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標題Title: Universality for random matrices and log-gases
作者Authors: La ́szlo ́ Erd ̋os
上傳單位Department: 電子工程系
上傳時間Date: 2012-12-8
上傳者Author: 傅俊結
審核單位Department: 電子工程系
審核老師Teacher: 傅俊結
檔案類型Categories: 論文Thesis
關鍵詞Keyword: random matrix
摘要Abstract: Eugene Wigner’s revolutionary vision predicted that the energy levels of large complex quan- tum systems exhibit a universal behavior: the statistics of energy gaps depend only on the basic symmetry type of the model. These universal statistics show strong correlations in the form of level repulsion and they seem to represent a new paradigm of point processes that are charac- teristically different from the Poisson statistics of independent points.
Simplified models of Wigner’s thesis have recently become mathematically accessible. For mean field models represented by large random matrices with independent entries, the celebrated Wigner-Dyson-Gaudin-Mehta (WDGM) conjecture asserts that the local eigenvalue statistics are universal. For invariant matrix models, the eigenvalue distributions are given by a log-gas with potential V and inverse temperature β = 1, 2, 4. corresponding to the orthogonal, unitary and symplectic ensembles. For β ̸∈ {1,2,4}, there is no natural random matrix ensemble behind this model, but the analogue of the WDGM conjecture asserts that the local statistics are independent of V .

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2012_12_03568211.pdf 814Kb pdf 112 48
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