通过大偏差求和规则：多项式势和多割域的推广
Sum rules via large deviations: extension to polynomial potentials and the multi-cut regime
摘要：
求和规则是将测度的熵与其正交多项式的构造中所涉及的系数（雅可比系数）联系起来的恒等式。我们的论文是Gamboa，Nagel和Rouault（2016）的扩展，其中我们仅使用概率工具（即大偏差理论）来展示求和规则。在这里，我们证明了酉不变随机矩阵的加权谱测度在两种一般情况下的大偏差原理：第一，当平衡测度不一定由单个区间支持时;第二，当势函数为非负多项式时.速率函数可以表示为雅可比系数的函数。这些新的大偏差结果导致了一个和多个切割区域的原始求和规则，也回答了Gamboa，Nagel和Rouault（2016）关于一般求和规则的猜想。
英文摘要：
A sum rule is an identity connecting the entropy of a measure with coefficients involved in the construction of its orthogonal polynomials (Jacobi coefficients). Our paper is an extension of Gamboa, Nagel and Rouault (2016), where we have showed sum rules by using only probabilistic tools (namely the large deviations theory). Here, we prove large deviation principles for the weighted spectral measure of unitarily invariant random matrices in two general situations: firstly, when the equilibrium measure is not necessarily supported by a single interval and secondly, when the potential is a nonnegative polynomial. The rate functions can be expressed as functions of the Jacobi coefficients. These new large deviation results lead to original sum rules both for the one and the multi-cut regime and also answer a conjecture stated in Gamboa, Nagel and Rouault (2016) concerning general sum rules.
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