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場所： 東京大学大学院数理科学研究科 数理科学研究科棟（駒場）１２６号室
講師: 白石 大典 氏
時間: 16:00 - 17:30
Title: 4次元単純ランダムウォークの交叉と長距離パーコレーションの関係
Abstract: よく知られているように、4次元ブラウン運動の軌跡は単純曲線になる。一方で4次元単純ランダムウォークの軌跡は、ループ除去ランダムウォークの長さに非自明な対数項が現れる程度のループを持つ。本講演では、4次元単純ランダムウォークの軌跡の交叉とある長距離パーコレーションモデルとの関係性について解説する。
場所： 早稲田大学西早稲田キャンパス５１号館１８階18-06号室
講師: Mathav Murugan 氏
時間: 16:00 - 17:30
Title: A simplified characterization of stable-like heat kernel estimates
Abstract: We obtain a characterization of two-sided stable-like heat kernel estimates, modelled on symmetric stable processes in Euclidean space. Our characterization confirms a conjecture of Grigor’yan, E. Hu, and J. Hu, and is based on two-sided bounds on the jump kernel together with upper bounds on capacity across annuli.
場所： 東京大学大学院数理科学研究科 数理科学研究科棟（駒場）１２６号室
講師: Clément Cosco 氏
時間: 14:00 - 15:30
Title: The maximum of 2d directed polymers. (Joint work with Shuta Nakajima and Ofer Zeitouni.)
Abstract: 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.
講師: Subhro Ghosh 氏
時間: 16:00 - 17:30
Title: Gaussian fluctuations for spin systems and point processes: near-optimal rates via quantitative Marcinkiewicz's theorem
Abstract: 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.