Renormarization of two dimensional random fields with rich symmetry
Project/Area Number  10640122 
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  Hiroshima University 
Principal Investigator 
IWATA Koichiro Hiroshima University, Faculty of Science, Associate Professor, 理学部, 助教授 (20241292)

CoInvestigator(Kenkyūbuntansha) 
NAKAMURA Munetaka Yamanashi University, Faculty of Education and Human Science, Associate Professor, 教育人間科学部, 助教授 (10227944)
KUBO Izumi Hiroshima University, Faculty of Science, Professor, 理学部, 教授 (70022621)

Project Period (FY) 
1998 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥1,800,000 (Direct Cost : ¥1,800,000)
Fiscal Year 1999 : ¥700,000 (Direct Cost : ¥700,000)
Fiscal Year 1998 : ¥1,100,000 (Direct Cost : ¥1,100,000)

Keywords  random fields / modular forms / CauchyRiemann equation / multiple Wiener integrals / 等角構造 / wick積 / コーシーリマン方程式 
Research Abstract 
The aim of the present research project is to study conformally invariant random fields which arises as unique solution of inhomogeneous CauchyRiemann equation. The upper half space in the complex plain parameterizes the conformal structure of the two dimensional torus on which the modular group acts. The latter action yields the modular covariance of the random fields. As a consequence, the moment functions of the random field evaluated at rational points are automorphic. In several cases one can prove that the moment functions are actually modular functions. With the help of the expression of the solution in terms of elliptic functions, it is possible to construct functionals, called the renormalized product, of the field with higher weight relative to the modular group action. This extends the class of modular functions which admits integral representation by the random field. This is related to the fact that there exist a large class of local functionals, called the Wick products, in two dimensional quantum field theory. Their existence reflects the logarithmic singularity of the Green function. The origin of the rather mild singularity is the conformal structure of the two dimensional space. So it is natural to ask how the conformal structure determines the renonnalized products and how the conformal structure determines class of modular functions which admits integral representation. A natural way to attack these problem is as follows : Study the action of the modular group on the configuration space of the pair of the evaluation points and the wight of the renormalized products and the action of the modular group on the cusps. So far the space of modular forms spanned by Eisenstein series are studied and it is proved that Eisenstein series are represented by moments of the random fields provided the weight is not greater than 10.

Report
(3results)
Research Output
(3results)