随机矩阵的不变2x2beta-系综

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摘要翻译：
引入并精确求解了一族依赖于一个参数eta的2×2随机矩阵，证明了旋转不变性与实Dyson指数beta不是不相容的性质。输入的概率密度包含一个权函数和一个多重迹-迹相互作用项，它对应于基于功率和的范德蒙平方耦合的表示。结果，系综的有效Dyson指数\beta_{eff}可以在一个区间内取任意实值。详细讨论了两种权函数（高斯和非高斯），并着重讨论了它们与Dumitriu-Edelmanβ系综和所谓Poisson-Wigner能级间距交叉的关系。不同对称等级的系综之间奇怪的光谱孪生现象被揭开。所提出的技术工具更通用地允许设计实际的矩阵模型，这些模型i）旋转不变；ii）具有真实的Dyson指数β{eff}；iii）具有预先指定的约束势或交替的能级间距分布。分析结果通过数值模拟得到了很好的验证。最后，我们讨论了可能的推广和进一步的研究方向。
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英文标题：
《On invariant 2x2 \beta-ensembles of random matrices》
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作者：
Pierpaolo Vivo, Satya N. Majumdar
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最新提交年份：
2008
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分类信息：

一级分类：Physics        物理学
二级分类：Mathematical Physics        数学物理
分类描述：Articles in this category focus on areas of research that illustrate the application of mathematics to problems in physics, develop mathematical methods for such applications, or provide mathematically rigorous formulations of existing physical theories. Submissions to math-ph should be of interest to both physically oriented mathematicians and mathematically oriented physicists; submissions which are primarily of interest to theoretical physicists or to mathematicians should probably be directed to the respective physics/math categories
这一类别的文章集中在说明数学在物理问题中的应用的研究领域，为这类应用开发数学方法，或提供现有物理理论的数学严格公式。提交的数学-PH应该对物理方向的数学家和数学方向的物理学家都感兴趣；主要对理论物理学家或数学家感兴趣的投稿可能应该指向各自的物理/数学类别
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一级分类：Physics        物理学
二级分类：Statistical Mechanics        统计力学
分类描述：Phase transitions, thermodynamics, field theory, non-equilibrium phenomena, renormalization group and scaling, integrable models, turbulence
相变，热力学，场论，非平衡现象，重整化群和标度，可积模型，湍流
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一级分类：Mathematics        数学
二级分类：Mathematical Physics        数学物理
分类描述：math.MP is an alias for math-ph. Articles in this category focus on areas of research that illustrate the application of mathematics to problems in physics, develop mathematical methods for such applications, or provide mathematically rigorous formulations of existing physical theories. Submissions to math-ph should be of interest to both physically oriented mathematicians and mathematically oriented physicists; submissions which are primarily of interest to theoretical physicists or to mathematicians should probably be directed to the respective physics/math categories
math.mp是math-ph的别名。这一类别的文章集中在说明数学在物理问题中的应用的研究领域，为这类应用开发数学方法，或提供现有物理理论的数学严格公式。提交的数学-PH应该对物理方向的数学家和数学方向的物理学家都感兴趣；主要对理论物理学家或数学家感兴趣的投稿可能应该指向各自的物理/数学类别
--

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英文摘要：
  We introduce and solve exactly a family of invariant 2x2 random matrices, depending on one parameter \eta, and we show that rotational invariance and real Dyson index \beta are not incompatible properties. The probability density for the entries contains a weight function and a multiple trace-trace interaction term, which corresponds to the representation of the Vandermonde-squared coupling on the basis of power sums. As a result, the effective Dyson index \beta_{eff} of the ensemble can take any real value in an interval. Two weight functions (Gaussian and non-Gaussian) are explored in detail and the connections with \beta-ensembles of Dumitriu-Edelman and the so-called Poisson-Wigner crossover for the level spacing are respectively highlighted. A curious spectral twinning between ensembles of different symmetry classes is unveiled. The proposed technical tool more generically allows for designing actual matrix models which i) are rotationally invariant; ii) have a real Dyson index \beta_{eff}; iii) have a pre-assigned confining potential or alternatively level-spacing profile. The analytical results have been checked through numerical simulations with an excellent agreement. Eventually, we discuss possible generalizations and further directions of research.
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PDF链接：
https://arxiv.org/pdf/706.2476

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关键词：beta Bet ETA Mathematical formulations random trace 实值 一族 Poisson