Workshop ‘Fluctuation and Correlation in Stochastic Systems’ October 15, 2014
数理物理の分野で活躍の G. Schehr 氏(Paris-Sud, CNRS)の来日
に合わせて、以下のような１日だけのワークショップを
中央大学理工学部（後楽園キャンパス理工３号館）で開きます。
アブストラクトも下に付けましたので、関心のある方は
是非参加ください。理論物理、実験物理、確率論、可積分系と
広く話題を集めましたので、学生の皆さんも是非いらしてください。
香取眞理（中央大学理工学部物理学科）
Workshop `Fluctuation and Correlation in Stochastic Systems'
October 15, 2014
Room 3300, Faculty of Science and Engineering, Building No.3 (3rd floor),
Chuo University (Korakuen Campus)
Organizers: Makoto KATORI (Chuo), Hiroyuki SUZUKI (Chuo),
Kazumasa A. TAKEUCHI (Tokyo), Tomohiro SASAMOTO (Tokyo Inst. Tech.)
[PROGRAM and ABSTRACTS]PROGRAM (version 1: 22/Sept/2014)
9:50-9:55 Makoto KATORI (Chuo Univ.) Opening address
10:00-10:30 Takashi IMAMURA (Chiba Univ.)
Combinatorial identities in the KPZ replica analysis
10:40-11:10 Saburo KAKEI (Rikkyo Univ.)
Hirota bilinear method and Hermite ensemble
11:20-11:50 Shinsuke M. NISHIGAKI (Shimane Univ.)
Individual eigenvalue distributions for chGSE-chGUE crossover and
low-energy constants in SU(2) $\times$ U(1) gauge theory
12:00-12:30 Shinsuke M. NISHIGAKI (Shimane Univ.)
Critical statistics at the mobility edge of QCD Dirac spectra
12:40-14:00 lunch
14:00-14:30 Gregory SCHEHR (Paris-Sud, CNRS)
The number of distinct and common sites visited by N random walkers
14:40-15:10 Gregory SCHEHR (Paris-Sud, CNRS)
The maximal height of N non-intersecting Brownian motions till their survival
15:20-15:40 coffee break
15:40-16:10 Jun-ichi WAKITA (Chuo Univ.)
Collective behavior of bacterial cells in interfacial environment
16:20-16:50 Kazumasa A. TAKEUCHI (Univ. of Tokyo)
Weak ergodicity breaking in KPZ-class interfaces
17:00-17:30 Tomohiro SASAMOTO (Tokyo Inst. Tech.)
Spectral theory for a q-boson zero range process and its generalization
18:00- Banquet at Room 3507 (5th floor of the same building)
Contact to: Makoto Katori
E-mail: katori@phys.chuo-u.ac.jp
Tel: 03-3817-1776
Fax: 03-3817-1792
Office: Room 1538, 5th floor, Building No.1,
Faculty of Science and Engineering,
Korakuen Campus, Chuo University,
1-13-27 Kasuga, Bunkyo-ku, Tokyo 112-8551
ABSTRACTS
Takashi IMAMURA (Chiba Univ.)
Combinatorial identities in the KPZ replica analysis
Recently much progress has been made on studies of height
fluctuation properties in the one-dimensional Kardar-Parisi-Zhang(KPZ)
equation and related integrable discrete models.
In particular, the replica method of the KPZ equation is
a powerful approach to get exact height distribution functions.
In this method combinatorial identities play a crucial role:
by them sum of messy terms is miraculously factorized.
In this talk we discuss some of these identities and their
role in the analyses of the KPZ equation and related models.
Saburo KAKEI (Rikkyo Univ.)
Hirota bilinear method and Hermite ensemble
It was shown that a Fredholm determinant associated with
the Hermite ensemble is related to a particular solution of
the fourth Painleve equation (Tracy-Widom, 1994).
In this talk, we reconsider this problem from the viewpoint of
Hirota's bilinear method in soliton theory.
Shinsuke M. NISHIGAKI (Shimane Univ.)
Individual eigenvalue distributions for chGSE-chGUE crossover
and low-energy constants in SU(2) $\times$U(1) gauge theory
We evaluate individual distributions of four smallest eigenvalues from
chiral random matrix ensembles interpolating chGSE and chGUE by
the quadrature method applied to the Fredholm Pfaffian of dynamical
Bessel kernel containing a crossover parameter. These distributions
are then fitted with the staggered Dirac spectra of the quenched SU(2)
lattice gauge theory in the presence of fluctuating or constant U(1) fields.
Combination of the four best-fitting crossover parameters from matching
each random matrix theory prediction to the corresponding histogram of
the k-th Dirac eigenvalue allows for an efficient and precise determination
of low-energy constants F and Sigma in the chiral Lagrangian of
Nambu-Goldstone bosons on the coset space SU(2n)/Sp(2n) from
relatively small lattices.
Shinsuke M. NISHIGAKI (Shimane Univ.)
