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2023 Fiscal Year Research-status Report

Effect of disorder on polymers and on real-world networks

Research Project

Project/Area Number 23K12984
Research InstitutionGakushuin University

Principal Investigator

JUNK StefanFlorian  学習院大学, 理学部, 助教 (90974907)

Project Period (FY) 2023-04-01 – 2027-03-31
KeywordsDirected polymer / Random network / Random environment / Phase transition / Degree correlation / Probability Theory
Outline of Annual Research Achievements

One part of the research project concerns the study of the so-called "directed polymer model". We were able to prove the "local limit theorem" in almost the whole weak disorder regime, which solves one of the questions that we wanted to address in the project. Namely, we showed that in the whole weak disorder regime (except for the critical point), the point-to-point partition function is well-approximated by a product of two point-to-line partition functions.
We also made an unexpected breakthrough and confirmed the equivalence of strong and very strong disorder for this model, which was a major open conjecture in the field for 20 years. This result is open because major structural properties of the model (e.g., full path localization) are only understood in the strong disorder phase and in the weak disorder phase, so it is important to exclude the possibility of an intermediate regime. Finally, our result also confirmed that weak disorder holds at the critical value, which resolved another major open question.
Both results are undergoing peer review and we have also presented the research at various national and international conferences.
In the second part of this project regarding the effect of long-range degree correlations in real-world networks, we are currently finalizing a mean-field approximation for such networks.

Current Status of Research Progress
Current Status of Research Progress

2: Research has progressed on the whole more than it was originally planned.

Reason

The research about the random polymer model is proceeding very well and has resulted in unexpected breakthroughs. The numerical investigation of real-world networks is proceeding well, but we are currently investigating some disagreements between the mean-field model and the data.

Strategy for Future Research Activity

Regarding the random polymer model, we will proceed with the second research question (the behavior in a heavy-tailed medium). We will also investigate the behavior of the model around the critical value, based on the new understanding provided by our proof of the equivalence of strong and very strong disorder.

For the numerical investigation of random networks, we will complete the mean-field approximation and publish the results. Based on this approximation, we will try to implement an algorithm that samples finite-size networks whose characteristics match the mean-field model.

Causes of Carryover

Due to the weakness of the Yen, the cost for oversea travel is significantly higher than anticipated and I had to reduce the number of conferences, which resulted in some leftover money. The money will now be used for travel expenses as originally intended.

  • Research Products

    (10 results)

All 2024 2023 Other

All Int'l Joint Research (1 results) Journal Article (1 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 1 results,  Open Access: 1 results) Presentation (7 results) (of which Int'l Joint Research: 5 results,  Invited: 6 results) Remarks (1 results)

  • [Int'l Joint Research] IMPA, Rio de Janero(ブラジル)

    • Country Name
      BRAZIL
    • Counterpart Institution
      IMPA, Rio de Janero
  • [Journal Article] Moment characterization of the weak disorder phase for directed polymers in a class of unbounded environments2023

    • Author(s)
      Fukushima Ryoki、Junk Stefan
    • Journal Title

      Electronic Communications in Probability

      Volume: 28 Pages: 1-9

    • DOI

      10.1214/23-ECP545

    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Presentation] The Directed Polymer Model in Weak Disorder beyond the L2-Phase2024

    • Author(s)
      Stefan Junk
    • Organizer
      Probability Seminar Series, Shanghai.
    • Int'l Joint Research / Invited
  • [Presentation] Strong disorder and very strong disorder are equivalent for directed polymers2024

    • Author(s)
      Stefan Junk
    • Organizer
      Probability Seminar, Hong Kong.
    • Int'l Joint Research / Invited
  • [Presentation] Strong disorder and very strong disorder are equivalent for directed polymers2024

    • Author(s)
      Stefan Junk
    • Organizer
      Probability Seminar, Singapore.
    • Int'l Joint Research / Invited
  • [Presentation] Fluctuations for the directed polymer model in weak disorder without L^2-boundedness2023

    • Author(s)
      Stefan Junk
    • Organizer
      Stochastic Processes and their Applications 2023, Lisbon
    • Int'l Joint Research
  • [Presentation] Local Limit Theorem for Directed Polymers beyond the L2 -phase2023

    • Author(s)
      Stefan Junk
    • Organizer
      Stochastic Analysis on Large Scale Interacting Systems, Kyoto
    • Invited
  • [Presentation] Local limit theorem for directed polymer in the delocalized phase2023

    • Author(s)
      Stefan Junk
    • Organizer
      Random Operators and Related Topics, Sendai.
    • Invited
  • [Presentation] Local limit theorem for directed polymer beyond the L 2-phase2023

    • Author(s)
      Stefan Junk
    • Organizer
      Workshop on random interacting systems, Singapore.
    • Int'l Joint Research / Invited
  • [Remarks] Homepage of Hubert Lacoin

    • URL

      https://w3.impa.br/~lacoin/

URL: 

Published: 2024-12-25  

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