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Tokyo Probability Seminar
Seminar information archive ～04/30｜Next seminar｜Future seminars 05/01～
| Date, time & place | Monday 16:00 - 17:30 126Room #126 (Graduate School of Math. Sci. Bldg.) |
|---|---|
| Organizer(s) | Makiko Sasada, Shuta Nakajima, Masato Hoshino |
Seminar information archive
2025/04/28
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.
Mitsuo Higaki (Kobe University)
ランダム粗面領域における粘性流体に対するナヴィエ壁法則
We are having teatime from 15:15 in the common room on the second floor. Please join us.
Mitsuo Higaki (Kobe University)
ランダム粗面領域における粘性流体に対するナヴィエ壁法則
[ Abstract ]
粗面を伴う領域における粘性流体運動の有効近似を得る経験的な手法として、工学分野では壁法則が知られてきた [cf. Nikuradse 1933]。筒状粗面領域における定常層流に対しては、壁法則により、ナヴィエ滑り境界条件に従う速度場が得られる (ナヴィエ壁法則)。本講演では、これが実際に有効近似を与えることを数学的に厳密に証明する。より正確には、粗面領域全体の標本空間を考えた際に、ある種のエルゴード性の仮定の下で、最適な近似率が得られることを報告する。証明の鍵は、粗面付近の流体運動を記述する境界層の確定的/確率的評価である。ここで我々は楕円型方程式に対する定量的確率均質化のアイディアを用いる [cf. Armstrong-Smart, Armstrong-Kuusi-Mourrat, Gloria-Neukamm-Otto, Shen]。ただし、係数行列ではなく粗面領域の標本空間を考えていることに注意されたい。なお、上述のエルゴード性としては、確率変数に対する関数不等式 (対数ソボレフ不等式やスペクトルギャップ不等式など) の成立を採用する。本講演の内容は Jinping Zhuge 氏 (Morningside Center of Mathematics, China)、Yulong Lu 氏 (University of Minnesota, USA) との共同研究に基づく。
粗面を伴う領域における粘性流体運動の有効近似を得る経験的な手法として、工学分野では壁法則が知られてきた [cf. Nikuradse 1933]。筒状粗面領域における定常層流に対しては、壁法則により、ナヴィエ滑り境界条件に従う速度場が得られる (ナヴィエ壁法則)。本講演では、これが実際に有効近似を与えることを数学的に厳密に証明する。より正確には、粗面領域全体の標本空間を考えた際に、ある種のエルゴード性の仮定の下で、最適な近似率が得られることを報告する。証明の鍵は、粗面付近の流体運動を記述する境界層の確定的/確率的評価である。ここで我々は楕円型方程式に対する定量的確率均質化のアイディアを用いる [cf. Armstrong-Smart, Armstrong-Kuusi-Mourrat, Gloria-Neukamm-Otto, Shen]。ただし、係数行列ではなく粗面領域の標本空間を考えていることに注意されたい。なお、上述のエルゴード性としては、確率変数に対する関数不等式 (対数ソボレフ不等式やスペクトルギャップ不等式など) の成立を採用する。本講演の内容は Jinping Zhuge 氏 (Morningside Center of Mathematics, China)、Yulong Lu 氏 (University of Minnesota, USA) との共同研究に基づく。
2025/04/21
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.
Masahisa Ebina (Kyoto Univercity)
Malliavin-Stein approach to local limit theorems
We are having teatime from 15:15 in the common room on the second floor. Please join us.
Masahisa Ebina (Kyoto Univercity)
Malliavin-Stein approach to local limit theorems
[ Abstract ]
Malliavin-Stein's method is a fruitful combination of the Malliavin calculus and Stein's method. It provides a powerful probabilistic technique for establishing the quantitative central limit theorems, particularly for functionals of Gaussian processes.
In this talk, we will see how the theory of generalized functionals in the Malliavin calculus can be combined with Malliavin-Stein's method to obtain quantitative local central limit theorems. If time allows, we will also discuss some applications to Wiener chaos. Part of this talk is based on the ongoing joint research with Ivan Nourdin and Giovanni Peccati.
Malliavin-Stein's method is a fruitful combination of the Malliavin calculus and Stein's method. It provides a powerful probabilistic technique for establishing the quantitative central limit theorems, particularly for functionals of Gaussian processes.
