We consider the $$d\ (\ge 3)$$ d ( ≥ 3 ) - dimensional lattice Gaussian free field on $$\varLambda _N :=[-N, N]^d\cap \mathbb {Z}^d$$ Λ N : = [ - N , N ] d ∩ Z d in the presence of a self-potential of the form $$U(r)= -b I(|r|\le a)$$ U ( r ) = - b I ( | r | ≤ a ) , $$a>0, b\in \mathbb {R}$$ a > 0 , b ∈ R . When $$b>0$$ b > 0 , the potential attracts the field to the level around zero and is called square-well pinning. It is known that the field turns to be localized and massive for every $$a>0$$ a > 0 and $$b>0$$ b > 0 . In this paper, we consider the situation that the parameter $$b<0$$ b < 0 and self-potentials are imposed on $$\varLambda _{\alpha N},\ \alpha \in (0, 1)$$ Λ α N , α ∈ ( 0 , 1 ) . We prove that once we impose this weak repulsive potential from the level $$[-a, a]$$ [ - a , a ] , most sites are located on the same side and the field is pushed to the same level when the original Gaussian field is conditioned to be positive everywhere, or negative everywhere with probability $$\frac{1}{2}$$ 1 2 , respectively. This result can be applied to show the similar path behavior for the disordered pinning model in the delocalized regime.

中文翻译：

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具有弱排斥势的格子高斯自由场的行为

我们考虑 $$d\ (\ge 3)$$ d ( ≥ 3 ) - $$\varLambda _N 上的维格高斯自由场 :=[-N, N]^d\cap \mathbb {Z}^d $$ Λ N : = [ - N , N ] d ∩ Z d 存在形式 $$U(r)= -b I(|r|\le a)$$ U ( r ) = - b I ( | r | ≤ a ) , $$a>0, b\in \mathbb {R}$$ a > 0 , b ∈ R 。当 $$b>0$$ b > 0 时，势将场吸引到零附近的水平，称为方阱钉扎。众所周知，对于每一个 $$a>0$$ a > 0 和 $$b>0$$ b > 0 ，该字段就会变得局部化和大规模。在本文中，我们考虑参数$$b<0$$ b < 0 和自势加在$$\varLambda _{\alpha N},\ \alpha \in (0, 1)$ $Λ α N , α ∈ ( 0 , 1 ) 。我们证明，一旦我们从 $$[-a, a]$$ [ - a , a ] 水平施加这种弱排斥潜力，大多数站点位于同一侧，当原始高斯场被条件为处处为正或处处为负时，场被推到同一水平，概率分别为 $$\frac{1}{2}$$ 1 2 。 . 该结果可用于显示离域状态下无序钉扎模型的类似路径行为。