Random polynomials, random matrices and L-functions

@article{Farmer2006RandomPR,
  title={Random polynomials, random matrices and L-functions},
  author={David W. Farmer and Francesco Mezzadri and Nina C. Snaith},
  journal={Nonlinearity},
  year={2006},
  volume={19},
  pages={919-936}
}
We show that the circular orthogonal ensemble of random matrices arises naturally from a family of random polynomials. This sheds light on the appearance of random matrix statistics in the zeros of the Riemann zeta-function. 
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Let $$Q_{n} (x) = \sum\nolimits_{i = 0}^{n} A_{i} x^{i}$$Qn(x)=∑i=0nAixi be a random algebraic polynomial where the coefficients $$A_{0} ,A_{1} , \cdots$$A0,A1,⋯ form a sequence of centered Gaussian
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TLDR
All matrices of physical relevance share in common that their form or entries are dependent on empirical observations taken on the system which they seek to describe, and any matrices built from them will be subject to error which could invalidate them as instruments of study.

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