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Tokyo Probability Seminar
Seminar information archive ～07/26｜Next seminar｜Future seminars 07/27～
| Date, time & place | Monday 16:00 - 17:30 126Room #126 (Graduate School of Math. Sci. Bldg.) |
|---|---|
| Organizer(s) | Makiko Sasada, Shuta Nakajima |
Seminar information archive
2024/07/08
16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)
We are having teatime from 15:15 in the common room on the second floor. Please join us.
Hironobu Sakagawa (Keio University)
Maximum of the Gaussian interface model in random external fields (日本語)
We are having teatime from 15:15 in the common room on the second floor. Please join us.
Hironobu Sakagawa (Keio University)
Maximum of the Gaussian interface model in random external fields (日本語)
[ Abstract ]
相分離の界面モデルの一つとして格子上のGauss型界面モデル（離散Gauss自由場）を取り上げ，そこにランダムな外場（化学ポテンシャル）を加えた（ランダムな）Gibbs測度の下での最大値について考える．特に，外場の確率変数の末尾確率の挙動に応じて最大値の挙動が変わることを示し，その主要項を特徴付ける．
相分離の界面モデルの一つとして格子上のGauss型界面モデル（離散Gauss自由場）を取り上げ，そこにランダムな外場（化学ポテンシャル）を加えた（ランダムな）Gibbs測度の下での最大値について考える．特に，外場の確率変数の末尾確率の挙動に応じて最大値の挙動が変わることを示し，その主要項を特徴付ける．
2024/06/24
16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)
We are having teatime from 15:15 in the common room on the second floor. Please join us.
Fumihiko Nakano (Tohoku University)
Temperley - Lieb 演算子の持ち上げとRazumov - Stroganov 予想について (日本語)
We are having teatime from 15:15 in the common room on the second floor. Please join us.
Fumihiko Nakano (Tohoku University)
Temperley - Lieb 演算子の持ち上げとRazumov - Stroganov 予想について (日本語)
[ Abstract ]
Razumov - Stroganov 予想とはリンクパターン上の生成する線型空間上のあるハミルトニアンの基底状態に対応するFPLの個数が現れるという予想で、２０１０年に解決されたが、O(1)-loop model, 交代符号行列を介して２次元統計力学の模型や組み合わせ論との様々なつながりがあり、今も注目されている。Temperley - Lieb 演算子の持ち上げを用いたRS予想のより平易な証明について議論する。
Razumov - Stroganov 予想とはリンクパターン上の生成する線型空間上のあるハミルトニアンの基底状態に対応するFPLの個数が現れるという予想で、２０１０年に解決されたが、O(1)-loop model, 交代符号行列を介して２次元統計力学の模型や組み合わせ論との様々なつながりがあり、今も注目されている。Temperley - Lieb 演算子の持ち上げを用いたRS予想のより平易な証明について議論する。
2024/06/17
15:40-17:45 Room #126 (Graduate School of Math. Sci. Bldg.)
Lectures and TeaTime start earlier. We are having teatime from 15:00 in the common room on the second floor. Please join us.
Kento Ueda (The University of Tokyo) 15:40-16:40
非整数ブラウン運動で駆動される確率微分方程式の数値解の漸近展開 (日本語)
Homogenization results for reflecting diffusions in a continuum percolation cluster (日本語)
Lectures and TeaTime start earlier. We are having teatime from 15:00 in the common room on the second floor. Please join us.
