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:2311.07800).

概要：We consider the supercritical bond percolation on the Z^d lattice and study the chemical distance, i.e., the graph distance on the infinite cluster. It is well-known from the Kingman subadditive ergodic theorem that there exists a deterministic constant μ(x) such that the chemical distance D(0, nx) between two connected points 0 and nx grows like nμ(x). Garet and Marchand prove that the probability of the upper tail large deviation event {D(0, nx) > μ(nx)(1 + ε), 0 ↔ nx} decays exponentially with n. In this talk, we discuss the existence of the rate function for the upper tail large deviation when d ≥ 3 and ε > 0 is small enough. Moreover, for d ≥ 3, we prove that the upper tail large deviation event is created by space-time cut-points (points that any geodesic from 0 to nx must cross at a given time) that forces the geodesics to go in a non-optimal direction or to wiggle significantly before reaching the cut-point, where the geodesics consume extra time. This enables us to express the rate function in terms of the rate function for a space-time cut-point. This talk is based on joint work with Barbara Dembin (ETH). 

概要：
人間関係からタンパク質相互作用まで、現実世界に存在する多種多様な複雑系はその構成要素をノード、要素間の相互作用をエッジとしたネットワークとして抽象化することができる。このような複雑ネットワークの性質を分析する際には、ある程度の性質を現実のネットワークに寄せつつもそれ以外は完全にランダムなネットワークを比較対象として用いることが多々ある。本講演では与えられた次数相関をもちながら、それ以外の構造はランダムであるようなネットワークについて説明する。

概要：
本講演では反応拡散模型に対する近年の研究について説明する。反応拡散模型はGlauber力学とKawasaki力学(排他過程)の重ね合わせで表されるMarkov過程であり、巨視的には反応拡散方程式に従って時間発展する粒子系の模型である。この模型は1986年に導入された模型であるが、近年この模型に対する大偏差原理や混合時間に関する問題について新たな進展がみられている。本講演では講演者の研究も含めてどのような研究が行われているか概説する。

概要：
Harmonic maps are critical points of the energy functional defined for maps between Riemannian manifolds. It is known that these maps can be characterized through Brownian motion and martingales on manifolds. In this talk, we will consider a probabilistic characterization of harmonic maps with respect to non-local Dirichlet forms. The most typical examples of such harmonic maps are fractional harmonic maps, which are defined as harmonic maps with respect to the fractional Laplacian. As a simple application of this approach, I will explain the relation between the singularities of martingales on manifolds and those of harmonic maps.

Abstract: The model of directed polymers describe the behavior of a long, directed chain that spreads among an inhomogeneous environment which may attract or repulse the polymer. When the spacial dimension is larger than three, a phase transition occurs between diffusivity (high temperature) and localization (low temperature). On the other hand, in dimensions one and two the polymer is always localized. Dimension two is however critical, as one can recover a phase transition by letting the temperature tend to infinity under a specific parametrization (Caravenna-Sun-Zygouras 17’).  In this talk, I will present some of the main results that are known about this scaling regime, and discuss the recent advances that have occurred in the past few years. In particular, I will describe some results that I have obtained with my coauthors (Anna Donadini, Shuta Nakajima and Ofer Zeitouni) on the diffusive phase and its relation to Gaussian logarithmically correlated fields. I will also discuss connexions of the model with the Kardar-Parisi-Zhang (KPZ) equation and the stochastic heat equation.

Abstract : For the white noise $\xi$ on ${\mathbb R}^2$, an operator corresponding to a limit of $-\Delta +\xi _{\varepsilon}-c_{\varepsilon}$ as $\varepsilon \to 0$ is realized as a self-adjoint operator, where, for each $\varepsilon >0$, $c_{\varepsilon}$ is a constant, $\xi _{\varepsilon}$ is a smooth approximation of $\xi$ defined by $\exp (\varepsilon ^2\Delta )\xi$, and $\Delta$ is the Laplacian. This result is a variant of result's obtained by Allez and Chouk, Mouzard, and Ugurcan. The proof in this paper is based on the heat semigroup approach of the paracontrolled calculus, referring the proof by Mouzard. For the obtained operator, the spectral set is shown to be ${\mathbb R}$.

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概要：This talk introduces a moment method to study eigenvalue processes related to three classical beta ensembles: Gaussian beta ensembles, beta Laguerre ensembles and beta Jacobi ensembles. Using the moment method, we can solve two fundamental problems: the convergence to a limit of the empirical measure process and fluctuations around the limit. The approach works in a classical regime where beta is fixed, in a high temperature regime and in a freezing regime. Here beta is regarded as an inverse temperature parameter.