In this talk, we will see how the theory of generalized functionals in the Malliavin calculus can be combined with Malliavin-Stein's method to obtain quantitative local central limit theorems. If time allows, we will also discuss some applications to Wiener chaos. Part of this talk is based on the ongoing joint research with Ivan Nourdin and Giovanni Peccati.
2025/04/14
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.
Masato Hoshino (Science Tokyo)
On the proofs of BPHZ theorem and future progress
We are having teatime from 15:15 in the common room on the second floor. Please join us.
Masato Hoshino (Science Tokyo)
On the proofs of BPHZ theorem and future progress
[ Abstract ]
Hairer’s theory of regularity structures (2014) provides a robust framework to guarantee the renormalizability of stochastic partial differential equations (SPDEs). This theory is established in several steps, among which the final and most technically involved step is the proof of the so-called "BPHZ theorem." There are two main approaches to this proof: a graph-theoretic approach developed by Chandra and Hairer (2016+), and a Malliavin calculus-based inductive approach introduced by Linares, Otto, Tempelmayr, and Tsatsoulis (2024). As for Gaussian noises, the latter is simpler and more inductive. While the language used by Otto and his coauthors is different from that of regularity structures, similar arguments have been formulated in the language of regularity structures by Hairer and Steele (2024) and Bailleul and Hoshino (2023+) by different approaches. In this talk, I will first give an overview of the theory of regularity structures, then compare the outlines of the proofs of BPHZ theorem. If time permits, I will also discuss some current researches and future problems.
Hairer’s theory of regularity structures (2014) provides a robust framework to guarantee the renormalizability of stochastic partial differential equations (SPDEs). This theory is established in several steps, among which the final and most technically involved step is the proof of the so-called "BPHZ theorem." There are two main approaches to this proof: a graph-theoretic approach developed by Chandra and Hairer (2016+), and a Malliavin calculus-based inductive approach introduced by Linares, Otto, Tempelmayr, and Tsatsoulis (2024). As for Gaussian noises, the latter is simpler and more inductive. While the language used by Otto and his coauthors is different from that of regularity structures, similar arguments have been formulated in the language of regularity structures by Hairer and Steele (2024) and Bailleul and Hoshino (2023+) by different approaches. In this talk, I will first give an overview of the theory of regularity structures, then compare the outlines of the proofs of BPHZ theorem. If time permits, I will also discuss some current researches and future problems.
2025/02/06
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.
Jacek Wesolowski (Warsaw University of Technology)
Asymptotics of random Motzkin paths
We are having teatime from 15:15 in the common room on the second floor. Please join us.
Jacek Wesolowski (Warsaw University of Technology)
Asymptotics of random Motzkin paths
[ Abstract ]
We study Motzkin paths of length L with general weights on the edges and end points. We investigate the limit behavior of the initial and final segments of the random Motzkin path viewed as a pair of processes starting from each of the two end points as L becomes large. Macroscopic limits of the resulting processes (under two different asymptotic regimes) appear to be non-Brownian parts of stationary measures for the KPZ equation and hypothetical KPZ fixed point on the half-line. The talk is based on a joint paper with W. Bryc (Univ. of Cincinnati) and A. Kuztetsov (York Univ., Toronto) - to appear in IMRN, available also on arXiv: https://arxiv.org/abs/2402.00265
We study Motzkin paths of length L with general weights on the edges and end points. We investigate the limit behavior of the initial and final segments of the random Motzkin path viewed as a pair of processes starting from each of the two end points as L becomes large. Macroscopic limits of the resulting processes (under two different asymptotic regimes) appear to be non-Brownian parts of stationary measures for the KPZ equation and hypothetical KPZ fixed point on the half-line. The talk is based on a joint paper with W. Bryc (Univ. of Cincinnati) and A. Kuztetsov (York Univ., Toronto) - to appear in IMRN, available also on arXiv: https://arxiv.org/abs/2402.00265
2025/01/20
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.
Arka Adhikari (University of Maryland)
Spectral measure for uniform d-regular digraphs
We are having teatime from 15:15 in the common room on the second floor. Please join us.
Arka Adhikari (University of Maryland)
Spectral measure for uniform d-regular digraphs
[ Abstract ]
Consider the matrix $\sfA_\GG$ chosen uniformly at random from the finite
set of all $N$-dimensional matrices of zero main-diagonal and binary entries,
having each row and column of $\sfA_\GG$ sum to $d$.
