推导出当$Z^d$上高斯自由场的水平集（d大于或等于3）低于给定水平时，该水平集使紧集A的离散爆破与包含A的盒子的离散爆破的边界断开的大偏差概率的渐近上界和下界。当该水平以上的高斯自由场的水平集处于强渗透状态时。这些界限大大加强了arXiv：1412.3960的结果，其中A是一个盒子，A的凸性在证明中起了重要作用。我们还得到了一个渐近上界的概率，高斯自由场的平均值以及内部的离散爆破的A是高于一定的水平时，发生断开。上界的推导使用了在A. S. Sznitman和作者处理一个类似的断开问题的空集的随机交织。如果高斯自由场的某些临界能级重合（这是目前的一个悬而未决的问题），我们得到的断开概率的渐近上界和下界在主序上匹配，并且断开条件使高斯自由场的平均值在A的离散爆破内降低，这可以理解为熵排斥。
英文摘要：
We derive asymptotic upper and lower bounds on the large deviation probability that the level set of the Gaussian free field on $Z^d$, d bigger or equal to three, below a given level disconnects the discrete blow-up of a compact set A from the boundary of the discrete blow-up of a box that contains A, when the level set of the Gaussian free field above this level is in a strongly percolative regime. These bounds substantially strengthen the results of arXiv:1412.3960, where A was a box and the convexity of A played an important role in the proof. We also derive an asymptotic upper bound on the probability that the average of the Gaussian free field well inside the discrete blow-up of A is above a certain level when disconnection occurs. The derivation of the upper bounds uses the solidification estimates for porous interfaces that were derived in the work arXiv:1706.07229 of A.-S. Sznitman and the author to treat a similar disconnection problem for the vacant set of random interlacements. If certain critical levels for the Gaussian free field coincide, an open question at the moment, the asymptotic upper and lower bounds that we obtain for the disconnection probability match in principal order, and conditioning on disconnection lowers the average of the Gaussian free field well inside the discrete blow-up of A, which can be understood as entropic repulsion.
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）
Entropic repulsion in $|\nabla ϕ|^p$ surfaces: a large deviation bound for all $p\geq 1$
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。