Abstract：平面 n 角形をランダム方向から射影したとき現れる線分の長さの偏差率を考察し特に最小値をとる多角形を決定したい。この問題は n が２の冪でなければ最小値とそれを attain する図形を決定でき K.Reihardt が別の最適化問題で考察した図形となる。関連する未解決問題も紹介したい。この研究は釜江哲朗先生との共同研究である。

Abstract：実数の10進展開の一般化として、beta展開が知られている。このbeta展開は、その後negative beta展開やrotational beta展開などへさらに一般化された。これらの展開を用いて代数的数を展開する際に現れるdigitの漸近的挙動に関して、本日は取り扱う。特に、代数的無理数の10進展開におけるdigitは一様であるというBorel予想の一般化などを扱う。本講演の内容は、川島誠氏との共同研究である。

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 an abrupt transition between localization and delocalization of eigenwaves. It is a work in collaboration with Amol Aggarwal (Columbia) and Patrick Lopatto (NYU).

談話会概要:
ループ除去ランダムウォークとはランダムウォークのパスからループを順々に取り除いて得られる確率過程のことである。1980年にGreg Lawlerにより導入されて以降、一様全域木やSchramm-Loewner evolution（SLE）等の確率モデルとの関係が見出され、活発に研究がなされてきた。本講演ではすでに示されていることと今後の課題をできるだけ分かりやすく説明する。

集中講義概要:
本講義では、ランダムウォークとループ測度の関係を講述する。本講義で考察するループ測度は、ループの集合上で定義された任意の測度である。典型的な例としては、各ループに「エネルギー」に相当する量を定義し、その定数倍を指数の肩に乗せたもの（ボルツマン因子）を重みとする測度を挙げることができる。このような測度は統計物理において基本的な研究対象である。本講義では「configurational」な視点の下でランダムウォークの性質を講述したい。

Abstract: We consider a one-dimensional discrete Dirac operator in a potential given by a family of i.i.d. random variables modulated by a decreasing envelope. In [1], we showed that these models exhibit a rich phase diagram in terms of their spectrum as a function of the rate of decay of the random potential where the spectrum of the operator

is absolutely continuous for fast decay

is pure point for slow decay

presents a spectral transition for critical decay

We show dynamical localization in the sub-critical region by means of the fractional moments method and provide control on the eigenfunctions. We studied spectral statistics in [2], we show that, in the fast decay case, the rescaled spectrum of the operator converges to the clock process while for critical decay, it converges to the Schroedinger point process. In this way, we recovered the results of Kritchevski, Valko and Virag established for the Anderson model. Our proof is based on the scaling limit of Prufer transform associated to the system and uses the monotonicity to deduce the convergence. In the slow decay, the spectral statistics are expected to be given by a Poisson process. 

References

[1] O. Bourget, G.R.Moreno Flores, A. Taarabt, 

One-dimensional Discrete Dirac Operators in a Decaying Random Potential I : Spectrum and Dynamics, 

Mathematical Physics, Analysis and Geometry, Volume 23, Auticle Number 20, 2020

[2] G.R.Moreno Flores, A. Taarabt, 

One-dimensional Discrete Dirac Operators in a Decaying Random Potential II : Clock, Schroedinger and Sine statistics, submitted. 

Abstract: 本講演では、確率過程の統計において、近年著しく発展している二つの分野「高頻度統計」と「多様体学習」について概観を述べ、その二つの分野をつなぐ手法としてのMalliavin-Mancino法と、その研究のフロンティアを紹介する。

Abstract: The supersymmetric hyperbolic sigma model is a model in mathematical physics that has found relations in a variety of areas, including statistical mechanics, reinforced random walk, Anderson localization, and stochastic calculus. In this talk, we will explore the model from these four different angles, or facets, and discuss recent results in each of these facets.

Throughout the talk, we will highlight the relations between these different facets and explain how they were discovered. We will also present some proofs of key results and discuss open questions and directions for future research.

Abstract: あるマルコフ連鎖は、Left Regular Bandと呼ばれる半群 を用いて表現することができる。Brown (2000) によってLeft Regular Band を用いて表現されるマルコフ連鎖の、推移確率行列の固有値と重複度が求められることが示された。本講演では、Left Regular Bandを用いて表現可能なマルコフ連鎖の紹介及びその推移確率行列の固有値と重複度について述べる。