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東京確率論セミナー
過去の記録 ～04/26｜次回の予定｜今後の予定 04/27～
| 開催情報 | 月曜日 16:00～17:30 数理科学研究科棟(駒場) 126号室 |
|---|---|
| 担当者 | 佐々田槙子、中島秀太（慶應義塾大学）、星野壮登（東京科学大学）、蛯名真久（東京科学大学） |
| セミナーURL | https://sites.google.com/view/tokyo-probability-seminar23/ |
今後の予定
2026年04月27日(月)
14:00-17:30 数理科学研究科棟(駒場) 126号室
講演の開始が早くなっています。15:30~ 教室126でTea timeを行います。ぜひこちらにもご参加ください。
Clément Cosco 氏 (Université Paris Dauphine) 14:00-15:30
The maximum of 2d directed polymers. (Joint work with Shuta Nakajima and Ofer Zeitouni.)
Gaussian fluctuations for spin systems and point processes: near-optimal rates via quantitative Marcinkiewicz's theorem
講演の開始が早くなっています。15:30~ 教室126でTea timeを行います。ぜひこちらにもご参加ください。
Clément Cosco 氏 (Université Paris Dauphine) 14:00-15:30
The maximum of 2d directed polymers. (Joint work with Shuta Nakajima and Ofer Zeitouni.)
[ 講演概要 ]
Directed polymers can be described as a tilting of the simple random walk, where some local random noise can attract or repel the trajectory of the walk. In the subcritical regime of the two-dimensional model, the partition function is known to be asymptotically approximated by a Gaussian log-correlated field. In a work in collaboration with Shuta Nakajima and Ofer Zeitouni, we could refine this result by proving that the maximum of the partition function field converges to that of a branching Brownian motion, which is the source of the log-correlation. In this talk, I will introduce the model as well as the objects related to it and present our result.
Subhro Ghosh 氏 (National University of Singapore) 16:00-17:30Directed polymers can be described as a tilting of the simple random walk, where some local random noise can attract or repel the trajectory of the walk. In the subcritical regime of the two-dimensional model, the partition function is known to be asymptotically approximated by a Gaussian log-correlated field. In a work in collaboration with Shuta Nakajima and Ofer Zeitouni, we could refine this result by proving that the maximum of the partition function field converges to that of a branching Brownian motion, which is the source of the log-correlation. In this talk, I will introduce the model as well as the objects related to it and present our result.
Gaussian fluctuations for spin systems and point processes: near-optimal rates via quantitative Marcinkiewicz's theorem
[ 講演概要 ]
We establish asymptotically Gaussian fluctuations for functionals of a large class of spin models and strongly correlated random point fields, achieving near-optimal rates.
For spin models, we demonstrate Gaussian asymptotics for the magnetization (i.e., the total spin) for a wide class of ferromagnetic spin systems on Euclidean lattices, in particular those with continuous spins. Specific applications include, in particular, the celebrated XY and Heisenberg models under ferromagnetic conditions, and more broadly, systems with very general rotationally invariant spins in arbitrary dimensions. We address both the setting of free boundary conditions and a large class of ferromagnetic boundary conditions, and our CLTs are endowed with near-optimal rate of O(log |Λ| · |Λ|−1/2) in the Kolmogorov-Smirnov distance, where the system size is |Λ|. Our approach leverages the classical Lee-Yang theory for the zeros of partition functions, and subsumes as a special case results of Lebowitz, Ruelle, Pittel and Speer on CLTs in discrete statistical mechanical models for which we obtain sharper convergence rates.
In a different direction, we obtain CLTs for linear statistics of a wide class of point processes known as α-determinantal point processes which interpolate between negatively and positively associated random point fields (including the usual determinantal, permanental and Poisson point processes).
We contribute a unified approach to CLTs in such models (agnostic to the parameter α that modulates the nature of association). Our methods are able to address a broad class of kernels including in particular those with slow spatial decay (such as the Bessel kernel in general dimensions). Significantly, our approach is able to analyse such processes in dimensions ≥ 3, where structural alternatives such as connections to random matrix theory are not available, and obtain explicit rates for fast convergence in a wide spectrum of models.
A key ingredient of our approach is a broad, quantitative extension of the classical Marcinkiewicz Theorem that holds under the limited condition that the characteristic function is non-vanishing only on a bounded disk. This technique complements classical work of Ostrovskii, Linnik, Zimogljad and others, as well as recent work of Michelen and Sahasrabudhe, and Eremenko and Fryntov. In spite of the general applicability of the results, including to heavy-tailed setups, our rates for the CLT match the classic Berry- Esseen bounds for independent sums up to a log factor.
Based on joint work with T.C. Dinh, H.S. Tran and M.H. Tran. Under revision at Annals of Applied Probability.
We establish asymptotically Gaussian fluctuations for functionals of a large class of spin models and strongly correlated random point fields, achieving near-optimal rates.
For spin models, we demonstrate Gaussian asymptotics for the magnetization (i.e., the total spin) for a wide class of ferromagnetic spin systems on Euclidean lattices, in particular those with continuous spins. Specific applications include, in particular, the celebrated XY and Heisenberg models under ferromagnetic conditions, and more broadly, systems with very general rotationally invariant spins in arbitrary dimensions. We address both the setting of free boundary conditions and a large class of ferromagnetic boundary conditions, and our CLTs are endowed with near-optimal rate of O(log |Λ| · |Λ|−1/2) in the Kolmogorov-Smirnov distance, where the system size is |Λ|. Our approach leverages the classical Lee-Yang theory for the zeros of partition functions, and subsumes as a special case results of Lebowitz, Ruelle, Pittel and Speer on CLTs in discrete statistical mechanical models for which we obtain sharper convergence rates.
In a different direction, we obtain CLTs for linear statistics of a wide class of point processes known as α-determinantal point processes which interpolate between negatively and positively associated random point fields (including the usual determinantal, permanental and Poisson point processes).
We contribute a unified approach to CLTs in such models (agnostic to the parameter α that modulates the nature of association). Our methods are able to address a broad class of kernels including in particular those with slow spatial decay (such as the Bessel kernel in general dimensions). Significantly, our approach is able to analyse such processes in dimensions ≥ 3, where structural alternatives such as connections to random matrix theory are not available, and obtain explicit rates for fast convergence in a wide spectrum of models.
A key ingredient of our approach is a broad, quantitative extension of the classical Marcinkiewicz Theorem that holds under the limited condition that the characteristic function is non-vanishing only on a bounded disk. This technique complements classical work of Ostrovskii, Linnik, Zimogljad and others, as well as recent work of Michelen and Sahasrabudhe, and Eremenko and Fryntov. In spite of the general applicability of the results, including to heavy-tailed setups, our rates for the CLT match the classic Berry- Esseen bounds for independent sums up to a log factor.
Based on joint work with T.C. Dinh, H.S. Tran and M.H. Tran. Under revision at Annals of Applied Probability.
