Abstract
Katsunori Fujie (藤江克徳), Hokkaido University
Title:
Analysis of eigenvalues of large random matrices from the viewpoint of free probability
Abstract:
Free probability theory was invented by Voiculescu in 1980s. One of its main successes is the application to random matrix theory. For example, we can generally say nothing about the eigenvalues of the sum of any pair of hermite matrices, however, if the randomness is added for the matrices, we can identify the eigenvalues of their sum in large dimension. This is an elementary application of free probability theory. After a brief overview of free probability and its application, the joint research work with Takahiro Hasebe will be mentioned shortly.
Mizuki Fujimoto (藤本 瑞希), Hokkaido University
Title :
Many body localization from the perspective of Anderson localization
Abstract :
The theory of typicality is that understanding equilibrium states from a microscopic perspective . However, the theory of typicality is based solely on the large dimension of energy shells and does not reflect the specific properties of the Hamiltonian. While typicality implies the absence of non-equilibrium states within energy shells, it gives an intuitive picture of the transition to equilibrium states through unitary time evolution.
However, examples of systems, such as those exhibiting many-body localization, have been observed where this thermalization does not occur. This necessitates the development of a formulation that can account for such cases. In this study, we attempt to construct non-typical states that do not reach thermal equilibrium using the notion of single-particle localization (Anderson localization). Specifically, we discuss the construction of non-equilibrium states from self-adjoint operators with certain spectra.
Kotaro Hata (幡 航太朗), Hokkaido University
Title :
Uniform Weak Convergence of Random Walks for Additive Processes.
Abstract :
An Additive process is a class of continuous in probability stochastic processes with independent increment, and one example of this is a Brownian motion. In 2020, Takahiro Hasebe(Hokkaido university) and Ikkei Hotta(Yamaguchi university) define the new convergence to establish a bijection from additive processes to multiplicative Loewner chains. In this talk, we call this convergence “uniform weak convergence” and show the limit theorem for additive processes in order to generalize Donsker’s theorem which states that the sequence of normalized random walks converges weakly to a Brownian motion in the space C, that is, the set of continuous functions equipped with uniform topology. Afterwards, we will consider an identical distributed case, an infinitely divisible case and so on. This talk is based on a joint work with Takahiro Hasebe.
Takahiro Kajisa (加治佐 貴大), Hokkaido University
Title:
On the Conditions for Establishing Equivalence of Ensembles in Phase Transition Systems
Abstract:
In statistical mechanics, phase transition occurs when two or more ergodic Gibbs states correspond to one energy density value. Under these circumstances, since the limit extreme microcanonical ensembles and the limit Gibbs ensembles have a one-to-many correspondence, equivalence of ensembles is expected to be broken which says that they coincide as probability measures. In this presentation, we show that ensemble equivalence holds under special situations for models with more than three ground states.
Noe Kawamoto (河本 野恵), Hokkaido University
Title:
Rate of convergence of the critical point of the memory-\tau self-avoiding walk in dimensions d>4
Abstract:
The memory-\tau walk is a model for walks with no loops of length \tau or less. The self-avoiding walk is defined as the \tau\uparrow\infty limit of the memory-\tau walk. The critical point of each model is defined as the radius of convergence of the susceptibility. The critical point of the memory-\tau walk converges to that of the self-avoiding walk, and the best estimate of the speed of convergence so far was of order \tau^{-1-\delta} with \delta < \min{1,(d-4)/2}, which was obtained by Madras and Slade (Birkh\”auser,1993). In this talk, we introduce the proof based on the lace expansion by which we obtain the improved speed of convergence: order \tau^{-(d-2)/2}. This talk is based on the ongoing work by this speaker.
Teruaki Nagasawa (長澤 輝明), Nagoya University
Title:
Observational entropy and macroscopic states
Abstract:
In the mathematical foundation of quantum mechanics, von Neumann derived the von Neumann entropy from thermodynamic discussions. The von Neumann entropy has played an important role in mathematics, physics, and information theory. At the same time, he proved the ergodic theorem and the H-theorem in quantum mechanics, and introduced the observational entropy, which is different from the von Neumann entropy, in the mathematical foundations of quantum mechanics. In recent years, the importance of observational entropy (from the viewpoint of thermostatistical dynamics) has been recognized again, and it has been actively studied. In this talk, we discuss the relationship between macroscopic states and observational entropy introduced by Buscemi et al. Its structure is considered using the theory of sufficiency introduced by Hiai, Petz et al. and quantum error correction.
Shunsuke Tomioka (富岡 駿允)， Hokkaido University
Title :
Bose-Einstein condensation and boundary conditions
Abstract :
In quantum statistical mechanics, various thermodynamic quantities are defined in the infinite volume limit. In the finite volume stage, there are many ways to take the boundary conditions of the Hamiltonian, but the infinite volume limit is considered to be independent of the choice of the boundary conditions because the contribution of the boundary becomes smaller as the volume increases.The purpose of our study is to examine this quantitatively. Currently, we are considering the quantitative evaluation of the effect of the choice of boundary conditions on the critical temperature and the particle number density of the lowest energy level in one dimensional Bose-Einstein condensation. In this presentation, we will present the results obtained at the present stage.