That is, the adjacency matrix for the uniformly random
$d$-regular simple digraph $\GG$. Fixing $d \ge 3$, it has long been conjectured
that as $N \to \infty$ the corresponding empirical eigenvalue distributions converge
weakly, in probability, to an explicit non-random limit,
given by the Brown measure of the free sum of $d$ Haar unitary operators.
We reduce this conjecture to bounding the decay in $N$ of the probability that
the minimal singular value of the shifted matrix $\sfA(w) = \sfA_\GG - w \sfI$
is very small. While the latter remains a challenging task, the required bound is
comparable to the recently established control on the singularity of $\sfA_\GG$.
The reduction is achieved here by sharp estimates
on the behavior at large $N$, near the real line, of the Green's function (aka resolvent)
of the Hermitization of $\sfA(w)$, which is of independent interest.
Joint w/ A. Dembo
Consider the matrix $\sfA_\GG$ chosen uniformly at random from the finite
set of all $N$-dimensional matrices of zero main-diagonal and binary entries,
having each row and column of $\sfA_\GG$ sum to $d$.
That is, the adjacency matrix for the uniformly random
$d$-regular simple digraph $\GG$. Fixing $d \ge 3$, it has long been conjectured
that as $N \to \infty$ the corresponding empirical eigenvalue distributions converge
weakly, in probability, to an explicit non-random limit,
given by the Brown measure of the free sum of $d$ Haar unitary operators.
We reduce this conjecture to bounding the decay in $N$ of the probability that
the minimal singular value of the shifted matrix $\sfA(w) = \sfA_\GG - w \sfI$
is very small. While the latter remains a challenging task, the required bound is
comparable to the recently established control on the singularity of $\sfA_\GG$.
The reduction is achieved here by sharp estimates
on the behavior at large $N$, near the real line, of the Green's function (aka resolvent)
of the Hermitization of $\sfA(w)$, which is of independent interest.
Joint w/ A. Dembo
2024/12/16
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.
Shu Kanazawa (Kyoto University)
Central limit theorem for linear eigenvalue statistics of the adjacency matrices of random simplicial complexes
We are having teatime from 15:15 in the common room on the second floor. Please join us.
Shu Kanazawa (Kyoto University)
Central limit theorem for linear eigenvalue statistics of the adjacency matrices of random simplicial complexes
[ Abstract ]
We consider the (higher-dimensional) adjacency matrix of the Linial-Meshulam complex model, which is a higher-dimensional generalization of the Erdős-Rényi random graph model. Recently, Knowles and Rosenthal proved that the empirical spectral distribution
of the adjacency matrix is asymptotically given by Wigner's semicircle law in a diluted regime. In this talk, I will present a central limit theorem for the linear eigenvalue statistics for test functions of polynomial growth that is of class C2 on a closed
interval. The proof is based on higher-dimensional combinatorial enumerations and concentration properties of random symmetric matrices. Furthermore, when the test function is a polynomial function, we obtain the explicit formula for the variance of the limiting
Gaussian distribution. This is joint work with Khanh Duy Trinh (Waseda University).
We consider the (higher-dimensional) adjacency matrix of the Linial-Meshulam complex model, which is a higher-dimensional generalization of the Erdős-Rényi random graph model. Recently, Knowles and Rosenthal proved that the empirical spectral distribution
of the adjacency matrix is asymptotically given by Wigner's semicircle law in a diluted regime. In this talk, I will present a central limit theorem for the linear eigenvalue statistics for test functions of polynomial growth that is of class C2 on a closed
interval. The proof is based on higher-dimensional combinatorial enumerations and concentration properties of random symmetric matrices. Furthermore, when the test function is a polynomial function, we obtain the explicit formula for the variance of the limiting
Gaussian distribution. This is joint work with Khanh Duy Trinh (Waseda University).
2024/12/09
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.
Hayate Suda (Institute of Science Tokyo)
Scaling limits of a tagged soliton in the randomized box-ball system
We are having teatime from 15:15 in the common room on the second floor. Please join us.
