Stochastic Processes and Random Matrices

doi:10.1093/oso/9780198797319.001.0001

2017

DOI: 10.1093/oso/9780198797319.001.0001

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Stochastic Processes and Random Matrices

Abstract: General rightsThis document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: http://www.bristol.ac.uk/pure/about/ebr-terms arXiv:1601.01822v1 [math-ph] 8 Jan 2016 IMPURITY MODELS AND PRODUCTS OF RANDOM MATRICES ALAIN COMTET AND YVES TOURIGNYAbstract. This is an extended version of lectures given at the Summer School on Stochastic Processes and Random Matrices, held at theÉcole de Physique, Les Houches, i… Show more

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Cited by 4 publications

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Dyson’s disordered linear chain from a random matrix theory viewpoint

Forrester

2021

Journal of Mathematical Physics

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The first work of Dyson relating to random matrix theory, “The dynamics of a disordered linear chain,” is reviewed. Contained in this work is an exact solution of the so-called type I chain in the case of the disorder variables being given by a gamma distribution. The exact solution exhibits a singularity in the density of states about the origin, which has since been shown to be universal for one-dimensional tight binding models with off diagonal disorder. We discuss this context and also point out some universal features of the weak disorder expansion of the exact solution near the band edge. Furthermore, a link between the exact solution and a tridiagonal formalism of anti-symmetric Gaussian β-ensembles with β proportional to 1/N is made.

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Dyson’s disordered linear chain from a random matrix theory viewpoint

Forrester

2021

Journal of Mathematical Physics

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Hausdorff dimension of collision times in one-dimensional log-gases

Hufnagel,

Andraus

2024

AIP Advances

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We consider systems of multiple Brownian particles in one dimension that repel mutually via a logarithmic potential on the real line, more specifically the Dyson model. These systems are characterized by a parameter that controls the strength of the interaction, k > 0. Despite being a one-dimensional system, this system is interesting due to the properties that arise from the long-range interaction between particles. It is a well-known fact that when k is small enough, particle collisions occur almost surely, while when k is large, collisions never occur. However, aside from this fact, there was no characterization of the collision times until now. In this paper, we derive the fractal (Hausdorff) dimension of the set of collision times by generalizing techniques introduced by L. Liu and Y. Xiao [Probab. Math. Stat. 18(2), 369–383 (1998)] to study the return times to the origin of self-similar Markov processes. In our case, we consider the return times to configurations where at least one collision occurs, which is a condition that defines unbounded sets as opposed to a single point, namely, the origin. We find that the fractal dimension characterizes the collision behavior of these systems and establishes a clear delimitation between the colliding and non-colliding regions in a way similar to that of a thermodynamic function.

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Statistical description of transport in multimode fibers with mode-dependent loss

Chiarawongse

Xiong

et al. 2018

New J. Phys.

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We analyze coherent wave transport in a new physical setting associated with multimode wave systems where reflection is completely suppressed and mode-dependent losses together with mode mixing are dictating the wave propagation. An additional physical constraint is the fact that in realistic circumstances the access to the scattering (or transmission) matrix is incomplete. We have addressed all these challenges by providing a statistical description of wave transport which fuses together a free probability theory approach with a filtered random matrix ensemble. Our theoretical predictions have been tested successfully against experimental data of light transport in multimode fibers. Recently, the interest in wave transport has extended to new physical settings with practical relevance, namely, a class of complex multimode systems where reflection processes are absent [16,17]. Obviously, the zero reflectivity condition, imposes new constraints to the wave scattering process, thus constituting the previous RMT predictions void. These type of transport problems have emerged naturally in the framework of multimode (or coupled multi-core) fiber optics. In these systems, fiber imperfections (core ellipticity and eccentricity) and external perturbations (index fluctuations and fiber bending) cause coupling and interference between propagating signals in different spatial modes and orthogonal polarizations. At the same time, the effect of mode-dependent loss (MDL) (or gain due to optical amplifiers) in wave propagation is another important feature whose ramifications are not yet completely understood [16][17][18][19]. In the framework of multimode fibers (MMFs), for example, it leads to fundamental limitations in their performance since extremely high MDL can reduce the number of propagating modes and thus the information capacity of MMFs. It is, therefore, imperative to develop statistical theories that take into consideration the modal and polarization mixing and MDL and provide a quantitative description of light transport in realistic MMFs (and other multimode systems that demonstrate similar challenges).

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Stochastic Processes and Random Matrices

Cited by 4 publications

References 40 publications

Dyson’s disordered linear chain from a random matrix theory viewpoint

Dyson’s disordered linear chain from a random matrix theory viewpoint

Hausdorff dimension of collision times in one-dimensional log-gases

Statistical description of transport in multimode fibers with mode-dependent loss

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