Mini-Symposium with Prof. G. Schehr
at Chuo University(version 1: 15/Aug/2023)

Dates/Time: Friday 18 and Monday 21, August 2023, 13:30–17:00
Venue: Room 3507, 5th ﬂoor, Building No.3, Korakuen Campus,
Department of Physics, Chuo University
Program:[Part 1: Friday 18 August 2023]
13:30–14:30 Grégory SCHEHR (Sorbonne, France)
When trapped fermions meet random matrix theory [slides]
14:45–15:45 Ryosuke SATO (Chuo, JSPS Research Fellowship)
Determinantal point processes and operator algebras
16:00–16:20 Yuya TANAKA (Chuo)
Rings-to-Disk Transitions in Complex Eigenvalue Processes
16:20–16:40 Ayana EZOE (Chuo)
Switching particle systems for foraging ants showing phase transitions in path selections
16:40–17:00 Saori MORIMOTO (Chuo)
Classiﬁcations and Generalizations of Traﬃc Flow Models in Statistical Mechanics(17:30– dinner)[Part 2: Monday 21 August 2023]
13:30–14:30 Grégory SCHEHR (Sorbonne, France)
Universal counting statistics in the Ginibre ensembles (complex,symplectic and real) [blackboard talk]
14:45–15:45 Tomoyuki SHIRAI (Kyushu, Japan)
Zeros of random power series with dependent Gaussian coeﬃcients
16:00–17:00 Ryosuke SATO (Chuo, JSPS Research Fellowship)
Dynamics on determinantal point processed from the viewpoint ofoperator algebras(17:30– dinner)
Contact：Makoto Katori
Department of Physics, Chuo University
Room 1538, 5th ﬂoor of Building No.1
e-mail: katori@phys.chuo-u.ac.jp
tel: (03) 3817-1776

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Abstracts
Grégory Schehr (Sorbonne, France)
When trapped fermions meet random matrix theory
I will pedagogically review recent progress in the study of fermionic quantum
many-body systems, both with and without interactions, and in the presence of an
external trapping potential, in their ground state. In the simplest instance of N
one-dimensional noninteracting fermions in the presence of a harmonic potential, Iwill show that the positions of these fermions can be mapped onto the eigenvaluesof the Gaussian Unitary Ensemble (GUE). In fact, by tuning the interactions
and/or changing the form of the trapping potential, it is possible to establish a
precise mapping between these fermionic quantum many-body systems and all theclassical random matrix ensembles, ranging from the Gaussian beta-ensembles tothe so-called Ginibre ensemble. If time permits I will discuss the consequencesof these mappings on the study of the full counting statistics as well as on thequantum entanglement entropy for these systems, in particular in the limit of alarge number of fermions.
Universal counting statistics in the Ginibre ensembles (complex, symplectic and real)In this talk, I will discuss the statistics of the number NR of eigenvalues in thecentred disk of radius R in the three classical Ginibre ensembles of random matrices of size N × N , with a special emphasis on the large N limit. In this limit,
the statistical properties of NR (characterized by its cumulants) display markedly
diﬀerent scaling behaviors depending on R: (i) the “origin scaling limit” for Rclose to the center, (ii) the “bulk scaling limit” for R large but still far from theedge of the spectrum and (iii) the edge scaling limit for R close to the edge of thespectrum. I will ﬁrst discuss these three diﬀerent regimes for the complex Ginibre ensemble, extending previous studies by T. Shirai beyond the origin scalinglimit. In that case, signiﬁcant progress can be made thanks to the determinantal structure of the correlations in the complex Ginibre ensemble. Then, I will
present very recent results obtained with G. Akemann, S. Byun and M. Ebke forthe symplectic Ginibre ensemble (for which the three aforementioned regimes canalso be studied in detail) as well as for the mean and variance of NR in the real
Ginibre ensemble (in the origin scaling limit). In these cases, progress were madepossible by exploiting the Pfaﬃan structure of the correlations in these matrix
models. I will also ﬁnally discuss the universality of these results beyond theGinibre ensembles, for more general normal matrix models (both complex andsympletic).

