论文标题
渐近奇异的安德森·哈密顿和抛物线模型中的相变
Phase Transitions in Asymptotically Singular Anderson Hamiltonian and Parabolic Model
论文作者
论文摘要
让$ξ$为$ \ mathbb r^d $上的高斯白噪声（$ d = 1,2,3 $）。令$（ξ_\ varepsilon）_ {\ varepsilon> 0} $成为连续的高斯过程，以便$ \ varepsilon \toξ$ as $ \ varepsilon \ to0 $，定义为通过针对mollifier卷动$ξ$来定义。我们考虑抛物线寄生机模型（PAM）的渐近学$ξ_ {\ varepsilon（t）} $在很大的时间内$ t \ gg1 $，以及带有潜在$之一的Anderson Hamiltonian（AH）的Dirichlet Eigenlet eigenlet eigenvalues，其中$之一{\ varepsilon（the） $ \ varepsilon（t）$消失为$ t \ to \ infty $。我们证明，所讨论的渐近学以$ \ varepsilon（t）$消失的速率表现出相变，这区分了在AH/PAM中与连续高斯噪声和白色噪声中观察到的行为。通过将我们的主要定理与AH/PAM的先前结果与白噪声进行比较，我们的结果表明，可以使用仅基本方法访问后者的某些渐近性，并且我们获得了有关白噪声的AH/PAM之间的差异及其连续提及的近似值的定量估计值，即$ t \ to \ t \ to \ fos fo \ iffty $。
Let $ξ$ be a Gaussian white noise on $\mathbb R^d$ ($d=1,2,3$). Let $(ξ_\varepsilon)_{\varepsilon>0}$ be continuous Gaussian processes such that $ξ_\varepsilon\toξ$ as $\varepsilon\to0$, defined by convolving $ξ$ against a mollifier. We consider the asymptotics of the parabolic Anderson model (PAM) with noise $ξ_{\varepsilon(t)}$ for large time $t\gg1$, and the Dirichlet eigenvalues of the Anderson Hamiltonian (AH) with potential $ξ_{\varepsilon(t)}$ on large boxes $(-t,t)^d$, where the parameter $\varepsilon(t)$ vanishes as $t\to\infty$. We prove that the asymptotics in question exhibit a phase transition in the rate at which $\varepsilon(t)$ vanishes, which distinguishes between the behavior observed in the AH/PAM with continuous Gaussian noise and white noise. By comparing our main theorems with previous results on the AH/PAM with white noise, our results show that some asymptotics of the latter can be accessed with solely elementary methods, and we obtain quantitative estimates on the difference between the AH/PAM with white noise and its continuous-noise approximations as $t\to\infty$.