Kento Ueda (The University of Tokyo) 15:40-16:40
非整数ブラウン運動で駆動される確率微分方程式の数値解の漸近展開 (日本語)
[ Abstract ]
本研究は非整数ブラウン運動(fBm)で駆動される確率微分方程式の数値解に対する極限定理(漸近誤差)に関する研究である。このfBmおよびそれによって駆動される方程式は非マルコフな時系列モデルとして用いられ、その数値解に対する極限定理は数学的興味のほか、数値シミュレーションの誤差の推定への応用が期待される。数値解の極限定理は駆動するfBmが1次元か否か、また1次元ならドリフト項が存在するか否か、さらにfBmのハースト指数、そして対象とする数値解法によって定理の主張も適用できる証明法も異なり、そのために条件ごとに様々な先行研究が存在する。このうち、本研究は1次元かつドリフト項が存在する場合に誤差分布の導出と正当化を行ったものであり、一般の数値解法に適用できる。同範囲の先行研究では高次ミルシュタイン法、クランク-ニコルソン法に対してハースト指数が1/3より大きい場合に関して漸近誤差を特定できるが、本研究では高次ミルシュタイン法の漸近誤差を任意のハースト指数に対して完全に決定するとともに、クランク-ニコルソン法に対してもハースト指数が1/4以上の場合に漸近誤差を特定している。なお、本講演では導出した誤差分布を視覚的に観察し、漸近誤差への直観的な理解を深められるよう、漸近誤差に対する数値実験の結果を詳しく説明する。
Yutaka Takeuchi ( Keio University) 16:45-17:45本研究は非整数ブラウン運動(fBm)で駆動される確率微分方程式の数値解に対する極限定理(漸近誤差)に関する研究である。このfBmおよびそれによって駆動される方程式は非マルコフな時系列モデルとして用いられ、その数値解に対する極限定理は数学的興味のほか、数値シミュレーションの誤差の推定への応用が期待される。数値解の極限定理は駆動するfBmが1次元か否か、また1次元ならドリフト項が存在するか否か、さらにfBmのハースト指数、そして対象とする数値解法によって定理の主張も適用できる証明法も異なり、そのために条件ごとに様々な先行研究が存在する。このうち、本研究は1次元かつドリフト項が存在する場合に誤差分布の導出と正当化を行ったものであり、一般の数値解法に適用できる。同範囲の先行研究では高次ミルシュタイン法、クランク-ニコルソン法に対してハースト指数が1/3より大きい場合に関して漸近誤差を特定できるが、本研究では高次ミルシュタイン法の漸近誤差を任意のハースト指数に対して完全に決定するとともに、クランク-ニコルソン法に対してもハースト指数が1/4以上の場合に漸近誤差を特定している。なお、本講演では導出した誤差分布を視覚的に観察し、漸近誤差への直観的な理解を深められるよう、漸近誤差に対する数値実験の結果を詳しく説明する。
Homogenization results for reflecting diffusions in a continuum percolation cluster (日本語)
[ Abstract ]
アブストラクト: ランダム媒質の研究において均一化は重要な問題の一つである. 均一化はいくつかの定式化が知られている, 本講演ではランダム媒質上の確率過程に関する極限定理であるquenched invariance principleと, その精密化である局所中心極限定理を考える. この様な定式化について, 離散的なモデルの場合には多くの結果が知られている. 連続的なモデルに関しても, random environment 上の拡散過程に関する結果は多く知られている. 一方拡散過程が反射壁を持つ場合に関しては, 境界の影響等により問題が複雑化するためquenchedな結果は知られていなかった. 本講演では連続パーコレーションが幾何的な条件を満たす場合, その上の反射壁を持つ拡散過程に関してquenched invariance principleと局所中心極限定理が成り立つという結果を紹介する.
アブストラクト: ランダム媒質の研究において均一化は重要な問題の一つである. 均一化はいくつかの定式化が知られている, 本講演ではランダム媒質上の確率過程に関する極限定理であるquenched invariance principleと, その精密化である局所中心極限定理を考える. この様な定式化について, 離散的なモデルの場合には多くの結果が知られている. 連続的なモデルに関しても, random environment 上の拡散過程に関する結果は多く知られている. 一方拡散過程が反射壁を持つ場合に関しては, 境界の影響等により問題が複雑化するためquenchedな結果は知られていなかった. 本講演では連続パーコレーションが幾何的な条件を満たす場合, その上の反射壁を持つ拡散過程に関してquenched invariance principleと局所中心極限定理が成り立つという結果を紹介する.
2024/06/10
16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)
We are having teatime from 15:15 in the common room on the second floor. Please join us.
Yukimi Goto (Gakushuin University)
Phase Transition in a Lattice Nambu–Jona-Lasinio Model (日本語)
We are having teatime from 15:15 in the common room on the second floor. Please join us.
Yukimi Goto (Gakushuin University)
Phase Transition in a Lattice Nambu–Jona-Lasinio Model (日本語)
[ Abstract ]
量子色力学で重要な概念としてカイラル対称性の破れとそれに伴うフェルミオンの質量生成があるが、その証明は困難が多い。その理解に格子上の量子色力学は成功していると見られているものの、数学的結果はいまだ限られている。
この講演では格子上のフェルミオンの定式化のひとつであるスタッガード・フェルミオンをもちいて、それらが4つのフェルミオンと相互作用する模型（lattice Nambu–Jona-Lasinio model）を考える。この模型は離散的なカイラル対称性しかもたないものの、質量が自発的に生成することと、それに伴う対称性の破れを証明できる。また、連続的なフレーバー対称性をもつ場合は南部・ゴールドストーン・モードと呼ばれるスペクトルにギャップのない無限系の基底状態が出現することを説明する。
本講演は高麗徹氏との共同研究にもとづく。
量子色力学で重要な概念としてカイラル対称性の破れとそれに伴うフェルミオンの質量生成があるが、その証明は困難が多い。その理解に格子上の量子色力学は成功していると見られているものの、数学的結果はいまだ限られている。
この講演では格子上のフェルミオンの定式化のひとつであるスタッガード・フェルミオンをもちいて、それらが4つのフェルミオンと相互作用する模型（lattice Nambu–Jona-Lasinio model）を考える。この模型は離散的なカイラル対称性しかもたないものの、質量が自発的に生成することと、それに伴う対称性の破れを証明できる。また、連続的なフレーバー対称性をもつ場合は南部・ゴールドストーン・モードと呼ばれるスペクトルにギャップのない無限系の基底状態が出現することを説明する。
本講演は高麗徹氏との共同研究にもとづく。
2024/05/27
16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)
We are having teatime from 15:15 in the common room on the second floor. Please join us.
