Abstract: We consider oriented percolation on $Z^d$, where each site is open or closed with probabilities $p$ and $1-p$. A classical result is that if $p$ is above a critical value, then there exists an open path from the origin to level $n$ with positive probability, uniformly in $n$. We show that, conditionally on percolation, the number of such open paths grows exponentially at a deterministic rate $alpha(p)$. We derive a quantitative estimate for the speed of convergence, which allows us to show that $alpha$ depends continuously on $p$. We also show that $alpha(p)$ is the limit of the free energy of the directed polymer model in Bernoulli environment as the inverse temperature $beta$ converges to zero.

Abstract: As you might remember from your high school physics class, electrical resistance results from electron scattering. At zero temperature the only scattering mechanism comes from disorder like charged impurities, so it seems like putting more disorder can only make resistance higher. Here we show a surprising case in two-dimensional small-bandgap semiconductor where more disorder actually leads to better conductivity. At some critical disorder, an insulator can even be turned into a metal, which looks counterintuitive at first glance. Our theoretical model offers a way to explain some experiments where the measured conductivity is much higher than expected. In this talk, I will explain how we should understand such an unconventional phenomenon. The key point is that disorder creates large electric field fluctuations, which help electrons quantum mechanically tunnel through the insulating gap and increase the conductivity.

Abstract: For the last few decades, many impressive results have been achieved for models in the Kardar-Parisi-Zhang(KPZ) universality class in one dimension. The current standard approach to study them exactly is to combine Markov duality and Bethe ansatz, and find a Fredholm determinant. However, subsequent asymptotic analysis is often rather complicated. Also studies of half-space models have been very much limited. Recently we discovered a direct connection between solvable models in the KPZ class and free fermionic models at positive temperature (or determinant point processes). The key ingredient in our theory is a new identity between marginals of the q-Whittaker measure and the periodic Schur measure, which is proved in bijective fashion by substantially generalizing the RSK algorithm. Once the connection is established, subsequent analysis becomes rather standard and it allows to study various half-space models for a much wider class of boundary conditions than before. In this talk we will explain these. The talk is based on collaborations with Takashi Imamura and Matteo Mucciconi.

Abstract: 酔歩の量⼦版として扱われる量⼦ウォークは，酔歩よりも伝播が速い⼀⽅で局在化しやすい，という違った顔を⾒せることからも，昨今様々なアプローチで解析がなされ，そこでは酔歩の粒⼦性・量⼦ウォークの波動性という対⽐がほのめかされている．当該講演では，多岐に渡る量⼦ウォークの表現や種類の中で，グラフ上の酔歩から誘導される量⼦ウォークである Szegedy walk を扱い，その「定常状態」をテーマとする．

Abstract: We consider a $p$-dimensional, centered normal population such that each variable has unit variance and any correlation coefficient between different variables is a given nonnegative constant $\rho<1$. Suppose that both the sample size $n$ and population dimension $p$ tend to infinity with $p/n \to c>0$. We prove that the empirical spectral distributions (ESDs) of the sample covariance matrices and the sample correlation matrices weakly converge to Mar\v{c}enko-Pastur distribution scaled by $1-\rho$. This scaling explains rigorously, for this population, a ``phase transition'' of the behavior of a well-used retention rule \emph{eigenvalue-greater-than-one rule} of principal component analysis and exploratory factor analysis depending on whether $\rho=0$ or not. We also show that the ESDs of \emph{Wigner matrices}  and \emph{Fisher matrices} weakly converge to corresponding limiting spectral distributions scaled by $1-\rho$.