Hayate Suda (Institute of Science Tokyo)
Scaling limits of a tagged soliton in the randomized box-ball system
[ Abstract ]
The box-ball system (BBS) is a cellular automaton that exhibits the solitonic behavior. In recent years, with the rapid progress in the study of the hydrodynamics of integrable systems, there has been a growing interest in BBS with random initial distribution. In this talk, we consider the scaling limits for a tagged soliton in the BBS starting from certain stationary distribution. This talk is based on a joint work with Stefano Olla and Makiko Sasada.
The box-ball system (BBS) is a cellular automaton that exhibits the solitonic behavior. In recent years, with the rapid progress in the study of the hydrodynamics of integrable systems, there has been a growing interest in BBS with random initial distribution. In this talk, we consider the scaling limits for a tagged soliton in the BBS starting from certain stationary distribution. This talk is based on a joint work with Stefano Olla and Makiko Sasada.
2024/12/02
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.
Shouhei Honda (The University of Tokyo)
Weyl’s law with Ricci curvature bounded below
We are having teatime from 15:15 in the common room on the second floor. Please join us.
Shouhei Honda (The University of Tokyo)
Weyl’s law with Ricci curvature bounded below
[ Abstract ]
Weyl’s law on a closed manifold gives an asymptotic behavior of eigenvalues of the Laplace operator in terms of the size of the manifold. It was conjectured by Luigi Ambrosio (Scuola Normale Superiore), David Tewodrose (Vrije Universiteit Brussel) and myself such that Weyl’s law is valid for Gromov-Hausdorff limit spaces with a restriction of Ricci curvature. A joint work with Xianzhe Dai (UC Santa Barbara), Jiayin Pan (UC Santa Cruz) and Guofang Wei
(UC Santa Barbara) disproved the conjecture. We will discuss about these topics in this talk.
Weyl’s law on a closed manifold gives an asymptotic behavior of eigenvalues of the Laplace operator in terms of the size of the manifold. It was conjectured by Luigi Ambrosio (Scuola Normale Superiore), David Tewodrose (Vrije Universiteit Brussel) and myself such that Weyl’s law is valid for Gromov-Hausdorff limit spaces with a restriction of Ricci curvature. A joint work with Xianzhe Dai (UC Santa Barbara), Jiayin Pan (UC Santa Cruz) and Guofang Wei
(UC Santa Barbara) disproved the conjecture. We will discuss about these topics in this talk.
2024/11/25
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.
Leon Frober (Grand Valley State University)
Free energy and ground state of the spiked SSK spin-glass model
We are having teatime from 15:15 in the common room on the second floor. Please join us.
Leon Frober (Grand Valley State University)
Free energy and ground state of the spiked SSK spin-glass model
[ Abstract ]
Spin-glasses are essentially mathematical models of particle interactions, and were originally describing magnetic states characterized by randomness in condensed matter physics. Due to the versatility of these types of models, however, they are now studied much more broadly for various complex systems such as statistical inference problems, weather/climate models or even neural networks. In this talk we will lay out the basic concepts of spin-glass models, while then focusing on the spiked SSK variant and its free energy as well as ground state energy. Furthermore we will discuss how one can determine these quantities including their lower order fluctuations with a so called "TAP approach" that was in this comprehensive form introduced in 2016 by N. Kistler and D. Belius, and what its benefits are compared to the earlier established "Parisi approach".
Spin-glasses are essentially mathematical models of particle interactions, and were originally describing magnetic states characterized by randomness in condensed matter physics. Due to the versatility of these types of models, however, they are now studied much more broadly for various complex systems such as statistical inference problems, weather/climate models or even neural networks. In this talk we will lay out the basic concepts of spin-glass models, while then focusing on the spiked SSK variant and its free energy as well as ground state energy. Furthermore we will discuss how one can determine these quantities including their lower order fluctuations with a so called "TAP approach" that was in this comprehensive form introduced in 2016 by N. Kistler and D. Belius, and what its benefits are compared to the earlier established "Parisi approach".
2024/10/28
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.
Kohki Sakamoto (The University of Tokyo)
Harmonic measures in invariant random graphs on Gromov-hyperbolic spaces (日本語)
We are having teatime from 15:15 in the common room on the second floor. Please join us.