Ryoichiro Noda (Kyoto University)
測度付き抵抗距離空間上の確率過程の局所時間のスケール極限について (日本語)
We are having teatime from 15:15 in the common room on the second floor. Please join us.
Ryoichiro Noda (Kyoto University)
測度付き抵抗距離空間上の確率過程の局所時間のスケール極限について (日本語)
[ Abstract ]
抵抗距離空間は電気回路の一般化であり，ディリクレ形式の理論により測度付き抵抗距離空間には確率過程が定まる．Croydon-Hambly-Kumagai (2017)は収束する抵抗距離空間が一様体積倍化条件を満たすならば対応する確率過程とその局所時間が収束することを示した．その後Croydon (2018)はより弱い条件である非爆発条件の下で確率過程の収束を示したが，局所時間の収束については未解決のままであった．本講演では非爆発条件及び距離エントロピーに関する適当な条件の下で確率過程とその局所時間の収束が従うこと，そしてこの結果の応用例について解説する．また同様の結果は離散時間マルコフ連鎖とその局所時間に対しても成立し，時間が許せばこの結果についても紹介する．
抵抗距離空間は電気回路の一般化であり，ディリクレ形式の理論により測度付き抵抗距離空間には確率過程が定まる．Croydon-Hambly-Kumagai (2017)は収束する抵抗距離空間が一様体積倍化条件を満たすならば対応する確率過程とその局所時間が収束することを示した．その後Croydon (2018)はより弱い条件である非爆発条件の下で確率過程の収束を示したが，局所時間の収束については未解決のままであった．本講演では非爆発条件及び距離エントロピーに関する適当な条件の下で確率過程とその局所時間の収束が従うこと，そしてこの結果の応用例について解説する．また同様の結果は離散時間マルコフ連鎖とその局所時間に対しても成立し，時間が許せばこの結果についても紹介する．
2024/05/20
16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)
We are having teatime from 15:15 in the common room on the second floor. Please join us.
Soma Nishino (Tokyo Metropolitan University)
2曲線間に制限されたパス空間上でのWiener測度に対する高階の部分積分公式 (日本語)
We are having teatime from 15:15 in the common room on the second floor. Please join us.
Soma Nishino (Tokyo Metropolitan University)
2曲線間に制限されたパス空間上でのWiener測度に対する高階の部分積分公式 (日本語)
[ Abstract ]
2曲線間に制限されたパス空間上でのWiener測度に対する1階微分の部分積分公式は既に知られている。本講演では、この結果を高階微分の部分積分公式に拡張する。高階微分の部分積分公式においては、従来の1階微分の場合にはない非自明な境界項が追加で現れ、さらに、その証明において、Brownian excursionやBrownian house-movingと呼ばれる確率過程のランダムウォーク近似による構成方法が新たに必要となる。また、証明の中で、1次および2次の無限小確率の概念を導入する。この概念を導入することで、部分積分公式の各項に現れる数式に対して確率論的な解釈が可能となり、部分積分公式を整理する上で有益な概念であることを説明する。なお、本講演内容は、東京都立大学の石谷謙介氏との共同研究(arXiv:2405.05595)に基づく。
2曲線間に制限されたパス空間上でのWiener測度に対する1階微分の部分積分公式は既に知られている。本講演では、この結果を高階微分の部分積分公式に拡張する。高階微分の部分積分公式においては、従来の1階微分の場合にはない非自明な境界項が追加で現れ、さらに、その証明において、Brownian excursionやBrownian house-movingと呼ばれる確率過程のランダムウォーク近似による構成方法が新たに必要となる。また、証明の中で、1次および2次の無限小確率の概念を導入する。この概念を導入することで、部分積分公式の各項に現れる数式に対して確率論的な解釈が可能となり、部分積分公式を整理する上で有益な概念であることを説明する。なお、本講演内容は、東京都立大学の石谷謙介氏との共同研究(arXiv:2405.05595)に基づく。
2024/05/13
16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)
We are having teatime from 15:15 in the common room on the second floor. Please join us.
Shuwen Lou (University of Illinois)
Brownian motion with darning and its related open problem (English)
We are having teatime from 15:15 in the common room on the second floor. Please join us.
