4 research outputs found
Regularity properties of bulk and edge current densities at positive temperature
We consider magnetic Schr\"odinger operators describing a quantum Hall effect
setup both in the plane and in the half-plane. First, we study the structure
and smoothness of the operator range of various powers of the half-plane
resolvent. Second, we provide a complete analysis of the diamagnetic current
density at positive temperature: we prove that bulk and edge current densities
are smooth functions and we show that the edge current density converges to the
bulk current density faster than any polynomial in the inverse distance from
the boundary. Our proofs are based on gauge covariant magnetic perturbation
theory and on a detailed analysis of the integral kernels of functions of
magnetic Schr\"odinger operators on the half-plane.Comment: 29 page
Singular distribution functions for random variables with stationary digits
Let be the cumulative distribution function (CDF) of the base-
expansion , where is an integer and
is a stationary stochastic process with state space
. In a previous paper we characterized the absolutely
continuous and the discrete components of . In this paper we study special
cases of models, including stationary Markov chains of any order and stationary
renewal point processes, where we establish a law of pure types: is then
either a uniform or a singular CDF on . Moreover, we study mixtures of
such models. In most cases expressions and plots of are given.Comment: This work extends some results of arXiv:2001.08492v