Statistical mechanics of shape ensembles and their limit theorems

由于杨氏图系综上的分布可以看作是整数的随机分拆规律，我们对它们进行了分类，并研究了它们之间的关系。下面列出的是三种类型的随机划分是众所周知的：随机划分出现在人口遗传学的抽样理论的背景下，吉布斯形式的随机划分出现在数论和组合学中，随机划分的行列式形式，其中发挥了重要作用的表示理论。Ewens分布，在人口遗传学中最重要的分布，是与平衡状态（可逆平稳分布）的赖特-费舍尔扩散模型。我们考虑他们的推广称为Gillespie-Sato扩散模型。扩散系数是广义的，因此该模型不是Wright-Fisher扩散模型的简单扰动。这就阻止了我们对一个显式形式的猜测。 关于我们 Gillespie-Sato扩散模型的可逆分布。尽管有这样的困难，我们得到了所有可能的可逆分布的明确形式，这些分布相对于某些狄利克雷分布是相互绝对连续的。乍一看，分配的明确形式相当复杂。然而，我们指出，该分布是吉布斯形式的潜在的一个合理的熵函数。结合极限定理，得到了一个对数Sobolev不等式.小仓讨论了某些随机集的极限定理，特别是中心极限定理和大偏差。与后一结果相关联的速率函数被证明是熵形式的泛函。Mitoma从无限维分析的角度考虑了随机振子积分。应用无限维扩展藤原的理论产生了它的渐近展开相对于振子水平。少

Since distributions on ensembles of Young diagrams can be regarded as laws of random partitions of integers, we classified them, and then studied the relationship to each other. Listed in the following are three types of such random partitions which are well known :Random partitions which appear in the context of the sampling theory of population genetics.Random partitions of Gibbsian form which appear in the number theory and combinatorics.Random partitions of determinantal form which play an important role in the representation theory.Ewens distributions, the most important distributions in population genetics, are associated with equilibrium states (reversible stationary distributions) of the Wright-Fisher diffusion models. We considered their generalization called Gillespie-Sato diffusion models. The diffusion coefficient is generalized, and therefore the model is not a simple perturbation of the Wright-Fisher diffusion model. This prevents us from any guess of an explicit form of … More the reversible distribution of the Gillespie-Sato diffusion model. In spite of this difficulty, we obtained explicit form of all possible reversible distributions, which turned out to be mutually absolutely continuous with respect to certain Dirichlet distributions. At first sight, the explicit form of the distribution is rather complicated. However, we point out that the distribution is of Gibbsian form with potential given by a reasonable entropy function. In connection with limit theorems, a logarithmic Sobolev inequality has been obtained. This implies a fast convergence to the equilibrium state.Ogura discussed limit theorems for certain random sets, especially central limit theorems and large deviations. The rate function associated with the latter result was shown to be a functional of entropy form.Mitoma considered stochastic oscillator integrals from a point of view of the infinite dimensional analysis. Applying an infinite dimensional extension of Fujiwara's theory yielded its asymptotic expansion with respect to the oscillator level. Less