Abstracts

Yuki Chino (NCTS, Taiwan)

Title:

Asymptotic behaviour of random walk in cooling random environment

Abstract:

One-dimensional Random Walk in Cooling Random Environment (RWCRE) is obtained as a patchwork of one-dimensional Random Walk in Random Environment (RWRE) by resampling the environment along a sequence of deterministic times. The RWCRE model can be seen as a model that interpolates between the classical static model and the model with i.i.d.~resamplings every unit of time. In this talk, we have two results about the asymptotic behaviour of RWCRE. First, we investigate how the recurrence versus transience criterion known for RWRE changes for RWCRE. Second, we explore the fluctuation for RWCRE when RWRE is either recurrent or satisfies a classical central limit theorem. In the previous work, we showed that SLLN and LDP for RWCRE were the same as those for RWRE under a certain condition for the resampling. However, two results in this talk are different from those for RWRE. They really depend in a delicate way on how we choose resampling. In particular, sub-diffusive scaling and convergence to mixtures of different limit laws are possible. This talk is based on a joint work with L. Avena (Leiden University), C. da Costa (Durham University) and F. den Hollander (Leiden University).

Takuya Mine (Kyoto Institute of Technology)

Title:

Schr\"odinger operator with point interactions of Poisson type

Abstract:

We study the Schr\"odinger operator with point interactions on d-dimensional Euclidean space (d=1,2,3). We assume the support of the interactions is the Poisson configuration, i.e., the support of the Poisson point process. We prove that the operator is self-adjoint, and the spectrum coincides with the whole real-line or the positive half-line, according to the sign of the interaction. We also give some numerical results for the integrated density of states.

Ryoki Fukushima (RIMS)

Title:

Biased random walk conditioned on survival among Bernoulli obstacles: sub-critical phase

Abstract:

We consider a discrete time biased random walk conditioned to avoid Bernoulli obstacles on the d-dimensional integer lattice. This model is known to undergo a phase transition: for a large bias, the walk is ballistic whereas for a small bias, it is sub-ballistic. This phase transition was proved by Sznitman and later, Ioffe and Velenik studied the ballistic phase in detail. In the sub-ballistic phase, physicists conjectured that the walk is localized in a sub-diffusive scale but it has not been proved. We prove this conjecture with a precise information on the behavior of whole path. Based on a joint work with Jian Ding, Rongfeng Sun and Changji Xu.

Naoki Kubota (Nihon University)

Title:

Continuity for the asymptotic shape in the frog model with random initial configurations

Abstract:

In this talk, we treat the so-called frog model with random initial configurations, which is described by the following evolution mechanism of simple random walks on the multidimensional cubic lattice: Some particles are randomly assigned to any site of the multidimensional cubic lattice. Initially, only particles at the origin are active and they independently perform simple random walks. The other particles are sleeping and do not move at first. When sleeping particles are hit by an active particle, they become active and start doing independent simple random walks. The target of this talk is the asymptotic shape of the set of all sites visited by active particles up to a certain time. In particular, we discuss the continuity for the asymptotic shape in the law of the initial configuration.

Satoshi Tsujimoto (Kyoto University)

Title:

ピットマン変換を用いた離散古典可積分系の解析について

Abstract:

ソリトン性を有する箱玉系に対して見出されたピットマン変換と離散古典可積分系との関係は， 離散可積分系と確率論の両面において新しい視点を与えている．従来，可積分では玉の個数を有限とする議論が専らであったが，無限個の玉を初期値に与え，不変測度などの議論を可能とした．また，実軸上のピットマン変換に対応して導入される「実軸上の箱玉系」と離散古典可積分系との関係など，最近の研究成果についても紹介する．

Takashi Imamura (Chiba University)

Title:

q-TASEP, stochastic vertex models and the q-Whittaker measures

Abstract:

The q-totally asymmetric simple exclusion process (q-TASEP) is an interacting particle process belonging to the Kardar-Parisi-Zhang(KPZ) universality class introduced in [1]. The q-TASEP has been playing an important role in the recent progress of the integrable probability since it has nice mathematical structures from which one can access detailed information about fluctuation properties. One typical feature is that the q-Laplace transform of the particle position distribution can be represented as a single Fredholm determinant. So far various approaches to get this relation have been developed such as the Macdonald process, stochastic duality/Bethe ansatz, Yang-Baxter equation etc. All of these are based on a common idea that they focus on the q-moments of the particle position distribution.

