is the surface tension at zero tilt, or step free energy, and $H_p(L)$ is the entropic repulsion height, that is the typical height of the field when a positivity constraint is imposed. This generalizes recent results obtained in \cite{CMT} for the standard SOS model ($p=1$).
Persistence and entropic repulsion of stationary Gaussian fields with spectral singularity at the origin
Bar-Ilan University（巴伊兰大学）；Einstein Institute of Mathematics, Hebrew University of Jerusalem（耶路撒冷希伯来大学爱因斯坦数学研究所）；School of Mathematics, Monash University（蒙纳士大学数学学院）
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