Kohki Sakamoto (The University of Tokyo)
Harmonic measures in invariant random graphs on Gromov-hyperbolic spaces (日本語)
[ Abstract ]
In discrete group theory, a Cayley graph is a fundamental concept to view a finitely generated group as a geometric object itself. For example, the planar lattice is constructed from the free abelian group Z^2, and the 4-regular tree is constructed from the free group F_2. A group acts naturally on its Cayley graph as translations, so Bernoulli percolations on the graph can be viewed as a random graph whose distribution is invariant under the group action. In this talk, after reviewing previous works on such group-invariant random graphs, I will present my result concerning random walks on group-invariant random graphs over Gromov-hyperbolic groups. If time permits, I would also like to talk about the analogue in continuous spaces, such as Lie groups or symmetric spaces.
In discrete group theory, a Cayley graph is a fundamental concept to view a finitely generated group as a geometric object itself. For example, the planar lattice is constructed from the free abelian group Z^2, and the 4-regular tree is constructed from the free group F_2. A group acts naturally on its Cayley graph as translations, so Bernoulli percolations on the graph can be viewed as a random graph whose distribution is invariant under the group action. In this talk, after reviewing previous works on such group-invariant random graphs, I will present my result concerning random walks on group-invariant random graphs over Gromov-hyperbolic groups. If time permits, I would also like to talk about the analogue in continuous spaces, such as Lie groups or symmetric spaces.
2024/10/21
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.
Kohei Noda (Institute for Industrial Mathematics, Kyushu University)
Scaling limits of non-Hermitian Wishart random matrices and their applications (日本語)
We are having teatime from 15:15 in the common room on the second floor. Please join us.
Kohei Noda (Institute for Industrial Mathematics, Kyushu University)
Scaling limits of non-Hermitian Wishart random matrices and their applications (日本語)
[ Abstract ]
This talk is based on joint work and an ongoing project with Sung-Soo Byun (Seoul National University) on the scaling limits of non-Hermitian Wishart random matrices, which were introduced in the context of quantum chromodynamics with a baryon chemical potential, and their probabilistic applications. We present a robust argument, a generalized Christoffel-Darboux type identity, to obtain the scaling limits of eigenvalue point processes (determinantal/Pfaffian point processes) for non-Hermitian Wishart ensembles. Additionally, I will discuss the fluctuation of real eigenvalues in non-Hermitian real Wishart ensembles.
This talk is based on joint work and an ongoing project with Sung-Soo Byun (Seoul National University) on the scaling limits of non-Hermitian Wishart random matrices, which were introduced in the context of quantum chromodynamics with a baryon chemical potential, and their probabilistic applications. We present a robust argument, a generalized Christoffel-Darboux type identity, to obtain the scaling limits of eigenvalue point processes (determinantal/Pfaffian point processes) for non-Hermitian Wishart ensembles. Additionally, I will discuss the fluctuation of real eigenvalues in non-Hermitian real Wishart ensembles.
2024/10/10
10:00-11:30 Room #122 (Graduate School of Math. Sci. Bldg.)
The lecture is in the morning. The classroom is 122. This is a joint seminar with the Infinite Analysis Seminar Tokyo. No teatime today.
Chiara Franceschini (University of Modena and Reggio Emilia)
Harmonic models out of equilibrium: duality relations and invariant measure (英語)
The lecture is in the morning. The classroom is 122. This is a joint seminar with the Infinite Analysis Seminar Tokyo. No teatime today.
Chiara Franceschini (University of Modena and Reggio Emilia)
Harmonic models out of equilibrium: duality relations and invariant measure (英語)
[ Abstract ]
Zero-range interacting systems of Harmonic type have been recently introduced by Frassek, Giardinà and Kurchan [JSP 2020] from the integrable XXX Hamiltonian with non compact spins. In this talk I will introduce this one parameter family of models on a one dimensional lattice with open boundary whose dynamics describes redistribution of energy or jump of particles between nearest neighbor sites. These models belong to the same macroscopic class of the KMP model, introduced in 1982 by Kipnis Marchioro and Presutti. First, I will show their similar algebraic structure as well as their duality relations. Second, I will present how to explicitly characterize the invariant measure out of equilibrium, a task that is, in general, quite difficult in this context and it has been achieved in very few cases, e.g. the well known exclusion process. As an application, thanks to this characterization, it is possible to compute formulas predicted by macroscopic fluctuation theory. This is from joint works with: Gioia Carinci, Rouven Frassek, Davide Gabrielli, Cirstian Giarinà, Frank Redig and Dimitrios Tsagkarogiannis.
