| 研究課題/領域番号 |
23K12984
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| 研究機関 | 学習院大学 |
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研究代表者 |
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| 研究期間 (年度) |
2023-04-01 – 2027-03-31
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| キーワード | Directed polymer / Random network / Random environment / Phase transition / Degree correlation / Probability Theory |
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| 研究実績の概要 |
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.
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
2: おおむね順調に進展している
理由
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.
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| 今後の研究の推進方策 |
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.
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| 次年度使用額が生じた理由 |
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.
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