We consider the (discrete) parabolic Anderson model $\partial u(t,x)/\partial t=\mathrm{\Delta }u(t,x)+{\xi }_t(x)u(t,x)$, $t\ge 0$, $x\in {\mathbb{Z}}^d$, where the ξ-field is $\mathbb{R}$-valued and plays the role of a dynamic random environment, and Δ is the discrete Laplacian. We focus on the case in which ξ is given by a properly rescaled symmetric simple exclusion process under which it converges to an Ornstein–Uhlenbeck process. Scaling the Laplacian diffusively and restricting ourselves to a torus, we show that in dimension $d=3$ upon considering a suitably renormalised version of the above equation, the sequence of solutions converges in law. As a by-product of our main result we obtain precise asymptotics for the survival probability of a simple random walk that is killed at a scale dependent rate when meeting an exclusion particle. Our proof relies on the discrete theory of regularity structures of Erhard and Hairer and on novel sharp estimates of joint cumulants of arbitrary large order for the exclusion process. We think that the latter is of independent interest and may find applications elsewhere.

中文翻译：

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具有排斥相互作用的抛物线安德森模型的缩放极限

我们考虑（离散）抛物线安德森模型$\partial 你（t,X）/\partial t=\mathrm{\Delta }你（t,X）+{\Psi }_t（X）你（t,X）$,$t\ge 0$,$X\epsilon {\mathbb{Z}}^d$，其中 xi 场是$\mathbb{右}$-值，扮演动态随机环境的角色，Δ是离散拉普拉斯算子。我们关注这样的情况，其中 xi 由适当重新调整的对称简单排除过程给出，在该过程下它收敛到 Ornstein-Uhlenbeck 过程。扩散地缩放拉普拉斯算子并将我们限制为环面，我们证明在维度上$d=3$在考虑上述方程的适当重整化版本后，解的序列按规律收敛。作为我们主要结果的副产品，我们获得了简单随机游走的生存概率的精确渐近，当遇到排除粒子时，该随机游走会以与尺度相关的速率被杀死。我们的证明依赖于 Erhard 和 Hairer 的正则结构的离散理论以及排除过程中任意大阶联合累积量的新颖锐估计。我们认为后者具有独立的利益，并且可能在其他地方找到应用。