Shuwen Lou (University of Illinois)
Brownian motion with darning and its related open problem (English)
[ Abstract ]
In this talk, I will discuss some existing results about Brownian motion with darning, including its HKE and discrete approximate by random walks, along with an open problem: What is the relationship between (a) subordinated BM with darning, and (b) the process obtained by darning together two subordinated reflected BM. This is an ongoing collaboration with Zhen-Qing Chen.
In this talk, I will discuss some existing results about Brownian motion with darning, including its HKE and discrete approximate by random walks, along with an open problem: What is the relationship between (a) subordinated BM with darning, and (b) the process obtained by darning together two subordinated reflected BM. This is an ongoing collaboration with Zhen-Qing Chen.
2024/04/15
16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)
We are having teatime from 15:15 in the common room on the second floor. Please join us.
Tomohiro Aya (Kyoto University)
Quantitative stochastic homogenization of elliptic equations with unbounded coefficients (日本語)
We are having teatime from 15:15 in the common room on the second floor. Please join us.
Tomohiro Aya (Kyoto University)
Quantitative stochastic homogenization of elliptic equations with unbounded coefficients (日本語)
2024/02/05
17:00-18:30 Room #126 (Graduate School of Math. Sci. Bldg.)
Sunder Sethuraman (University of Arizona)
Atypical behaviors of a tagged particle in asymmetric simple exclusion (English)
Sunder Sethuraman (University of Arizona)
Atypical behaviors of a tagged particle in asymmetric simple exclusion (English)
[ Abstract ]
Informally, the one dimensional asymmetric simple exclusion process follows a collection of continuous time random walks on Z interacting as follows: When a clock rings, the particle jumps to the nearest right or left with probabilities p or q=1-p, if that location is unoccupied. If occupied, the jump is suppressed and clocks start again.
In this system, seen as a toy model of `traffic', the motion of a distinguished or `tagged' particle is of interest. Starting from a stationary state, we study the `typical' behavior of a tagged particle, conditioned to deviate to an `atypical' position at time Nt, for a t>0 fixed. In the course of results, an `upper tail' large deviation principle, in scale N, is established for the position of the tagged particle. Also, with respect to `lower tail' events, in the totally asymmetric version, a connection is made with a `nonentropy' solution of the associated hydrodynamic Burgers equation. This is work with S.R.S. Varadhan (arXiv:2311.0780).
Informally, the one dimensional asymmetric simple exclusion process follows a collection of continuous time random walks on Z interacting as follows: When a clock rings, the particle jumps to the nearest right or left with probabilities p or q=1-p, if that location is unoccupied. If occupied, the jump is suppressed and clocks start again.
In this system, seen as a toy model of `traffic', the motion of a distinguished or `tagged' particle is of interest. Starting from a stationary state, we study the `typical' behavior of a tagged particle, conditioned to deviate to an `atypical' position at time Nt, for a t>0 fixed. In the course of results, an `upper tail' large deviation principle, in scale N, is established for the position of the tagged particle. Also, with respect to `lower tail' events, in the totally asymmetric version, a connection is made with a `nonentropy' solution of the associated hydrodynamic Burgers equation. This is work with S.R.S. Varadhan (arXiv:2311.0780).
2023/11/27
17:00-18:30 Room #126 (Graduate School of Math. Sci. Bldg.)
Stefan Junk (学習院大学)
Local limit theorem for directed polymer in (almost) the whole weak disorder regime (English)
Stefan Junk (学習院大学)
Local limit theorem for directed polymer in (almost) the whole weak disorder regime (English)
[ Abstract ]
We consider the directed polymer model in the weak disorder (high temperature) phase in spatial dimension d>2. In the case where the (normalized) partition function is L^2-bounded it is known for that time
polymer measure satisfies a local limit theorem, i.e., that the point-to-point partition function can be approximated by two point-to-plane partition functions at the start- and endpoint. We show
that this result continues to hold true if the partition function is L^p-bounded for some p>1+2/d. We furthermore show that for environments with finite support the required L^p -boundedness holds in the whole weak disorder phase, except possibly for the critical value itself.
We consider the directed polymer model in the weak disorder (high temperature) phase in spatial dimension d>2. In the case where the (normalized) partition function is L^2-bounded it is known for that time
polymer measure satisfies a local limit theorem, i.e., that the point-to-point partition function can be approximated by two point-to-plane partition functions at the start- and endpoint. We show
that this result continues to hold true if the partition function is L^p-bounded for some p>1+2/d. We furthermore show that for environments with finite support the required L^p -boundedness holds in the whole weak disorder phase, except possibly for the critical value itself.
2023/11/20
17:00-18:30 Room #126 (Graduate School of Math. Sci. Bldg.)
Jun Kigami (Kyoto University)
Yet another construction of “Sobolev” spaces on metric spaces (日本語)
Jun Kigami (Kyoto University)
Yet another construction of “Sobolev” spaces on metric spaces (日本語)
2023/10/30
16:00-18:50 Room #126 (Graduate School of Math. Sci. Bldg.)
