Towards deeper understanding of renormalization group and lace expansion
Project/Area Number 
16540102

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  Kyushu University 
Principal Investigator 
HARA Takashi Kyushu University, Faculty of Mathematics, Professor (20228620)

Project Period (FY) 
2004 – 2007

Project Status 
Completed (Fiscal Year 2007)

Budget Amount *help 
¥3,540,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥240,000)
Fiscal Year 2007: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2006: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2005: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2004: ¥900,000 (Direct Cost: ¥900,000)

Keywords  lace expansion / renormalization group / critical phenomena / selfavoiding walk / percolation / pq model / twopoint function / hierarchical model / 階層モデル / イジングモデル 
Research Abstract 
My research results can be classified into three: (1) Analysis of critical twopoint functions in selfavoiding walk, percolation, and lattice trees and animals. I used the lace expansion technique to the extreme, and derived the asymptotic form of twopoint functions of these models in a mathematically rigorous manner, in high dimensions. I also derived a detailed estimates on subleading behavior of twopoint functions. (2) Analysis of the critical behavior of pq model. Pq model has two parameters, p and q, and coincides with percolation when q=1p, and coincides with lattice animals when q=1. This model is interesting, because percolation has the critical dimension six (6) and lattice animal has eight (8); which of the critical dimension does the pq model exhibit? We first established the lace expansion and van den BergKesten inequalities for this model. We then proved that the model exhibits the same critical behavior as that of lattice animals in high dimensions. Our analysis also suggests that the critical dimension of the pq model is right (8), not six(6). (3) Analysis of hierarchical selfavoiding walk in four dimensions. We investigated the critical behavior of hierarchical selfavoiding walk model in four dimensions. By making use of hierarchical structure, we could derive a renormalization group recursion which can be rigorously analyzed. As a result, we proved the model exhibits the socalled logarithmic corrections.

Report
(5 results)
Research Products
(5 results)