Zufällige Schrödinger-Operatoren aus dem Blickwinkel verstärkter Zufallsprozess
Zusammenfassung der Projektergebnisse
A reinforced random walk on a graph is a random walk that is more likely to visit edges or vertices that have already been visited in the past. It is therefore a process with memory, and can serve to model real-life experiences that involve learning. In recent years, reinforced random processes have attracted much interest from the scientific community due to the discovery of connections with different areas of mathematics that have allowed for rigorous proofs of phase transitions in these models. This proposal deals with a recently discovered connection between two types of reinforced random process (the linearly edge reinforced random walk and the vertex reinforced jump process) with a random Schrödinger operator (RSO) of the form [ST15, STZ17, SZ19]. Hω = −∆W + Vω (1) where −∆W is the lattice Laplacian on a weighted graph with weights Wij = W i = j ∈ Λ, W > 0, and Vω is the multiplication by the random variable 2ωi , i ∈ Λ. Precisely Λ −Wij i=j (Hω )ij = 2ωi i=j. This type of operator is used to model the propagation of electrons in disordered media. In dimension d ≥ 3, these reinforced processes exhibit a phase transition between a recurrent phase at strong reinforcement and a transient phase at weak reinforcement [DS2010, DSZ2010, ST15, DST2015]. In [ST15] the authors conjectured that a recurrence/transience transition for the reinforced random process corresponds to a localization/delocalization transition in the random Schrödinger model, which is a major open problem in the theory of disordered solids [Mott]. Our goal is to combine the well developed theory of quantum disordered systems with the new connection to random processes to contribute to the understanding of phase transitions in disordered quantum systems. In this project we have achieved to prove a phase transition for the Integrated Density of States of the operator (1) that corresponds to the transition existing in reinforced random processes. This is a first result in the direction of a deeper understanding of phase transitions in this type of random Schrödinger operators. The study of localization properties are part of follow-up research.
