Research on strongly correlated random fields related to stochastic processes

2021 Fiscal Year Final Research Report

• Project/Area Number: 18K13429
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 12010:Basic analysis-related
• Research Institution: Chiba University
• Principal Investigator: Abe Yoshihiro 千葉大学, 大学院理学研究院, 講師 (50814109)
• Project Period (FY): 2018-04-01 – 2022-03-31
• Keywords: 被覆時間 / 離散ガウス自由場 / thick point / late point / random interlacement / 局所時間

Outline of Final Research Achievements

I studied statistical properties of local times for the simple random walk (SRW) on the two-dimensional lattice. Local times are total accumulated times SRW spends on vertices. I especially focused on thick points (points whose local times are close to the maximum), thin points (points whose local times are close to the minimum), light points (points whose local times are of order one), late point (point whose local times are zero). As joint works with Marek Biskup (UCLA) and Sangchul Lee (USC), we proved that statistics of these points are characterized by random measures called Liouville quantum gravity measure. We wrote two papers and submit them to journals.

Free Research Field

確率論

Academic Significance and Societal Importance of the Research Achievements

Thick pointやavoided pointの研究はErdos-Taylor (1960)、Aldous (1989)まで遡り、いくつかの問題提起がなされた。その後、Dembo-Peres-Rosen-Zeitouni (2001、 2006)が問題を解決した。同時に、新たな問題提起をして、現在も未解決な問題は多い。本研究はthick pointやlate pointが2次元量子重力測度と密接に関連していることを明らかにし、これらの問題に新たな知見を与えた。これにより、今後Dembo氏らが提起した問題解決に向けた研究のより一層の進展が期待される。