Zero-range interacting systems of Harmonic type have been recently introduced by Frassek, Giardinà and Kurchan [JSP 2020] from the integrable XXX Hamiltonian with non compact spins. In this talk I will introduce this one parameter family of models on a one dimensional lattice with open boundary whose dynamics describes redistribution of energy or jump of particles between nearest neighbor sites. These models belong to the same macroscopic class of the KMP model, introduced in 1982 by Kipnis Marchioro and Presutti. First, I will show their similar algebraic structure as well as their duality relations. Second, I will present how to explicitly characterize the invariant measure out of equilibrium, a task that is, in general, quite difficult in this context and it has been achieved in very few cases, e.g. the well known exclusion process. As an application, thanks to this characterization, it is possible to compute formulas predicted by macroscopic fluctuation theory. This is from joint works with: Gioia Carinci, Rouven Frassek, Davide Gabrielli, Cirstian Giarinà, Frank Redig and Dimitrios Tsagkarogiannis.
2024/10/01
16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)
The classroom is 128. This is a joint seminar with the Tuesday Seminar of Analysis. No TeaTime today.
Patricia Goncalves (Instituto Superior Técnico)
Hydrodynamics, fluctuations, and universality of exclusion processes (英語)
The classroom is 128. This is a joint seminar with the Tuesday Seminar of Analysis. No TeaTime today.
Patricia Goncalves (Instituto Superior Técnico)
Hydrodynamics, fluctuations, and universality of exclusion processes (英語)
[ Abstract ]
In the seventies, Frank Spitzer introduced interacting particle systems to the mathematics community. These systems consist of particles evolving randomly according to Markovian dynamics that conserve certain quantities. Interacting particle systems were already known in the physics and biophysics communities and served as toy models for a variety of interesting phenomena. One of the most classical interacting particle systems is the exclusion process, where particles evolve in a discrete space according to a transition probability, but at each site, only one particle is allowed. One of the goals of studying these models is to derive their hydrodynamic limit, i.e., to deduce the macroscopic equations governing the space-time evolution of the conserved quantities of the system from the underlying random motion of the microscopic particles.
In this talk, I will review the derivation of these limits for the exclusion process. I will also discuss their equilibrium fluctuations, i.e., the fluctuations around the typical profile when the system starts from the invariant measure. Our focus will then shift to the two-species exclusion process, a system with two conservation laws, namely particles of type A and B. We will see that for proper linear combinations of the conserved quantities, their evolution is autonomous. This advances our understanding of the universal behavior of these systems. This presentation is based on joint work with G. Cannizzaro, R. Misturini, and A. Occelli.
In the seventies, Frank Spitzer introduced interacting particle systems to the mathematics community. These systems consist of particles evolving randomly according to Markovian dynamics that conserve certain quantities. Interacting particle systems were already known in the physics and biophysics communities and served as toy models for a variety of interesting phenomena. One of the most classical interacting particle systems is the exclusion process, where particles evolve in a discrete space according to a transition probability, but at each site, only one particle is allowed. One of the goals of studying these models is to derive their hydrodynamic limit, i.e., to deduce the macroscopic equations governing the space-time evolution of the conserved quantities of the system from the underlying random motion of the microscopic particles.
In this talk, I will review the derivation of these limits for the exclusion process. I will also discuss their equilibrium fluctuations, i.e., the fluctuations around the typical profile when the system starts from the invariant measure. Our focus will then shift to the two-species exclusion process, a system with two conservation laws, namely particles of type A and B. We will see that for proper linear combinations of the conserved quantities, their evolution is autonomous. This advances our understanding of the universal behavior of these systems. This presentation is based on joint work with G. Cannizzaro, R. Misturini, and A. Occelli.
2024/09/30
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.
Naotaka Kajino (Kyoto University)
Heat kernel estimates for boundary traces of reflected diffusions on uniform domains
We are having teatime from 15:15 in the common room on the second floor. Please join us.
