## Tokyo Probability Seminar

Seminar information archive ～05/28｜Next seminar｜Future seminars 05/29～

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**

### 2024/05/27

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.

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.

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.

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.

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.)

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.)

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.)

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.)

Quantitative homogenization of interacting particle systems (English)

https://chenlin-gu.github.io/index.html

Wasserstein geometry and Ricci curvature bounds for Poisson spaces (English)

https://lzdsmath.github.io

Curvature Bound of the Dyson Brownian Motion (English)

https://www.durham.ac.uk/staff/kohei-suzuki/

**Chenlin Gu**(Tsinghua University) 16:00-16:50Quantitative 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)) 17:00-17:50Wasserstein 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) 18:00-18:50Curvature 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.)

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.)

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.)

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.)

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.)

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.)

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.)

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.)

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.)

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.)

(JAPANESE)

**Yosuke Kawamoto**(FUKUOKA DENTAL COLLEGE)(JAPANESE)

### 2018/12/10

17:00-18:00 Room # (Graduate School of Math. Sci. Bldg.)

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.)

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.)

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/

### 2018/10/29

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

On Hydrodynamic Limits of Young Diagrams (ENGLISH)

http://math.arizona.edu/~sethuram/

**Sunder Sethuraman**(University of Arizona)On Hydrodynamic Limits of Young Diagrams (ENGLISH)

[ Abstract ]

We consider a family of stochastic models of evolving two-dimensional Young diagrams, given in terms of certain energies, with Gibbs invariant measures. `Static' scaling limits of the shape functions, under these Gibbs measures, have been shown by several over the years. The purpose of this article is to study corresponding `dynamical' limits of which less is understood. We show that the hydrodynamic scaling limits of the diagram shape functions may be described by different types of parabolic PDEs, depending on the energy structure.

The talk will be based on the article: https://arxiv.org/abs/1809.03592

[ Reference URL ]We consider a family of stochastic models of evolving two-dimensional Young diagrams, given in terms of certain energies, with Gibbs invariant measures. `Static' scaling limits of the shape functions, under these Gibbs measures, have been shown by several over the years. The purpose of this article is to study corresponding `dynamical' limits of which less is understood. We show that the hydrodynamic scaling limits of the diagram shape functions may be described by different types of parabolic PDEs, depending on the energy structure.

The talk will be based on the article: https://arxiv.org/abs/1809.03592

http://math.arizona.edu/~sethuram/

### 2018/10/22

16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Limit theorems for random geometric complexes in the critical regime (ENGLISH)

**Trinh Khanh Duy**(Tohoku University)Limit theorems for random geometric complexes in the critical regime (ENGLISH)

[ Abstract ]

Geometric complexes (eg. Cech complexes or Rips complexes) are simplicial complexes defined on a finite set of points in a Euclidean space together with a radius parameter, which can be viewed as a higher dimensional generalization of geometric graphs. This talk concerns with random geometric complexes built over binomial point processes (collections of iid points). Like random geometric graphs, there are three regimes (subcritical(or dust, sparse) regime, critical (or thermodynamic) regime and supercritical regime) which are divided according the growth of the radius parameters in which the limiting behavior of random geometric complexes is totally different. This talk introduces some results on the strong law of large numbers and a central limit theorem in the critical regime.

Geometric complexes (eg. Cech complexes or Rips complexes) are simplicial complexes defined on a finite set of points in a Euclidean space together with a radius parameter, which can be viewed as a higher dimensional generalization of geometric graphs. This talk concerns with random geometric complexes built over binomial point processes (collections of iid points). Like random geometric graphs, there are three regimes (subcritical(or dust, sparse) regime, critical (or thermodynamic) regime and supercritical regime) which are divided according the growth of the radius parameters in which the limiting behavior of random geometric complexes is totally different. This talk introduces some results on the strong law of large numbers and a central limit theorem in the critical regime.

### 2018/07/30

16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Parameter estimation of random matrix models via free probability theory (JAPANESE)

https://www.ms.u-tokyo.ac.jp/~hayase/

**Tomohiro Hayase**(Graduate School of Mathematical Sciences, The University of Tokyo)Parameter estimation of random matrix models via free probability theory (JAPANESE)

[ Abstract ]

For random matrix models, the parameter estimation based on the likelihood is not straightforward in particular when there is only one sample matrix. We introduce a new parameter optimization method of random matrix models which works even in such a case not based on the likelihood, instead based on the spectral distribution. We use the spectral distribution perturbed by Cauchy noises because the free deterministic equivalent, which is a tool in free probability theory, allows us to approximate it by a smooth and accessible density function.

In addition, we propose a new rank recovery method for the signal-plus-noise model, and experimentally demonstrate that it recovers the true rank even if the rank is not low; It is a simultaneous rank recovery and parameter estimation procedure.

[ Reference URL ]For random matrix models, the parameter estimation based on the likelihood is not straightforward in particular when there is only one sample matrix. We introduce a new parameter optimization method of random matrix models which works even in such a case not based on the likelihood, instead based on the spectral distribution. We use the spectral distribution perturbed by Cauchy noises because the free deterministic equivalent, which is a tool in free probability theory, allows us to approximate it by a smooth and accessible density function.

In addition, we propose a new rank recovery method for the signal-plus-noise model, and experimentally demonstrate that it recovers the true rank even if the rank is not low; It is a simultaneous rank recovery and parameter estimation procedure.

https://www.ms.u-tokyo.ac.jp/~hayase/

### 2018/07/02

16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Distributional limit theorems for intermittent maps (JAPANESE)

**Toru SERA**(Graduate School of Science, Kyoto University)Distributional limit theorems for intermittent maps (JAPANESE)