Ryosuke SATO (Chuo, JSPS Research Fellowship)
Determinantal point processes and operator algebras
In this talk, we will ﬁrst summarize several fundamental aspects of operator
algebra theory. More speciﬁcally, we will discuss operator algebras deﬁned by thecanonical anti-commutation relations, which are referred to as (gauge invariant)
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CAR algebras. Furthermore, we will explore their connection to determinantal
point processes.
Dynamics on determinantal point processed from the viewpoint of operator algebras
After my previous talk, we will discuss determinantal point processes with correlation kernels given by diﬀerence operators and study stochastic dynamics onthem based on operator algebras. First, we construct “noncommutative” stochastic dynamics on (gauge invariant) CAR algebras and demonstrate how they induce
stochastic dynamics on determinantal point processes.

Tomoyuki SHIRAI (Kyushu, Japan)
Zeros of random power series with dependent Gaussian coeﬃcients
The zeros of random power series with i.i.d. complex Gaussian coeﬃcients form
the DPP associated with Bergman kernel. As a generalization of this model,
we are concerned with zeros of random power series with coeﬃcients being astationary, centered, complex Gaussian process. We discuss the expected numberof zeros in a disk and compare it with the i.i.d. coeﬃcients case. When the spectral
density of the Gaussian process of coeﬃcients is a trigonometric polynomial, we
discuss the precise asymptotics of the expected number of zeros inside the disk ofradius r centered at the origin as r tends to the radius of convergence. This talkis based on a joint work with Kohei Noda (Kyushu Univ.).

Yuya TANAKA (Chuo)
Rings-to-Disk Transitions in Complex Eigenvalue Processes
First I show two movies of complex eigenvalue processes of the non-Hermitianmatrix-valued Brownian motions starting from two diﬀerent initial matrices. Theﬁrst one shows a simple growing of a disk starting from a point source. The
second one shows a transition from a single ring to a disk. I am interested inthe latter case and studying its generalization as explained in this talk. I brieﬂyreview the hydrodynamic equations of eigenvalue processes following the paper
by Burda et al. (Nucl. Phys. B 897 (2015) 421). The solutions of the hydrodynamic equations describe both cases well. We found the initial matrices showing
transitions from double rings to a disk of eigenvalue distributions. I report thedouble-ring solutions of the hydrodynamic equations. Finally I discuss the generalcase starting from m-multiple rings with an arbitrary integer m.

Anaya EZOE (Chuo)
Switching particle systems for foraging ants showing phase transitions
in path selections
Switching interacting particle systems studied in probability theory are thestochastic processes of hopping particles on a lattice consisting of slow particles
and fast particles, where the switching between these two types of particles happens randomly with a given transition rate. In the present paper, we show thatsuch stochastic processes of many particles are useful in modeling of group be3

haviors of ants. Recently the situation-depending switching between two distinct
types of primarily relied cues for ants in selecting foraging paths has been experimentally studied by the research group of Prof. H. Nishimori (Meiji University).
We propose the interacting random-walk models in which two kinds of hopping
rules are included. We report the numerical simulation results which exhibit theglobal changes of path patterns depending on the model parameters controllingthe switching. By introducing two kinds of order parameters, we discuss suchglobal changes as phase transitions realized in path selections of ants.

Saori MORIMOTO (Chuo)
Classiﬁcations and Generalizations of Traﬃc Flow Models in Statistical
Mechanics
One of the most serious issues in modern society is a traﬃc jam. Starting from asimple cellular automaton (CA) model named Rule-184, a series of improvementsof traﬃc ﬂow models have been reported in statistical mechanics. The main
purpose of the improvements was to reproduce the fundamental diagram (thedensity-ﬂow diagram) obtained by empirical data. Following ‘a general stochastic
traﬃc CA model’ proposed by Sakai et al. (J. Phys. A: Math. Gen. 39 (2006)
15327), I have programmed in Python my own ‘generalized CA model’ which
includes the quick-start model and the slow-to-start model. In the preset talk,I will explain a variety of models and demonstrate the numerical simulations
showing dependence of the obtained fundamental diagrams on the parameters of
my model. I expect that the traﬃc ﬂow models studied so far can be classiﬁed
into two categories by the following two factors: drivers’ individual factor and theeivironmental factor.

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