Critical statistics at the mobility edge of QCD Dirac spectra
We examine statistical fluctuation of eigenvalues from the near-edge bulk
of QCD Dirac spectra above the critical temperature. We start by reviewing
on the scale-invariant intermediate spectral statistics at the mobility edge of
Anderson tight-binding Hamiltonians. By fitting the level spacing distributions,
Stieltjes-Wigert random matrix ensembles are shown to provide an excellent
effective description for such a critical statistics. Next we carry over the above
strategy for the Anderson Hamiltonians to the Dirac spectra. For the staggered
Dirac operators of QCD with 2+1 flavors of dynamical quarks at the physical
point and of SU(2) quenched gauge theory, we identify the precise location of
the mobility edge as the scale-invariant fixed point of the level spacing distribution.
The eigenvalues around the mobility edge are shown to obey critical statistics
described by the aforementioned deformed random matrix ensembles of unitary
and symplectic classes.
Gregory SCHEHR (Paris-Sud, CNRS)
The number of distinct and common sites visited by N random walkers
I will present an analytical study of the number of distinct sites $S_N(t)$
and common sites $W_N(t)$ visited by $N$ independent one dimensional
random walkers, all starting at the origin, after $t$ time steps. One can show
that these two random variables can be mapped onto extreme value quantities
associated to $N$ independent random walkers. Using this mapping, one computes
exactly their probability distributions $P_N^d(S,t)$ and $P_N^c(W,t)$ for any value
of $N$ in the limit of large time $t$, where the random walkers can be described
by Brownian motions. In the large $N$ limit, $P_N^d(S,t)$ and $P_N^c(W,t)$
are described by non trivial scaling functions which are computed exactly.
Gregory SCHEHR (Paris-Sud, CNRS)
The maximal height of N non-intersecting Brownian motions till their survival
I will consider $N$ Brownian particles moving on a line starting from initial positions
$u \equiv {u_1,u_2,\dots u_N}$ such that $0<u_1 < u_2 < \cdots < u_N$.
Their motion gets stopped at time $t_s$ when either two of them collide or when
the particle closest to the origin hits the origin for the first time. For $N=2$,
I will present an exact computation of the probability distribution function
$p_1(m|u)$ and $p_2(m|u)$ of the maximal distance travelled by the 1st and 2nd
walker till $t_s$. For general $N$ particles with identical diffusion constants $D$,
one can show that the probability distribution $p_N(m|u)$ of the global maximum $m_N$,
has a power law tail $p_i(m|u) \sim {N^2B_N\mathcal{F}_{N}(u)}/{m^{\nu_N}}$
with exponent $\nu_N =N^2+1$. I will present explicit expressions of the function
$\mathcal{F}_{N}(u)$ and of the $N$ dependent amplitude $B_N$ which can be
analyzed for large $N$ using techniques borrowed from random matrix theory.
Jun-ichi WAKITA (Chuo Univ.)
Collective behavior of bacterial cells in interfacial environment
Collective behavior of biological organisms is generally considered to be complex,
since individual organisms are supposed to behave in complex ways. However, that of
unicellular organisms such as bacterial cells may be not so complex under some
conditions.Bacterial colony formation on agar plate surface is an example, since the
colonies expand their territories simply by the motility and the multiplication of
individual cells. Althoughsome characteristic types of colony patterns have been
found about bacterial speciesBacillus subtilis, Proteus mirabilis, Serratia marcescens
and Escherichia coli by changingagar and nutrient concentrations, it is expected that
they can be understood from a physicalview point through the variances of cell motility
and cell multiplication. While collective behavior of bacterial cells in a two-dimensional
circular pool which is not so large compared with the size of bacterial cells is another
example. We have investigated the collective behavior of Bacillus subtilis by varying cell
size and cell density. Then the behavior has been found to be classified into six distinct
types: rotational motion with randomness,rotational laminar flow (single layer), rotational
laminar flow (two layers), turbulent flow,intermittent motion and random motion.
Furthermore, not the cell sizes but the ratios ofan averaged cell size to a pool size
have been found to be essential for the collective behavior of bacterial cells.
Kazumasa A. TAKEUCHI (Univ. of Tokyo)
Weak ergodicity breaking in KPZ-class interfaces
The last fifteen years have witnessed remarkable progress on the (1+1)-dimensional
KPZ class: some universal fluctuation properties of growing interfaces, especially their
height distribution and spatial correlation, were derived rigorously and confirmed
experimentally.Yet the same level of understanding is not reached on their time
correlation, mainly because of the absence of exact expressions. Here I show,
on the basis of experiments on turbulent liquidcrystal, that time correlation is
similarly intriguing, showing qualitatively different behavior between flat and circular
interfaces. I will argue that the notion called weak ergodicity breaking may be a key
concept to resolve this difference, and present an ongoing attempt in this direction.
This is joint work with Takuma Akimoto in Keio University.
Tomohiro SASAMOTO (Tokyo Inst. Tech.)
Spectral theory for a q-boson zero range process and its generalization
The q-boson totally asymmetric zero range process (q-TAZRP) is a discretization
of the KPZ equation. We develop a spectral theory for this process and explain
how one can obtain the current distribution. We also discuss a few generalization.
