Phase Transitions in Asymptotically Singular Anderson Hamiltonian and Parabolic Model

Abstract

Let $\xi$ be a Gaussian white noise on ${\mathbb{R}}^d$ ($d=1,2,3$). Let $({\xi }_{\epsilon }{)}_{\epsilon >0}$ be continuous Gaussian processes such that ${\xi }_{\epsilon }\to \xi$ as $\epsilon \to 0$, defined by convolving $\xi$ against a mollifier. We consider the asymptotics of the parabolic Anderson model (PAM) with noise ${\xi }_{\epsilon (t)}$ for large time $t\gg 1$, and the Dirichlet eigenvalues of the Anderson Hamiltonian (AH) with potential ${\xi }_{\epsilon (t)}$ on large boxes $(-t,t{)}^d$, where the parameter $\epsilon (t)$ vanishes as $t\to \infty$. We prove that the asymptotics in question exhibit a phase transition in the rate at which $\epsilon (t)$ 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 $t\to \infty$.

arXiv commentMajor modifications: Incorporating referee comments, including substantial improvements to the presentation and the correction of errors in the previous version

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Cited by in corpus (1)

• Longtime asymptotics of the two-dimensional parabolic Anderson model with white-noise potential