Studies on random fractals and integrable stochastic processes

2019 Fiscal Year Final Research Report

• Project/Area Number: 16K05468
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical physics/Fundamental condensed matter physics
• Research Institution: Tokyo University of Science
• Principal Investigator: Sakai Kazumitsu 東京理科大学, 理学部第二部物理学科, 准教授 (10397028)
• Co-Investigator(Kenkyū-buntansha): 茂木 康平 東京海洋大学, 学術研究院, 准教授 (30583033)
• Project Period (FY): 2016-04-01 – 2020-03-31
• Keywords: Schramm-Loewner発展 / 可解模型 / 量子可積分系 / 共形場理論 / ベーテ仮説 / 輸送特性 / 対称多項式 / Grothendieck多項式

Outline of Final Research Achievements

Geometric properties in the 2D critical phenomena can be classified by the Schramm-Loewner evolution. We have studied a relationship between SLE's and (1+1) quantum integrable systems. In particular, we found that SLE's with spin degrees of freedom can be associated with some quantum integrable systems. We have also investigated random fractals appearing in the critical 4-state Potts model and found a characteristic logarithmic behavior of crossing probabilities.

The partition functions of type B and C ice models have been investigated by using the quantum inverse scattering method. We have shown that the wavefunctions are expressed using generalizations of the symplectic Schur functions.

Free Research Field

数理物理学

Academic Significance and Societal Importance of the Research Achievements

２次元臨界現象にはフラクタルと呼ばれる特有の幾何構造が現れることが知られている．Schramm-Loewner発展(SLE)によって，これらのフラクタルは数学的に分類することができることがわかってきた．スピン自由度を付随させたSLEが(1+1)次元の量子可積分系と結びついている点を見出したことは意義がある．

また，数学的に重要な対称多項式の研究をを物理学における可解格子模型とその波動関数の研究に帰着させて発展させた点にも意義がある．