In this talk, we report a different approach to the q-TASEP without using the q-moments [2]. By using the result by Cauchy, Ramanujan and Frobenius, we clarify determinantal structures of the q-Whittaker measure, a probability measure written as a product of two q-Whittaker functions. From this, we obtain a different Fredholm determinant formula of the q-Laplace transform. Using this formula, we can also analyze the q-TASEP under the non-equilibrium stationary state, where the previous approaches have a difficulty that the q-moments diverge. Furthermore this approach can be applicable to the higher spin stochastic vertex models [3], which can be regarded as generalized stochastic processes to the q-TASEP. This is a joint work with Tomohiro Sasamoto and Metteo Mucciconi.

References

[1] A. Borodin and I. Corwin, Probability Theory and Related Fields, 158, 225-400, 2014 (arXiv:1111.4408)

[2] T. Imamura and T. Sasamoto, Probability Theory and Related Fields, 174, 647-730, 2019 (arXiv:1701.05991)

[3]T. Imamura, M. Mucciconi, and T. Sasamoto, arXiv: 1901.08381

Yoshiko Ogata (University of Tokyo)

Title :

Classification of symmetry protected topological phases in quantum spin chains

Abstract :

For the classification of SPT phases, defining an index is a central problem. In the famous paper, Pollmann, Tuner, Berg, and Oshikawa introduced ${\mathbb Z}_2$-indices for injective matrix products states (MPS) which have either ${\mathbb Z}_2\times {\mathbb Z}_2$ dihedral group (of $\pi$-rotations about $x$, $y$, and $z$-axes) symmetry, time-reversal symmetry, or reflection symmetry. we introduce an index which generalizes the index by Pollmann et.al. The index is an invariant of the $C^1$-classification of SPT phases.

Rongfeng Sun (National University of Singapore)

Title:

The 2-dimensional KPZ equation in the entire subcritical regime

Abstract:

We consider the KPZ equation in space dimension 2 driven by space-time white noise. Previously, we showed that if the noise is mollified in space on scale epsilon and the noise strength scaled appropriately, then a phase transition occurs. Chatterjee and Dunlap then showed that the solution admits non-trivial sub-sequential scaling limits deep within the subcritical regime. In recent joint work with F. Caravenna and N. Zygouras, we showed that the limit exists in the entire subcritical regime, and we identify it as the solution of an additive Stochastic Heat Equation, establishing so-called Edwards-Wilkinson fluctuations. Deep within the subcritical regime, the same result was obtained independently by Y. Gu.

Kenkichi Tsunoda (Osaka University)

Title:

Incompressible limit for the weakly asymmetric simple exclusion process

Abstract:

We consider in this talk the so-called incompressible limit for the weakly asymmetric simple exclusion process. One of fundamental questions in mathematical physics is the derivation of the master equation of fluid such as the Burgers equation or the Navier-Stokes equation. By the celebrated Varadhan’s non-gradient method, Esposito, Marra and Yau (1996) and Quastel and Yau (1998) have derived the Navier-Stokes equation as incompressible limit in dimensions strictly larger than 2. However, the derivation of these equations in low dimension is achieved only from a lattice gas which admits mesoscopically long jumps. In this talk, we discuss the incompressible limit for the weakly asymmetric simple exclusion process together with some notable entropy estimate developed by Jara and Menezes (2019+). This talk is based on joint work with Milton Jara (IMPA) and Claudio Landim (IMPA).

Cristian Giardina (University of Modena and Reggio Emilia)

Title:

Boundary-driven processes and integrable spin chains

Abstract:

We show that a large class of Markov processes with unbounded state space admits an algebraic description via a quantum spin chains with non-compact spins [1]. The class includes the "Multi-particle Asymmetric Diffusion Model" introduced by Sasamoto-Wadati (for which the corresponding spin chain is integrable) and the "KMP model" introduced by Kipnis-Marchioro-Presutti. We are mostly interested in the boundary-driven version of these processes, for which we show the existence a dual process that simplifies the computation of correlation functions in the non-equilibrium steady states. The issue remains if integrability can be fully exploited in the boundary driven setting, as it has been successfully done in the asymmetric context.

Reference:

[1] Non-compact quantum spin chains as integrable stochastic particle processes, R Frassek, C Giardinà, J Kurchan - arXiv preprint arXiv:1904.01048

Takuya Machida (Nihon University)

Title:

Introduction to Quantum Walks

Abstract:

Quantum walks are quantum analogs of random walks in mathematics, and they are also considered as discretizations of Dirac equations in quantum physics. Most importantly, the quantum walks are quantum algorithms as themselves in computer science. The study of quantum walks, hence, has been appreciated since 2000 because quantum computers got large focus around the year.

