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SCIENTIA SINICA Mathematica, Volume 49 , Issue 3 : 327(2019) https://doi.org/10.1360/N012017-00269

Mathematical topics motivated from statistical physics —Interaction of phase transition and mathematics

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  • ReceivedDec 27, 2017
  • AcceptedMar 7, 2018
  • PublishedMar 18, 2019

Abstract


Funding

国家自然科学基金(11771046)


Acknowledgment

本文是笔者发表在《中国科学: 数学》庆贺侯振挺教 授80华诞专辑[38]和 庆贺严士健教授90华诞专辑文章[2]的续篇. 这三篇序列 文章概略地回顾了笔者在严士健、侯振挺两位导师的指导和王梓坤先生的 指点下所走过的研究历程. 这些回忆一方面让笔者深感惭愧, 仔细一算, 没有做出多少好成果, 有愧师恩; 另一方面, 也 深深怀念老师们几十年的栽培之情. 仅以此拙文表达对王梓 坤老师90华诞的衷心祝贺本文中所用到的笔者的论文, 应当都可从其主页http://math0.bnu.edu.cn/ chenmf 中找到. 特别是中间的论文集: Vol.1–Vol.4.


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  • Table 1   生灭过程10种遍历性的显式判别准则
    性质 判断标准
    唯一性 $\sum_{n\ge~0}{\hat\nu}_n \mu[0,~n]~=~\infty~~~(*)$
    常返性 ${\hat\nu}[0,\infty)=\infty$
    遍历性 $(*)~~\&~~\mu[0,~\infty)<~\infty$
    指数遍历性 $(*)~\&~\sup~_{n\ge~1}~\mu[n,~\infty)~{\hat\nu}[0,~n-1]~<\infty$
    Poincaré 不等式
    离散谱 $(*)~\&~~\lim~_{n\to\infty}\sup~_{k\ge~n+1}~\mu[k, \infty)~{\hat\nu}~[n,k-1]~=0$
    对数 Sobolev 不等式 $(*)~\&~\sup~_{n\ge~1}\mu[n,\infty) ~[\log\frac{1}{\mu[n,\infty)}~]~{\hat\nu}~[0,n-1] ~~~~<\infty~$
    关于相对熵指数收敛?
    强遍历性 $(*)~~\&~\sum~_{n\ge~0}{\hat\nu}_n~\mu[n+1,\infty)~=~\sum~_{n\ge~1}\mu_n~{\hat\nu}~[0,n-1]<\infty$
    $L^1$ 指数收敛性
    Nash 不等式 $(*)~~\&~\sup~_{n\ge~1}~\mu[n,\infty)^{(p^*-2)/(p^*-1)}{\hat\nu}~[0,n-1]<\infty~$
    其中第一行的条件 $(*)$ 用于随后的6种情形
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