2004 Fiscal Year Final Research Report Summary
Statistical mechanics of shape ensembles and their limit theorems
| Project/Area Number |
14540128
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Saga University |
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Principal Investigator |
HANDA Kenji Saga University, Faculty of Sci. & Eng., Associate prof., 理工学部, 助教授 (10238214)
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| Co-Investigator(Kenkyū-buntansha) |
OGURA Yukio Saga University, Faculty of Sci. & Eng., Professor, 理工学部, 教授 (00037847)
MITOMA Itaru Saga University, Faculty of Sci. & Eng., Professor, 理工学部, 教授 (40112289)
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| Project Period (FY) |
2002 – 2004
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| Keywords | Young diagram / sampling theory / population genetics / Ewens distribution / Gillespie-Sato diffusion / random sets / stochastic oscillator integral |
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| Research Abstract |
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
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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
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