On the Absolute-Value Integral of a Brownian Motion with Drift: Exact and Asymptotic Formulae

The present paper is concerned with the integral of the absolute value of a Brownian motion with drift. By establishing an asymptotic expansion of the space Laplace transform, we obtain series representations for the probability density function and cumulative distribution function of the integral, making use of Meijer's G-function. A functional recursive formula is derived for the moments, which is shown to yield only exponentials and Gauss' error function up to arbitrary orders, permitting exact computations. To obtain sharp asymptotic estimates for small- and large-deviation probabilities, we employ a marginal space-time Laplace transform and apply a newly developed generalization of Laplace's method to exponential Airy integrals. The impact of drift on the complete distribution of the integral is explored in depth. The resultant new formulae complement existing ones in the standard Brownian motion case to great extent in terms of both theoretical generality and modeling capacity and have been presented for easy implementation, which numerical experiments demonstrate.Comment: 35 pages, 4 tables, 5 figures; added reference

Non-Abelian inverse Anderson transitions

Inverse Anderson transitions, where the flat-band localization is destroyed by disorder, have been wildly investigated in quantum and classical systems in the presence of Abelian gauge fields. Here, we report the first investigation on inverse Anderson transitions in the system with non-Abelian gauge fields. It is found that pseudospin-dependent localized and delocalized eigenstates coexist in the disordered non-Abelian Aharonov-Bohm cage, making inverse Anderson transitions depend on the relative phase of two internal pseudospins. Such an exotic phenomenon induced by the interplay between non-Abelian gauge fields and disorder has no Abelian analogy. Furthermore, we theoretically design and experimentally fabricate nonAbelian Aharonov-Bohm topolectrical circuits to observe the non-Abelian inverse Anderson transition. Through the direct measurements of frequency-dependent impedance responses and voltage dynamics, the pseudospin-dependent non-Abelian inverse Anderson transitions are observed. Our results establish the connection between inverse Anderson transitions and non-Abelian gauge fields, and thus comprise a new insight on the fundamental aspects of localization in disordered non-Abelian flat-band systems

Observation of inverse Anderson transitions in Aharonov-Bohm topolectrical circuits

It is well known that Anderson transition is a disorder-induced metal-insulator transition.Contrary to this conventional wisdom, some investigations have shown that disorders could destroy the phase coherence of localized modes in flatbands, making the localized states melt into extended states. This phenomenon is called the inverse Anderson transition. While, to date, the experimental observation of inverse Anderson transitions is still lacking. In this work, we report the implementation of inverse Anderson transitions based on Aharonov-Bohm topolectrical circuits. Different types of disorders, including symmetric-correlated, antisymmetric-correlated and uncorrelated disorders, can be easily implemented in Aharonov-Bohm circuits by engineering the spatial distribution of ground settings. Through the direct measurements of frequency-dependent impedance responses and time-domain voltage dynamics, the inverse Anderson transitions induced by antisymmetric-correlated disorders are clearly observed. Moreover, the flat bands and associated spatial localizations are also fulfilled in clean Aharonov-Bohm circuits or Aharonov-Bohm circuits sustaining symmetric-correlated and uncorrelated disorders, respectively. Our proposal provides a flexible platform to investigate the interplay between the geometric localization and Anderson localization, and could have potential applications in electronic signal control.Comment: 12 pages, 4 figure