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Algebraic area distribution of two-dimensional random walks and the Hofstadter model

Abstract : This thesis is about the Hofstadter model, i.e., a single electron moving on a two-dimensional lattice coupled to a perpendicular homogeneous magnetic field. Its spectrum is one of the famous fractals in quantum mechanics, known as the Hofstadter's butterfly. There are two main subjects in this thesis: the first is the study of the deep connection between the Hofstadter model and the distribution of the algebraic area enclosed by two-dimensional random walks. The second focuses on the distinctive features of the Hofstadter's butterfly and the study of the bandwidth of the spectrum. We found an exact expression for the trace of the Hofstadter Hamiltonian in terms of the Kreft coefficients, and for the higher moments of the bandwidth.This thesis is organized as follows. In chapter 1, we begin with the motivation of our work and a general introduction to the Hofstadter model as well as to random walks will be presented. In chapter 2, we will show how to use the connection between random walks and the Hofstadter model. A method to calculate the generating function of the algebraic area distribution enclosed by planar random walks will be explained in details. In chapter 3, we will present another method to study these issues, by using the point spectrum traces to recover the full Hofstadter trace. Moreover, the advantage of this construction is that it can be generalized to the almost Mathieu operator. In chapter 4, we will introduce the method which was initially developed by D.J.Thouless to calculate the bandwidth of the Hofstadter spectrum. By following the same logic, I will show how to generalize the Thouless bandwidth formula to its n-th moment, to be more precisely defined later.
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Shuang Wu. Algebraic area distribution of two-dimensional random walks and the Hofstadter model. Mathematical Physics [math-ph]. Université Paris-Saclay, 2018. English. ⟨NNT : 2018SACLS459⟩. ⟨tel-01974028⟩

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