Naotaka Kajino (Kyoto University)
Heat kernel estimates for boundary traces of reflected diffusions on uniform domains
[ Abstract ]
This talk is aimed at presenting the results of the speaker's recent joint work (arXiv:2312.08546) with Mathav Murugan (University of British Columbia) on the boundary trace processes of reflected diffusions on uniform domains. We obtain stable-like heat kernel estimates for such a boundary trace process when the diffusion on the underlying ambient space satisfies sub-Gaussian heat kernel estimates. Our arguments rely on new results of independent interest such as sharp two-sided estimates and the volume doubling property of the harmonic measure, the existence of a continuous extension of the Na\"im kernel to the topological boundary, and the Doob--Na\"im formula identifying the Dirichlet form of the boundary trace process as the pure-jump Dirichlet form whose jump kernel with respect to the harmonic measure is exactly (the continuous extension of) the Na\"im kernel.
This talk is aimed at presenting the results of the speaker's recent joint work (arXiv:2312.08546) with Mathav Murugan (University of British Columbia) on the boundary trace processes of reflected diffusions on uniform domains. We obtain stable-like heat kernel estimates for such a boundary trace process when the diffusion on the underlying ambient space satisfies sub-Gaussian heat kernel estimates. Our arguments rely on new results of independent interest such as sharp two-sided estimates and the volume doubling property of the harmonic measure, the existence of a continuous extension of the Na\"im kernel to the topological boundary, and the Doob--Na\"im formula identifying the Dirichlet form of the boundary trace process as the pure-jump Dirichlet form whose jump kernel with respect to the harmonic measure is exactly (the continuous extension of) the Na\"im kernel.
2024/07/29
15:00-17:50 Room #122 (Graduate School of Math. Sci. Bldg.)
Lectures start earlier. The classroom is 122. No teatime today.
Yoshinori Kamijima (Toyo University) 15:00-15:50
時空間でのランダムカレント表現に基づくIsing模型に対するレース展開の導出 (日本語)
強局所なp-エネルギーに付随するp-エネルギー測度の構成について (日本語)
Liouville Brown運動とLiouville Cauchy過程 (日本語)
Lectures start earlier. The classroom is 122. No teatime today.
Yoshinori Kamijima (Toyo University) 15:00-15:50
時空間でのランダムカレント表現に基づくIsing模型に対するレース展開の導出 (日本語)
[ Abstract ]
レース展開は平均場臨界現象を解析する為の強力な手法の一つである．レース展開を用いると，例えば臨界点の漸近展開が得られ，それは現在までに自己回避歩行・無向パーコレーション・有効パーコレーション・コンタクトプロセス等で示されている．本研究の目的は，量子Ising模型に対するレース展開を導出し，それによって量子Ising模型の臨界点の評価を得ることである．頂点集合 $\Lambda$ 上のスピン配置 $\vec{\sigma} \in \{-1, +1\}^{\Lambda}$ がGibbs分布に従って実現されるという数理模型を古典Ising模型という．量子Ising模型とは，その古典Ising模型のスピン配置空間の代わりに対応するテンソル空間 $(\mathbb{C}^2)^{\otimes \Lambda}$ を考え，更に強さ $q$ の横磁場を印加した数理模型である．横磁場の為に温度のみの時とは異なる種の相転移が起こる．また，$d$ 次元量子Ising模型は空間に時間と呼ばれる別の座標軸を加えた時空間を考えることによって，$d+1$ 次元の特殊な古典Ising模型と等価であることが知られている．
