Statistical mechanics of shape ensembles and their limit theorems

由于杨图的系综上的分布可以看作是整数的随机划分定律，我们对它们进行了分类，并研究了它们之间的关系。下面列出了三种众所周知的随机划分：出现在种群遗传学抽样理论背景下的随机划分；出现在数论和组合学中的吉布斯形式的随机划分；在表示论中起重要作用的行列式形式的随机划分。Ewens分布是种群遗传学中最重要的分布，它与Wright-Fisher扩散模型的平衡态(可逆平稳分布)有关。我们考虑了它们的推广，称为Gillesbie-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。这防止了我们猜测…的显式形式更多的是吉列斯皮-佐藤扩散模型的可逆分布。尽管有这个困难，我们还是得到了所有可能的可逆分布的显式形式，证明了它们对于某些Dirichlet分布是相互绝对连续的。乍一看，该分布的显式形式相当复杂。然而，我们指出，该分布是吉布斯形式的，势由合理的熵函数给出。结合极限定理，得到了一个对数Sobolev不等式。这意味着快速收敛到平衡态。Ogura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。与后一结果相关的速率函数被证明是一个熵形式的泛函。Mitoma从无限维分析的角度考虑了随机振子积分。应用Fujiwara理论的无限维扩展，得到了它关于振子能级的渐近展开式。较少

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