Level Statistics for Ensembles of Random Matrices with Different Symmetry Properties

Das, Adway Kumar (2019) Level Statistics for Ensembles of Random Matrices with Different Symmetry Properties. Masters thesis, Indian Institute of Science Education and Research Kolkata.

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Abstract

Random Matrix Theory (RMT) has been a pivotal statistical technique in diverse disciplines ranging from nuclear spectroscopy to neural networking. The most prevalent universality classes in these systems are Wigner and Poisson ensembles. However, experimental data often show statistics intermediate between above symmetry classes. In this work, we have studied Rosenzweig-Porter Ensemble (RPE), which gives a natural generalization of above ensembles, thus allowing to treat mixed dynamics as well. Here we have observed some interesting phenomena, e.g. breaking of integrability in symmetric RPE and anti-unitary symmetry in hermitian RPE as rank of matrices is increased from 2 to 3, existence of three different phases for rank 2 hermitian RPE etc. For small matrices we have tried to obtain analytical expression, whereas for large ones, we have relied on numerical data. We have produced some interpolating functions to quantify transition from one behaviour to another. We have also studied some unconventional symmetries like centrosymmetry, persymmetry etc. Some interesting features are present in these ensembles, e.g. gradual breaking of integrability as thermodynamic limit is approached, oscillation of clustering strength as rank is increased, deviation of average Density of states from Wigner's semicircle law, combination of two independent symmetries giving rise to level clustering despite the fact that both parent ensembles show level repulsion etc.

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