本講演では量子Ising模型に対するレース展開を導出する試みの一端として，古典Ising模型 ($q=0$ の場合の量子Ising模型) に対する新しいレース展開の導出方法を解説する．それ自体はランダムカレント表現を用いて [Sakai (2007) \textit{Commun. Math. Phys.}] [Sakai (2022) \textit{Commun. Math. Phys.}] で既に得られている．ランダムカレント表現は簡単に言えばスピンの言葉をボンドの言葉に翻訳する手法の一種である．本講演では，量子Ising模型で使われる，時空間でのランダムカレント表現 [Bj\"{o}rnberg and Grimmett (2009) \textit{J. Stat. Phys.}] [Crawford and Ioffe (2010) \textit{Commun. Math. Phys.}] を用いる点が先行研究と異なる．横磁場有り ($q > 0$) の場合の研究は現在進行中である．時間に余裕があれば，その現状についても言及する．
本研究は坂井哲（北海道大学）との共同研究である．
Kohei Sasaya (The University of Tokyo) 16:00-16:50レース展開は平均場臨界現象を解析する為の強力な手法の一つである．レース展開を用いると，例えば臨界点の漸近展開が得られ，それは現在までに自己回避歩行・無向パーコレーション・有効パーコレーション・コンタクトプロセス等で示されている．本研究の目的は，量子Ising模型に対するレース展開を導出し，それによって量子Ising模型の臨界点の評価を得ることである．頂点集合 $\Lambda$ 上のスピン配置 $\vec{\sigma} \in \{-1, +1\}^{\Lambda}$ がGibbs分布に従って実現されるという数理模型を古典Ising模型という．量子Ising模型とは，その古典Ising模型のスピン配置空間の代わりに対応するテンソル空間 $(\mathbb{C}^2)^{\otimes \Lambda}$ を考え，更に強さ $q$ の横磁場を印加した数理模型である．横磁場の為に温度のみの時とは異なる種の相転移が起こる．また，$d$ 次元量子Ising模型は空間に時間と呼ばれる別の座標軸を加えた時空間を考えることによって，$d+1$ 次元の特殊な古典Ising模型と等価であることが知られている．
本講演では量子Ising模型に対するレース展開を導出する試みの一端として，古典Ising模型 ($q=0$ の場合の量子Ising模型) に対する新しいレース展開の導出方法を解説する．それ自体はランダムカレント表現を用いて [Sakai (2007) \textit{Commun. Math. Phys.}] [Sakai (2022) \textit{Commun. Math. Phys.}] で既に得られている．ランダムカレント表現は簡単に言えばスピンの言葉をボンドの言葉に翻訳する手法の一種である．本講演では，量子Ising模型で使われる，時空間でのランダムカレント表現 [Bj\"{o}rnberg and Grimmett (2009) \textit{J. Stat. Phys.}] [Crawford and Ioffe (2010) \textit{Commun. Math. Phys.}] を用いる点が先行研究と異なる．横磁場有り ($q > 0$) の場合の研究は現在進行中である．時間に余裕があれば，その現状についても言及する．
本研究は坂井哲（北海道大学）との共同研究である．
強局所なp-エネルギーに付随するp-エネルギー測度の構成について (日本語)
[ Abstract ]
本講演におけるp-エネルギー(E,F)とは, Dirichlet形式のL^p空間における対応物のことを指す. 近年, このp-エネルギーはフラクタル上の(1,p)-Sobolev空間の対応物を考えるという動機のもとで研究が進められている.
本講演では, 幾何的な対称性や自己相似性といった仮定を底空間に課さない, 強局所, 正則なp-エネルギーに対応するp-エネルギー測度(Dirichlet形式でのエネルギー測度に対応するもの)の構成について述べる. さらに, セミノルムE^(1/p)で定義される商ノルム空間F/~が可分であれば, このエネルギー測度に付随する非対称p次形式がチェインルール, Leibnizルールを満たすことを示す.
Takumu Ooi (Tokyo University of Science) 17:00-17:50本講演におけるp-エネルギー(E,F)とは, Dirichlet形式のL^p空間における対応物のことを指す. 近年, このp-エネルギーはフラクタル上の(1,p)-Sobolev空間の対応物を考えるという動機のもとで研究が進められている.
本講演では, 幾何的な対称性や自己相似性といった仮定を底空間に課さない, 強局所, 正則なp-エネルギーに対応するp-エネルギー測度(Dirichlet形式でのエネルギー測度に対応するもの)の構成について述べる. さらに, セミノルムE^(1/p)で定義される商ノルム空間F/~が可分であれば, このエネルギー測度に付随する非対称p次形式がチェインルール, Leibnizルールを満たすことを示す.
Liouville Brown運動とLiouville Cauchy過程 (日本語)
[ Abstract ]
2次元Brown運動をLiouville測度によって時間変更してできた確率過程であるLiouville Brown運動は、Liouville量子重力と呼ばれるランダム曲面上の自然な拡散過程である。また、その1次元の対応物としてLioville Cauchy過程がBaverez(2021)によって構成されている。本講演では、Liouville Brown運動とLioville Cauchy過程との関係や、これらへの収束などの性質について説明する。
2次元Brown運動をLiouville測度によって時間変更してできた確率過程であるLiouville Brown運動は、Liouville量子重力と呼ばれるランダム曲面上の自然な拡散過程である。また、その1次元の対応物としてLioville Cauchy過程がBaverez(2021)によって構成されている。本講演では、Liouville Brown運動とLioville Cauchy過程との関係や、これらへの収束などの性質について説明する。
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.
