Probabilistic Analysis of Gibbs measures associated with the integrable system
Project/Area Number 
14540110

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  Kwansei Gakuin University 
Principal Investigator 
CHIYONOBU Taizo Kwansei Gakuin Univ., School of Science and Technology, Ass.Professor, 理工学部, 助教授 (50197638)

CoInvestigator(Kenkyūbuntansha) 
YABUTA Kozo Kwansei Gakuin Univ., School of Science and Technology, Professor, 理工学部, 教授 (30004435)
YAMANE Hideshi Kwansei Gakuin Univ., School of Science and Technology, Ass.Professor, 理工学部, 助教授 (80286145)
ICHIHARA Kanji Kansai Uni., Faculty of Engineering, Professor, 工学部, 教授 (00112293)
TAMURA Yozo Keio Univ., Science and Technology, Ass.Professor, 理工学部, 助教授 (50171905)
OSADA Hirofumi Kyushu Univ., Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (20177207)

Project Period (FY) 
2002 – 2005

Project Status 
Completed (Fiscal Year 2005)

Budget Amount *help 
¥2,100,000 (Direct Cost: ¥2,100,000)
Fiscal Year 2005: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2004: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2003: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2002: ¥600,000 (Direct Cost: ¥600,000)

Keywords  integrable system / Gibbs measure / limit theorems / network / ポリヤの壺 / 長距離可積分系 / 大偏差原理 / ラプラス法 
Research Abstract 
We have obtained results on two topics. First, we study the asymptotic behavior of the partition function of a finete Gibbs measure with repulsive interaction potentials. The function is a modification of the one studied by G.Gallavotti & C.Marchioro. While their approach depends on the integrability of the model, we apply the Laplace method to give an logarithmic asymptotic of the partition function. This is of some interest for the theory of the large deviation principle in the sense that the partition function has the different order of asymptotics from the one given by Varadhan's asymptotic formula. Our result is weaker than theirs in the sense that it gives only the logarithmic asymptotics. But we believe it is still worth noting down here, because we view the integal from a totally different perspective ; namely, we treat as a Laplace asymptotics and thus our method here is applicable to not only the potentials which constitutes the integrable Hamiltonian system, but to more gene
… More
ral singular potentials. Theorem is of interest from the point of view of the theory of large deviations because it gives the different order 1/n^2 from the ordinary order 1/n, which is the one when Varadhan's asymptotic formula can be applied. Secondly, we study on networks. Many real networks in the world have the powerlaw degree distribution One of the most famous model that generates SF networks is ‘BA model' proposed by Barab'asi and Albert. BA model has two properties. One is the growth property and the other is the it preferential attachment property such that a new added vertex is connected to an old vertex randomly chosen with the probability proportional to its degree. However, all of realworld networks are not generated by these properties. This network model based on thresholding of the summed vertex weights, which belongs to the subclass of a model such that the connection by edges are determined by interactions of vertices that are endowed with intrinstic weights. We focused on the mechanism by which the weight distribution is generated. We proved the weight distribution asymptotically tends to the geometric distribution which generates on the preferential attachment property, and we show by the numerical experiments that a graph generated by the threshold model when the weight distribution is geometric distribution is a SF network. Less

Report
(5 results)
Research Products
(16 results)