## Tokyo Probability Seminar

Seminar information archive ～05/29｜Next seminar｜Future seminars 05/30～

Date, time & place | Monday 17:00 - 18:30 126Room #126 (Graduate School of Math. Sci. Bldg.) |
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Organizer(s) | Hiroshi Kawabi, Shuta Nakajima, Makiko Sasada |

**Seminar information archive**

### 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)

### 2018/06/25

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

BSDEs driven by cylindrical martingales with application to approximate hedging in bond markets (JAPANESE)

**Yushi HAMAGUCHI**(Graduate School of Science, Kyoto University)BSDEs driven by cylindrical martingales with application to approximate hedging in bond markets (JAPANESE)

### 2018/06/18

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

(JAPANESE)

**Hideki TANEMURA**(Department of Mathematics, Keio University)(JAPANESE)

### 2018/06/04

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

Uniqueness set for degenerate Hamilton-Jacobi equations (JAPANESE)

**Hiroyoshi MITAKE**(Graduate School of Mathematical Sciences, The University of Tokyo)Uniqueness set for degenerate Hamilton-Jacobi equations (JAPANESE)

### 2018/05/14

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

Chordal Komatu-Loewner equation for a family of continuously growing hulls (JAPANESE)

**Takuya MURAYAMA**(Graduate School of Science, Kyoto University)Chordal Komatu-Loewner equation for a family of continuously growing hulls (JAPANESE)

### 2018/05/07

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

(JAPANESE)

[ Reference URL ]

http://www.taksagawa.com

**Takahiro SAGAWA**(Faculty of Engineering, The University of Tokyo)(JAPANESE)

[ Reference URL ]

http://www.taksagawa.com

### 2018/04/23

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

Functional central limit theorems for non-symmetric random walks on nilpotent covering graphs (JAPANESE)

**Hiroshi KAWABI**(Faculty of Economics, Keio University)Functional central limit theorems for non-symmetric random walks on nilpotent covering graphs (JAPANESE)

### 2018/01/29

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

(JAPANESE)

**Kazuhiro Kuwae**(Department of Applied Mathematics, Faculty of Science, Fukuoka University)(JAPANESE)

### 2018/01/22

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

(JAPANESE)

**Makiko Sasada**(Graduate School of Mathematical Science, the University of Tokyo)(JAPANESE)

### 2017/12/04

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

Some results for range of random walk on graph with spectral dimension two (JAPANESE)

**Kazuki Okamura**(Research Institute for Mathematical Sciences, Kyoto University)Some results for range of random walk on graph with spectral dimension two (JAPANESE)

[ Abstract ]

We consider the range of random walk on graphs with spectral dimension two. We show that a certain weak law of large numbers hold if a recurrent graph satisfies a uniform condition. We construct a recurrent graph such that the uniform condition holds but appropriately scaled expectations fluctuate. Our result is applicable to showing LILs for lamplighter random walks in the case that the spectral dimension of the underlying graph is two.

We consider the range of random walk on graphs with spectral dimension two. We show that a certain weak law of large numbers hold if a recurrent graph satisfies a uniform condition. We construct a recurrent graph such that the uniform condition holds but appropriately scaled expectations fluctuate. Our result is applicable to showing LILs for lamplighter random walks in the case that the spectral dimension of the underlying graph is two.

### 2017/11/27

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

Random Recursive Tree, Branching Markov Chains and Urn Models (ENGLISH)

**Antar Bandyopadhyay**(Indian Statistical Institute)Random Recursive Tree, Branching Markov Chains and Urn Models (ENGLISH)

[ Abstract ]

In this talk, we will establish a connection between random recursive tree, branching Markov chain and urn model. Exploring the connection further we will derive fairly general scaling limits for urn models with colors indexed by a Polish Space and show that several exiting results on classical/non-classical urn schemes can be easily derived out of such general asymptotic. We will further show that the connection can be used to derive exact asymptotic for the sizes of the connected components of a "random recursive forest", obtained by removing the root of a random recursive tree.

[This is a joint work with Debleena Thacker]

In this talk, we will establish a connection between random recursive tree, branching Markov chain and urn model. Exploring the connection further we will derive fairly general scaling limits for urn models with colors indexed by a Polish Space and show that several exiting results on classical/non-classical urn schemes can be easily derived out of such general asymptotic. We will further show that the connection can be used to derive exact asymptotic for the sizes of the connected components of a "random recursive forest", obtained by removing the root of a random recursive tree.

[This is a joint work with Debleena Thacker]

### 2017/11/13

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

(JAPANESE)

**Masaki Wada**(Faculty of Human Development and Culture, Fukushima University)(JAPANESE)

### 2017/10/30

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

Integration of controlled rough paths via fractional calculus (JAPANESE)

**Yu Ito**(Department of Mathematics, Faculty of Science, Kyoto Sangyo University)Integration of controlled rough paths via fractional calculus (JAPANESE)

### 2017/10/23

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

(JAPANESE)

**Yoshihiro Abe**(Department of Mathematics, Gakushuin University)(JAPANESE)