June 25, 2022
9:50 Opening
10:00 John Cardy Interacting walkers - friendly and otherwise
10:30 Tomohiro Sasamoto KPZ in half-space
11:00 Ikkei Hotta Infinite-slit limits of multiple Radial/Chordal SLEs
11:30 Tomoyuki Shirai $\alpha$行列式点過程のポアソン表現
12:00 — 13:30 Lunch
13:30 Takayuki Mizuno A walker arriving at the U.S.-China confrontation
14:00 Masato Takei Random walks with step-reinforcement
14:30 Norio Konno 無限粒子系から量子ウォーク，ゼータ対応へ
(From Interacting Particle Systems to Quantum Walks and Zeta Correspondence)
15:00 — 15:30 Break
15:30 Hiraku Nishimori Modelling Autonomous Workload Distribution in Ant Colonies
16:00 Christian Krattenthaler Proofs of Borwein Conjectures
16:30 Makoto Katori Matrix-valued Brownian motions and two-dimensional processes
17:00 Closing
I review some results on interacting random walks: friendly, vicious, and predatory, using the Fock space and path integral formulation of the master equation.
After finishing PhD in 2000 which was mainly on stationary open ASEP, I was struggling to shift my research to its time dependent properties. After a few years I could write the first paper on the subject with Takashi Imamura[1], which was about multi point distribution of TASEP in half-space. This work would have never been possible without strong influence by previous works of Makoto Katori, Hideki Tanemura and Taro Nagao[2].
For the KPZ equation, rigorous understandings of half-space models have been quite limited, compared to the case of whole space. In this talk we employ a symmetrized version of the bijection we discovered last year, which connects KPZ models to free fermions at a positive temperature, and present a Fredholm Pfaffian formula for KPZ models in half-space. Generalization to multi-point distribution is still open.
The talk is based on collaborations with T. Imamura and M. Mucciconi[3].
[1] T. Sasamoto and T. Imamura, Fluctuations of the one-dimensional polynuclear growth model in half-space. Journal of Statistical Physics, 115:749–803 (2004).
[2] T. Nagao, M. Katori, and H. Tanemura, Dynamical correlations among vicious random walkers, Phys. Lett. A 307:29–35 (2003).
[3] T. Imamura, M. Mucciconi, T. Sasamoto, Solvable models in the KPZ class: approach through periodic and free boundary Schur measure, arXiv: 2204.0842.
通常SLEでは複素平面上の（単連結）領域内に伸びる1本のランダムスリットを考え，スリットを除いた領域を標準領域へと写す等角写像族とその族が満たす微分方程式を考察する．この拡張として複数本のスリットを扱うモデルがMultiple SLEである．本講演では「Multiple SLE」および「スリットの本数を無限にした極限」に関する結果とその展望について概括する．
行列式点過程は種々の応用をもつ確率過程の重要なクラスである．相関関数がある積分核の行列式で表されることが名前の由来であるが，行列式に$1$パラメータを導入して得られる$\alpha$行列式を相関関数とする点過程は$\alpha$行列式点過程とよばれる．本講演では，$\alpha$行列式点過程のポアソン表現について議論する．
A student who studied traffic flow modeling in Professor Katori's lab walked through economics and information science to international political science. When statistical physics, economics, information science, and political science are fused, the struggle for global hegemony can be understood scientifically. Democracy is a nonlinear phenomenon. Using nonlinear interactions among economic actors, ultimate beneficiaries hold the casting board to control the economic activities of many people. In this presentation, we observe corporate control power by foreign governments by estimating the voting power that flows through a real ownership network, which includes about 100 million stock companies and about 500 million shareholders.
[1] Mizuno T, Doi S, Kurizaki S (2020) The power of corporate control in the global ownership network. PLoS ONE 15(8): e0237862.
1次元のstep-reinforced random walksの中でも代表的な，elephant random walkとcorrelated random walkの極限挙動に関する最近の研究成果を紹介する．
私の1990年前後の無限粒子系に関する研究を出発点として，その後の量子ウォーク，そして最近のゼータ対応シリーズに関する研究について概観したい．
Ants have evolved to the present forms from the same ancestor with bees, through which evolution process they have simplified their own structure and the behavior of each, whereas cooperative behavior as a mass of them has got more and more complex and sophisticated. In the present study, we introduce a new experimental method to analyze the statistical behavior of colonies of ants using both i)very-tiny RFID tags attached to each body of all ants and ii)sensors attached to gates connecting a nest and foraging arenas. Then, we found various kinds of statistical structure of the ant society in which sophisticated task allocation among ants and its dynamical reorganization took place[1]. In particular, our data indicates that the response threshold model[2] which has widely been believed to explain the task allocation dynamics of ants should be reconsidered.
References
[1] O. Yamanaka, M. Shiraishi, A. Awazu, H. Nishimori: Verification of mathematical models of response threshold through statistical characterization of the foraging activity in ant societies’, Scientific Reports 9 (1), 8845 (2019).
[2] E.Bonabeau and Marco Drigo and Guy Theraulaz, SWARM INTELLIGENCE From Natural to Artificial Systems (Oxford University Press,1999).
The (so-called) "Borwein Conjecture" arose around 1990 and states that the coefficients in the polynomial $$(1-q)(1-q^2)(1-q^4)(1-q^5)\cdots(1-q^{3n-2})(1-q^{3n-1})$$ have the sign pattern $+--+--\dots$. To be correct, this is the "First Borwein Conjecture" since there are two more similarly looking conjectures.
I shall present these conjectures and then review the history of the conjectures and the various attempts that have been made to prove them.
I shall then outline a proof plan that is (in principle) applicable to all these conjectures. Indeed, this leads to a proof of the "First Borwein Conjecture" and of the "Second Borwein Conjecture", and to a partial proof of another Borwein-like conjecture.
This is joint work with Chen Wang (from 2022).
I have studied Dyson’s Brownian motion model with Hideki Tanemura and others, which is a Hermitian matrix-valued Brownian motion. The eigenvalue process is determinantal in space and time and the hydrodynamic limit is described by the complex Burgers equation in the inviscid limit. In the present talk, I review the recent work by Burda et al. (2015), Grela and Warchoł (2018), Bourgade and Dubach (2020), Akemann et al. (2020), and others on a non-Hermitian matrix-valued Brownian motion. I will explain new aspects of its eigenvalue process on the complex plane: It is dynamically coupled with the left and right eigenvectors and hence essentially different from Dyson’s process on the real line. Nevertheless, the inviscid Burgers equation and the determinantal structure appear associated with the eigenvector processes. Some numerical results obtained by my students will be also demonstrated.