摘要翻译：
我们考虑了一类由n次大整数d索引的实随机多项式，重点讨论了这类随机多项式的实根个数。这类多项式在[0,1]区间内无实根的概率以幂律n^{-θ(d)}的形式衰减，其中θ(d)>0是空间维d中具有随机初始条件的扩散方程的持续概率衰减的指数。对于n偶数，这样的多项式在全实轴上无根的概率衰减为n^{-2(\theta(d)+\theta(2))}。对于d=1，这种联系允许实随机多项式的物理实现。我们进一步证明了这类多项式在[0,1]中精确有k个实根的概率有一个由n^{-\tilde\phi(k/\log n）}给出的不寻常的标度形式，其中\tilde\phi(x)是一个普遍的大偏差函数。
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英文标题：
《Statistics of the Number of Zero Crossings : from Random Polynomials to
  Diffusion Equation》
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作者：
Gregory Schehr, Satya N. Majumdar
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最新提交年份：
2007
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分类信息：

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：Physics        物理学
二级分类：Disordered Systems and Neural Networks        无序系统与神经网络
分类描述：Glasses and spin glasses; properties of random, aperiodic and quasiperiodic systems; transport in disordered media; localization; phenomena mediated by defects and disorder; neural networks
眼镜和旋转眼镜；随机、非周期和准周期系统的性质；无序介质中的传输；本地化；由缺陷和无序介导的现象；神经网络
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一级分类：Mathematics        数学
二级分类：Probability        概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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英文摘要：
  We consider a class of real random polynomials, indexed by an integer d, of large degree n and focus on the number of real roots of such random polynomials. The probability that such polynomials have no real root in the interval [0,1] decays as a power law n^{-\theta(d)} where \theta(d)>0 is the exponent associated to the decay of the persistence probability for the diffusion equation with random initial conditions in space dimension d. For n even, the probability that such polynomials have no root on the full real axis decays as n^{-2(\theta(d) + \theta(2))}. For d=1, this connection allows for a physical realization of real random polynomials. We further show that the probability that such polynomials have exactly k real roots in [0,1] has an unusual scaling form given by n^{-\tilde \phi(k/\log n)} where \tilde \phi(x) is a universal large deviation function.
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PDF链接：
https://arxiv.org/pdf/705.2648