Kazumitsu Sakai (Tokyo University of Science)

Title:

The spin transport in integrable quantum systems

Abstract:

In this talk, the spin transport in integrable quantum spin systems will be discussed, mainly focusing on the spin Drude weight for the spin-1/2 XXZ chain. We combine the thermodynamic Bethe ansatz with the functional relations satisfied by the transfer matrices. This makes it possible to evaluate the asymptotic behavior of the finite temperature spin Drude weight with respect to the system size. As a result, the Drude weight converges to the results obtained by Zotos (Phys. Rev. Lett. 82, 1764 (1999)), however with very slow convergence upon increase of the system size. This strong size dependence may explain that extrapolations from various numerical approaches yield conflicting results. This talk is based on joint work with Andreas Kluemper (University of Wuppertal).

[1] A. Kluemper and K. Sakai, arXiv:1904.11253

Shinji Koshida (Chuo University)

Title:

Multiple backward Schramm—Loewner evolution and coupling with Gaussian free field

Abstract:

It is known that a backward Schramm–Loewner evolution (SLE) is coupled with a free boundary Gaussian free field (GFF) with boundary perturbation to give a conformal welding of a quantum surface. Motivated by a generalization of conformal welding for a quantum surface with multiple marked boundary points, we propose a notion of multiple backward SLE. To this aim, we investigate a commutation relation between two backward Loewner chains, and consequently, we find that the driving process of each backward Loewner chain has to have a drift term given by logarithmic derivative of a partition function, which is determined by a system of Belavin–Polyakov–Zamolodchikov-like equations so that these Loewner chains are commutative. After this observation, we define a multiple backward SLE as a tuple of mutually commutative backward Loewner chains. It immediately follows that each backward Loewner chain in a multiple backward SLE is obtained as a Girsanov transform of a backward SLE. We also discuss coupling of a multiple backward SLE with a GFF with boundary perturbation and find that a partition function and a boundary perturbation are uniquely determined so that they are coupled with each other. This talk is based on arXiv:1908.07180.

Yosuke Kubota (RIKEN)

Title:

Operator algebra K-theory and topological phases

Abstract:

Operator algebra K-theory is useful for the study of topological phases (of one-particle Hamiltonians) in several senses. Firstly, it enables us to treat Hamiltonians which are not translation invariant. Secondly, it provides a technique to determines the set of topological phases with a given symmetry. Thirdly, it plays a fundamental role in the proof of the bulk-boundary correspondence. In this talk I summarize these strength from the viewpoint of coarse geometry.

Hosho Katsura (University of Tokyo)

Title:

Ferromagnetism in the SU(n) Hubbard model with a nearly flat band

Abstract:

In this talk, I will present rigorous results for the SU(n) Fermi-Hubbard model with a completely or nearly flat band [1]. The model can be thought of as a natural generalization of the SU(2) Hubbard model on a one-dimensional Tasaki lattice, for which ferromagnetism at quarter filling has been established rigorously [2,3]. In the talk, I will first give an overview of previous work on flat-band ferromagnetism in the SU(2) Hubbard model. I will next introduce an SU(n) model with a flat band at the bottom of the single-particle spectrum and prove that the ground states exhibit SU(n) ferromagnetism when the total fermion number is half the number of lattice sites. Then I will perturb the model by adding extra hopping terms and make the flat band dispersive. Under the same filling condition, it is proved that the ground states of the perturbed model remain SU(n) ferromagnetic when the bottom band is nearly flat. This is the first rigorous example of the ferromagnetism in non-singular SU(n) Hubbard models in which both the single-particle density of states and the on-site repulsive interaction are finite.

[1] K. Tamura and H. Katsura, arXiv:1908.06286 [cond-mat.str-el] (2019).

[2] H. Tasaki, Phys. Rev. Lett.75, 4678 (1995).

[3] H. Tasaki, Prog. Theor. Phys.99, 489 (1997).

Federico Camia (NYU Abu Dhabi)

Title:

A Gaussian process related to the mass spectrum of the near-critical planar Ising model

Abstract:

The Ising model, introduced by Lenz in 1920 to describe ferromagnetism, is one the most studied models of statistical mechanics. Its two dimensional version has played a special role in rigorous statistical mechanics since Peierls’ proof of a phase transition in 1936 and Onsager’s derivation of the free energy in 1944. The version of the model with a nonzero external magnetic field has never been solved, but in 1989 Zamolodchikov proposed a solution of the associated relativistic quantum field theory. After introducing the model and discussing its connection to quantum field theory, I will describe a Gaussian process constructed using the Ising magnetization field and I will show that this process captures information about the mass spectrum of the Zamolodchikov field theory. (Based on joint work with J. Jiang and C.M. Newman.)