Chenlin Gu (Tsinghua University) 16:00-16:50
Quantitative homogenization of interacting particle systems (English)
https://chenlin-gu.github.io/index.html
Lorenzo Dello-Schiavio (Institute of Science and Technology Austria (ISTA)) 17:00-17:50
Wasserstein geometry and Ricci curvature bounds for Poisson spaces (English)
https://lzdsmath.github.io
Kohei Suzuki (Durham University) 18:00-18:50
Curvature Bound of the Dyson Brownian Motion (English)
https://www.durham.ac.uk/staff/kohei-suzuki/
Chenlin Gu (Tsinghua University) 16:00-16:50
Quantitative homogenization of interacting particle systems (English)
[ Abstract ]
This talk presents that, for a class of interacting particle systems in continuous space, the finite-volume approximations of the bulk diffusion matrix converge at an algebraic rate. The models we consider are reversible with respect to the Poisson measures with constant density, and are of non-gradient type. This approach is inspired by recent progress in the quantitative homogenization of elliptic equations. Along the way, a modified Caccioppoli inequality and a multiscale Poincare inequality are developed, which are of independent interest. The talk is based on a joint work with Arianna Giunti and Jean-Christophe Mourrat.
[ Reference URL ]This talk presents that, for a class of interacting particle systems in continuous space, the finite-volume approximations of the bulk diffusion matrix converge at an algebraic rate. The models we consider are reversible with respect to the Poisson measures with constant density, and are of non-gradient type. This approach is inspired by recent progress in the quantitative homogenization of elliptic equations. Along the way, a modified Caccioppoli inequality and a multiscale Poincare inequality are developed, which are of independent interest. The talk is based on a joint work with Arianna Giunti and Jean-Christophe Mourrat.
https://chenlin-gu.github.io/index.html
Lorenzo Dello-Schiavio (Institute of Science and Technology Austria (ISTA)) 17:00-17:50
Wasserstein geometry and Ricci curvature bounds for Poisson spaces (English)
[ Abstract ]
Let Υ be the configuration space over a complete and separable metric base space, endowed with the Poisson measure π. We study the geometry of Υ from the point of view of optimal transport and Ricci-lower bounds. To do so, we define a formal Riemannian structure on P_1(Y), the space of probability measures over Υ with finite first moment, and we construct an extended distance W on P_1(Y). The distance W corresponds, in our setting, to the Benamou–Brenier variational formulation of the Wasserstein distance. Our main technical tool is a non-local continuity equation defined via the difference operator on the Poisson space. We show that the closure of the domain of the relative entropy is a complete geodesic space, when endowed with W. We establish non-local infinite-dimensional analogues of results regarding the geometry of the Wasserstein space over a metric measure space with synthetic Ricci curvature bounded below. In particular, we obtain that: (a) the Ornstein–Uhlenbeck semi-group is the gradient flow of the relative entropy; (b) the Poisson space has Ricci curvature bounded below by 1 in the entropic sense; (c) the distance W satisfies an HWI inequality.
Base on joint work arXiv:2303.00398 with Ronan Herry (Rennes 1) and Kohei Suzuki (Durham)
[ Reference URL ]Let Υ be the configuration space over a complete and separable metric base space, endowed with the Poisson measure π. We study the geometry of Υ from the point of view of optimal transport and Ricci-lower bounds. To do so, we define a formal Riemannian structure on P_1(Y), the space of probability measures over Υ with finite first moment, and we construct an extended distance W on P_1(Y). The distance W corresponds, in our setting, to the Benamou–Brenier variational formulation of the Wasserstein distance. Our main technical tool is a non-local continuity equation defined via the difference operator on the Poisson space. We show that the closure of the domain of the relative entropy is a complete geodesic space, when endowed with W. We establish non-local infinite-dimensional analogues of results regarding the geometry of the Wasserstein space over a metric measure space with synthetic Ricci curvature bounded below. In particular, we obtain that: (a) the Ornstein–Uhlenbeck semi-group is the gradient flow of the relative entropy; (b) the Poisson space has Ricci curvature bounded below by 1 in the entropic sense; (c) the distance W satisfies an HWI inequality.
Base on joint work arXiv:2303.00398 with Ronan Herry (Rennes 1) and Kohei Suzuki (Durham)
https://lzdsmath.github.io
Kohei Suzuki (Durham University) 18:00-18:50
Curvature Bound of the Dyson Brownian Motion (English)
[ Abstract ]
The Dyson Brownian Motion (DBM) is an eigenvalue process of a particular Hermitian matrix-valued Brownian motion introduced by Freeman Dyson in 1962, which has been one of the central subjects in the random matrix theory. In this talk, we study the DBM from a geometric perspective. We show that the infinite particle DBM possesses a lower bound of the Ricci curvature à la Bakry-Émery. As a consequence, we obtain various quantitative estimates of the transition probability of the DBM (e.g., the local spectral gap, the local log-Sobolev, and the dimension-free Harnack inequalities) as well as the characterisation of the DBM as the gradient flow of the Boltzmann entropy in a particular Wasserstein-type space, the latter of which provides a new viewpoint of the Dyson Brownian motion.
