Random mass flow through random potential

• Applicant: Professor Dr. Wolfgang König
• Subject Area: Mathematics
• Term: from 2014 to 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 252472933

• Final Report Year: 2018

Final Report Abstract

Originally introduced in solid state physics to model amorphous materials and alloys exhibiting disorder induced metal-insulator transitions, the Anderson Hamiltonian and its phenomenology has become a paradigmatic example for the relevance of effects resulting from a random background. A popular model for the random motion governed by this operator is the parabolic Anderson model (PAM), the discrete heat equation with random sources and sinks modelled by the Anderson Hamiltonian. A characteristic property of its solutions is the occurrence of intermittency peaks in the large time limit. They determine the thermodynamic observables extensively studied in the probabilistic literature using path integral methods and the theory of large deviations. In more probabilistic language, the random mass flow through the random potential concentrates on small islands, which are separated far from each other. Since the beginning of the 1990s, for some very special potential distributions, the intermittency picture and many detailed properties of the islands have been proved in recent years. These distributions all had in common that they make the relevant islands trivial, i.e., just singletons. In the current project, however, a paradigmatic example of a potential distribution was handled that makes the islands and the relevant spectral objects (eigenvalues and eigenvectors) interesting objects that are characterised in terms of variational formulas. For this distribution, the strongest assertions about intermittency have been proved: a spatial spectral expansion for the random Schrödinger operator and the asymptotic concentration of the total mass in just one of the island. Additionally, ageing and metastability properties of the location of this island were identified, and we studied other detailed questions about the timeevolution of the random mass flow. Furthermore, the knowledge about the PAM gained in recent years has was applied in the project to answer ambitious questions about more complex models, like the PAM with random diffusion rates or for some particular (and interesting) random potential that is in a sense critical, since it has to be renormalised. The general principles underlying the effects have been brought to the surface. We studied the PAM after replacing the diffusive part of the Anderson Hamiltonian, the Laplace operator, by the generator of a random walk among random conductances. Furthermore, we analysed the largetime asymptotics in a critical case in the continuous-space version for a renormalised Poisson potential, a potential for which even a proof of the well-definedness of the solution to the PAM requires crucial insight and technical work. We also planned to study a critical phase in the transition of the model from intermittent to diffusive behaviour (adding a small prefactor in front of the potential).

Publications

• Eigenvalue order statistics for random Schrödinger operators with doubly-exponential tails, Commun. Math. Phys. 341:1, 179-218 (2016)
  M. Biskup and W. König
  
• The parabolic Anderson model. Random walk in random potential, Birkhäuser, Pathways in Mathematics (2016)
  W. König
  
• Brownian motion in attenuated or renormalized inversesquare Poisson potential
  P. Nelson and R.S. dos Santos
  
• Mass concentration and aging in the parabolic Anderson model with doubly-exponential tails, Probab. Theory Relat. Fields 171, 251–331 (2018)
  M. Biskup, W. König and R.S. dos Santos
  
• (2019): The Bouchaud–Anderson model with double-exponential potential. In: Ann. Appl. Probab. 29
  S. Muirhead, R. Pymar and R.S. dos Santos