Abstract: Dunkl processes are multidimensional Markov processes defined through the use of Dunkl operators. Their paths show discontinuities, and so they can be separated into their continuous (radial) part, and their discontinuous (jump) part. Radial Dunkl processes, also called multivariate Bessel processes, have been studied thoroughly due to their relationship with families of stochastic, log-interacting particle systems such as the Dyson model and Wishart–Laguerre processes. These systems are indexed by a parameter, $\beta$, which serves as a coupling constant of interaction, but which can also be understood as the inverse temperature. In this talk, we make a survey of the main results in the freezing limit, namely $\beta\to\infty$, the fluctuations around it, and we discuss the connections that appear with classical orthogonal polynomials. We also give a quick look at the jump part of Dunkl processes, we study their dynamical properties and their dependence on the radial part, and we find that the jump processes corresponding to the Dyson model and the Wishart-Laguerre processes undergo a phase transition when $\beta$ decreases towards one in the bulk scaling limit ($t~N$). Finally, we discuss the connection between this phase transition and particle collisions in these systems.

Abstract: In this talk we will review some recent results on the fractional Anderson model, a random Schrödinger operator driven by a fractional laplacian. The interest on the latter lies in their association to stable Levy processes, random walks with long jumps and anomalous diffusion. We discuss in this talk the interplay between the non-locality of the fractional laplacian and the localization properties of the random potential in the fractional Anderson model, in both the continuous and discrete settings. In the discrete setting we study the integrated density of states and show a fractional version of Lifshitz tails. This coincides with results obtained in the continuous setting by the probability community. This is based on joint work with M. Gebert (LMU Munich).

Abstract: 2次元格⼦内の有限部分集合上を動く単純ランダムウォーク(SRW)を考える. このSRWは境界に到達したら⼀様ランダムに選んだ境界辺を通って再び集合内に戻るとし, 被覆時間(すべての点を訪問し尽くすまでの時間)の定数倍だけ⾛るとする. 本講演では, thick point (SRWが頻繁に訪問する点)とlate point (SRWが全く訪問していない点)の統計的性質を紹介する. これらは2次元ガウス⾃由場の極値理論に現れるLiouville ランダム測度と密接に関連する. 本講演はMarek Biskup ⽒と Sangchul Lee ⽒ との共同研究に基づく.

Abstract: The Schrödinger operator with point interactions on the Poisson configuration (Poisson point interactions) is an interesting model in the theory of random Schrödinger operators, because the spectral properties can be analyzed more easily than the usual random Schrödinger operators. However, it was studied only in the one-dimensional case up to 2019, since the self-adjointness was not known until then. In the present talk, we introduce some recent results about the Poisson point interactions in two or three-dimensional space. The proof of the results are closely related to the theory of continuum percolation. This is a joint work with Professor Masahiro Kaminaga (Tohoku Gakuin University) and Professor Fumihiko Nakano (Tohoku University).

Abstract: Linear response theory is a tool with which one can study systems that are driven out of equilibrium by external perturbations. It has been used to give a first-principles derivation of Ohm’s law, where the current depends linearly on the applied external electric field. Simply put, the conductivity coefficients that quantify the response can be obtained from a “Taylor expansion”. Justifying  this Taylor expansion of the current density in the electric field has been the subject of a lot of research over the last few decades, starting with the work of Green and Kubo. And making linear response theory rigorous has been a steady source of inspiration for mathematicians. The approach to linear response theory I will discuss in this talk combines elements from functional analysis with insights from operator algebras. This is advantageous, because it is not tailored to a specific model, works for operators on the continuum or the discrete alike and can deal with disorder.

Abstract: I talk about a non-minimal quasi-stationary distribution (QSD) for one-dimensional diffusions. I give a method of reducing convergence to non-minimal QSDs to the tail behavior of the lifetime via a property which I call the first hitting uniqueness. As an application of the result, for Kummer diffusions with negative drifts I give a class of initial distributions converging to each non-minimal quasi-stationary distribution.

Abstract: 本講演では，グラフェン（六⾓格⼦）に対するバルクハミルトニアンとエッジハミルトニアンを量⼦グラフの⼿法で構成し，それらのスペクトル構造の⽐較を⾏います。特に，全空間の場合には固有値でないエネルギー準位のうち，ジグザグ型境界を有する場合には固有値となるものが導出され，エッジ状態の存在に関する結果が得られましたのでご報告致します。なお，講演スライドを，講演前⽇に下記の URL にアップロードする予定です:www.maebashi-it.ac.jp/~niikuni/slide/20210423.pdf