[ Reference URL ]The Dyson Brownian Motion (DBM) is an eigenvalue process of a particular Hermitian matrix-valued Brownian motion introduced by Freeman Dyson in 1962, which has been one of the central subjects in the random matrix theory. In this talk, we study the DBM from a geometric perspective. We show that the infinite particle DBM possesses a lower bound of the Ricci curvature à la Bakry-Émery. As a consequence, we obtain various quantitative estimates of the transition probability of the DBM (e.g., the local spectral gap, the local log-Sobolev, and the dimension-free Harnack inequalities) as well as the characterisation of the DBM as the gradient flow of the Boltzmann entropy in a particular Wasserstein-type space, the latter of which provides a new viewpoint of the Dyson Brownian motion.
https://www.durham.ac.uk/staff/kohei-suzuki/
2023/09/25
17:00-18:30 Room #126 (Graduate School of Math. Sci. Bldg.)
Jimmy He (MIT)
Boundary current fluctuations for the half space ASEP (English)
Jimmy He (MIT)
Boundary current fluctuations for the half space ASEP (English)
[ Abstract ]
The half space asymmetric simple exclusion process (ASEP) is an interacting particle system on the half line, with particles allowed to enter/exit at the boundary. I will discuss recent work on understanding fluctuations for the number of particles in the half space ASEP started with no particles, which exhibits the Baik-Rains phase transition between GSE, GOE, and Gaussian fluctuations as the boundary rates vary. As part of the proof, we find new distributional identities relating this system to two other models, the half space Hall-Littlewood process, and the free boundary Schur process, which allows exact formulas to be computed.
The half space asymmetric simple exclusion process (ASEP) is an interacting particle system on the half line, with particles allowed to enter/exit at the boundary. I will discuss recent work on understanding fluctuations for the number of particles in the half space ASEP started with no particles, which exhibits the Baik-Rains phase transition between GSE, GOE, and Gaussian fluctuations as the boundary rates vary. As part of the proof, we find new distributional identities relating this system to two other models, the half space Hall-Littlewood process, and the free boundary Schur process, which allows exact formulas to be computed.
2023/08/07
17:00-18:30 Room #123 (Graduate School of Math. Sci. Bldg.)
Freddy Delbaen (Professor emeritus at ETH Zurich)
Approximation of Random Variables by Elements that are independent of a given sigma algebra (English)
Freddy Delbaen (Professor emeritus at ETH Zurich)
Approximation of Random Variables by Elements that are independent of a given sigma algebra (English)
[ Abstract ]
Given a square integrable m-dimensional random variable $X$ on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and a sub sigma algebra $\mathcal{A}$, we show that there exists another m-dimensional random variable $Y$, independent of $\mathcal{A}$ and minimising the $L^2$ distance to $X$. Such results have an importance to fairness and bias reduction in Artificial Intelligence, Machine Learning and Network Theory. The proof needs elements from transportation theory, a parametric version due to Dudley and Blackwell of the Skorohod theorem, selection theorems, … The problem also triggers other approximation problems. (joint work with C. Majumdar)
Given a square integrable m-dimensional random variable $X$ on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and a sub sigma algebra $\mathcal{A}$, we show that there exists another m-dimensional random variable $Y$, independent of $\mathcal{A}$ and minimising the $L^2$ distance to $X$. Such results have an importance to fairness and bias reduction in Artificial Intelligence, Machine Learning and Network Theory. The proof needs elements from transportation theory, a parametric version due to Dudley and Blackwell of the Skorohod theorem, selection theorems, … The problem also triggers other approximation problems. (joint work with C. Majumdar)
2023/07/10
17:00-18:30 Room #126 (Graduate School of Math. Sci. Bldg.)
松井 千尋 (東京大学大学院数理科学研究科)
孤立量子系の熱化と緩和 (日本語)
松井 千尋 (東京大学大学院数理科学研究科)
孤立量子系の熱化と緩和 (日本語)
2023/06/26
17:00-18:30 Room #126 (Graduate School of Math. Sci. Bldg.)
簗島 瞬 (東京都立大学)
δ次元Bessel引越過程の構成方法，サンプルパス生成方法，および汎関数期待値の数値計算法について (日本語)
簗島 瞬 (東京都立大学)
δ次元Bessel引越過程の構成方法，サンプルパス生成方法，および汎関数期待値の数値計算法について (日本語)
2023/06/05
17:00-18:30 Room #126 (Graduate School of Math. Sci. Bldg.)
福山克司 (神戸大学)
大きな公比を持つ等比数列の差異量の重複対数の法則について (日本語)
福山克司 (神戸大学)
大きな公比を持つ等比数列の差異量の重複対数の法則について (日本語)
2023/05/15
17:00-18:30 Room #126 (Graduate School of Math. Sci. Bldg.)
岡田いず海 (千葉大学)
Capacity of the range of random walk (JAPANESE)
岡田いず海 (千葉大学)
Capacity of the range of random walk (JAPANESE)
[ Abstract ]
We study the capacity of the range of a simple random walk in three and higher dimensions. It is known that the order of the capacity of the random walk range in n dimensions is similar to that of the volume of the random walk range in n-2 dimensions. We show that this correspondence breaks down for the law of the iterated logarithm for the capacity of the random walk range in three dimensions. We also prove the law of the iterated logarithm in higher dimensions. This is joint work with Amir Dembo.
We study the capacity of the range of a simple random walk in three and higher dimensions. It is known that the order of the capacity of the random walk range in n dimensions is similar to that of the volume of the random walk range in n-2 dimensions. We show that this correspondence breaks down for the law of the iterated logarithm for the capacity of the random walk range in three dimensions. We also prove the law of the iterated logarithm in higher dimensions. This is joint work with Amir Dembo.
2023/05/08
17:00-18:30 Room #126 (Graduate School of Math. Sci. Bldg.)
新井裕太 (千葉商科大学)
On the Chapman-Kolmogorov equation for LPP (JAPANESE)
新井裕太 (千葉商科大学)
On the Chapman-Kolmogorov equation for LPP (JAPANESE)
[ Abstract ]
KPZ普遍クラスに属するいくつかのモデルにおいて，その推移確率等が複素積分形の関数で書き表せることが知られている．しかしながら，複素積分を用いた計算は複雑となることも多く，KPZ普遍クラスに属するモデルにとって重要な確率論的性質を証明するのが困難となっていた．近年，この問題を解決するものとして対称多項式等を用いた組合せ論的手法に注目が集まってきている．本講演では，最先端の組合せ論的アプローチを用いることで，KPZ普遍クラスの基礎的なモデルであるLast Passage Percolation（LPP）において， Chapman-Kolmogorov equationが容易に得られることを紹介する.
KPZ普遍クラスに属するいくつかのモデルにおいて，その推移確率等が複素積分形の関数で書き表せることが知られている．しかしながら，複素積分を用いた計算は複雑となることも多く，KPZ普遍クラスに属するモデルにとって重要な確率論的性質を証明するのが困難となっていた．近年，この問題を解決するものとして対称多項式等を用いた組合せ論的手法に注目が集まってきている．本講演では，最先端の組合せ論的アプローチを用いることで，KPZ普遍クラスの基礎的なモデルであるLast Passage Percolation（LPP）において， Chapman-Kolmogorov equationが容易に得られることを紹介する.
2023/04/24
17:00-18:30 Room #126 (Graduate School of Math. Sci. Bldg.)
Charles Bordenave (Institut de Mathématiques de Marseille)
Mobility edge, the Poisson Infinite weighted tree of Aldous and Lévy Matrices (English)
Charles Bordenave (Institut de Mathématiques de Marseille)
Mobility edge, the Poisson Infinite weighted tree of Aldous and Lévy Matrices (English)
[ Abstract ]
Anderson's 1958 paper on wave scattering in disordered media is still of central importance in contemporary mathematical physics. In this talk, we will present recent progress in understanding the phenomena of localization / delocalization of eigenwaves for some random operators. These operators are built on random trees introduced by Aldous and these are the scaling limits of heavy-tailed random matrices, the Lévy matrices. The focus will be put on the existence of a mobility edge, that is to say of かn abrupt transition between localization and delocalization of eigenwaves. It is a work in collaboration with Amol Aggarwal (Columbia) and Patrick Lopatto (NYU).
Anderson's 1958 paper on wave scattering in disordered media is still of central importance in contemporary mathematical physics. In this talk, we will present recent progress in understanding the phenomena of localization / delocalization of eigenwaves for some random operators. These operators are built on random trees introduced by Aldous and these are the scaling limits of heavy-tailed random matrices, the Lévy matrices. The focus will be put on the existence of a mobility edge, that is to say of かn abrupt transition between localization and delocalization of eigenwaves. It is a work in collaboration with Amol Aggarwal (Columbia) and Patrick Lopatto (NYU).
2023/04/17
17:00-18:30 Room #126 (Graduate School of Math. Sci. Bldg.)
清水良輔 (早稲田大学)
Construction of Sobolev spaces and energies on the Sierpinski carpet (Japanese)
清水良輔 (早稲田大学)
Construction of Sobolev spaces and energies on the Sierpinski carpet (Japanese)
2019/01/28
16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)
Yosuke Kawamoto (FUKUOKA DENTAL COLLEGE)
(JAPANESE)
Yosuke Kawamoto (FUKUOKA DENTAL COLLEGE)
(JAPANESE)
2018/12/10
17:00-18:00 Room # (Graduate School of Math. Sci. Bldg.)
Nikolaos Zygouras (University of Warwick)
Random polymer models and classical groups (ENGLISH)
https://warwick.ac.uk/fac/sci/statistics/staff/academic-research/zygouras/
Nikolaos Zygouras (University of Warwick)
Random polymer models and classical groups (ENGLISH)
[ Abstract ]
The relation between polymer models at zero temperature and characters of the general linear group GL_n(R) has been known since the first breakthroughs in the field around the KPZ universality through the works of Johansson, Baik, Rains, Okounkov and others. Later on, geometric liftings of the GL_n(R) characters appeared in the study of positive temperature polymer models in the form of GL_n(R)-Whittaker functions. In this talk I will describe joint works with E. Bisi where we have established that Whittaker functions associated to the orthogonal group SO_{2n+1}(R) can be used to describe laws of positive temperature polymers when their end point is free to lie on a line. Going back to zero temperature, we will also see that characters of other classical groups such as SO_{2n+1}(R); Sp_{2n}(R); SO_{2n}(R) do play a role in describing laws of polymers in various geometries. This occurence might be surprising given the length of time these models have been studied.
[ Reference URL ]The relation between polymer models at zero temperature and characters of the general linear group GL_n(R) has been known since the first breakthroughs in the field around the KPZ universality through the works of Johansson, Baik, Rains, Okounkov and others. Later on, geometric liftings of the GL_n(R) characters appeared in the study of positive temperature polymer models in the form of GL_n(R)-Whittaker functions. In this talk I will describe joint works with E. Bisi where we have established that Whittaker functions associated to the orthogonal group SO_{2n+1}(R) can be used to describe laws of positive temperature polymers when their end point is free to lie on a line. Going back to zero temperature, we will also see that characters of other classical groups such as SO_{2n+1}(R); Sp_{2n}(R); SO_{2n}(R) do play a role in describing laws of polymers in various geometries. This occurence might be surprising given the length of time these models have been studied.
https://warwick.ac.uk/fac/sci/statistics/staff/academic-research/zygouras/
2018/11/19
16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)
Fabio Toninelli (University Lyon 1)
Two-dimensional stochastic interface growth (ENGLISH)
http://math.univ-lyon1.fr/~toninelli/
Fabio Toninelli (University Lyon 1)
Two-dimensional stochastic interface growth (ENGLISH)
[ Abstract ]
I will discuss stochastic growth of two-dimensional, discrete interfaces, especially models in the so-called Anisotropic KPZ (AKPZ) class, that has the same large-scale behavior as the Stochastic Heat equation with additive noise. I will focus in particular on: 1) the relation between AKPZ exponents, convexity properties of the speed of growth and the preservation of the Gibbs property; and 2) the relation between singularities of the speed of growth and the occurrence of "smooth" (i.e. non-rough) stationary states.
[ Reference URL ]I will discuss stochastic growth of two-dimensional, discrete interfaces, especially models in the so-called Anisotropic KPZ (AKPZ) class, that has the same large-scale behavior as the Stochastic Heat equation with additive noise. I will focus in particular on: 1) the relation between AKPZ exponents, convexity properties of the speed of growth and the preservation of the Gibbs property; and 2) the relation between singularities of the speed of growth and the occurrence of "smooth" (i.e. non-rough) stationary states.
http://math.univ-lyon1.fr/~toninelli/
2018/11/12
16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)
Alejandro Ramirez (Pontificia Universidad Catolica de Chile)
Random walk at weak and strong disorder (ENGLISH)
http://www.mat.uc.cl/~aramirez/
Alejandro Ramirez (Pontificia Universidad Catolica de Chile)
Random walk at weak and strong disorder (ENGLISH)
[ Abstract ]
We consider random walks at low disorder on $\mathbb Z^d$. For dimensions $d\ge 4$, we exhibit a phase transition on the strength of the disorder expressed as an equality between the quenched and annealed rate functions. In dimension $d=2$ we exhibit a universal scaling limit to the stochastic heat equation. This talk is based on joint works with Bazaes, Mukherjee and Saglietti, and with Moreno and Quastel.
[ Reference URL ]We consider random walks at low disorder on $\mathbb Z^d$. For dimensions $d\ge 4$, we exhibit a phase transition on the strength of the disorder expressed as an equality between the quenched and annealed rate functions. In dimension $d=2$ we exhibit a universal scaling limit to the stochastic heat equation. This talk is based on joint works with Bazaes, Mukherjee and Saglietti, and with Moreno and Quastel.
http://www.mat.uc.cl/~aramirez/
