Program

Abstract: We will discuss new methods to, in principle, obtain all cumulants of von Neumann entropy over different models of random states. The new methods uncover the structures of cumulants in terms of lower-order joint cumulants involving families of ancillary linear statistics that we have constructed. Importantly, the new methods avoid the tedious tasks of simplifying nested summations that prevent existing methods in the literature to obtain higher-order cumulants. This talk is based on an ongoing joint work with Youyi Huang. 

Abstract: The quantum de Finetti theorem is a fundamental theorem in quantum information theory, and it asserts that the separable states of a composite quantum system are infinitely and symmetrically extendable. In this talk, we will discuss a new proof of the theorem inspired by the non-commutative Choquet--Deny theory due to Biane. Moreover, this new approach leads us to a new understanding of separable states based on non-commutative Martin boundaries of non-commutative random walk on dual spaces of compact groups. This talk is based on a collaboration with B. Collins and T. Giordano.

Abstract: Motivated by the recent study of the non-Hermitian matrix-valued Brownian motion, we propose a family of matrix-valued dynamical systems generated by non-Hermitian and non-normal Toeplitz matrices with additive perturbations. We first show complicated structures and motions of "eigenvalues" found in numerical calculations. Then we derive the specific equations determining the non-zero exact-eigenvalue processes. We use the theory of symbol curves of the corresponding Toeplitz operators represented by semi-infinite matrices and characterize the numerical results. We report a new phenomenon in our processes such that at each time the outermost closed simple curve in the symbol curve is realized as the exact eigenvalues, but the inner part of the symbol curve is reduced in size and embedded in the pseudospectrum surrounding the origin. The multiplicity of the eigenvalue at the origin is increasing in time and the matrices in our processes become defective (non-diagonalizable). But we found a systematic procedure to construct the generalized eigenspaces associated with the degenerated zero-eigenvalue. The Jordan-block decompositions of the resolvents of our matrices are studied to describe the pseudospectrum processes, where the generalized eigenvector-overlaps play important roles. This talk is based on the joint work with Makoto Katori (Chuo) and Tomoyuki Shirai (Kyushu). A preprint is available at https://arxiv.org/abs/2401.08129.

Abstract:  Switching interacting particle systems, introduced and extensively studied by den Hollander and his collaborators, are the stochastic processes of hopping particles on a lattice made up of slow and fast particles, in which the switching between these types of particles occurs randomly at a given transition rate. In this talk, we will show how such stochastic processes involving multiple particles can model group behaviors of ants. Recent experimental research by the Nishimori group (Meiji Univ., Japan) has investigated how ants switch between two types of primarily relied cues to select foraging paths based on the current situation. Here, we propose a discrete-time interacting random walk model on a square lattice, incorporating two types of hopping rules. Numerical simulation results demonstrate global changes in selected homing paths, transitioning from trailing paths of the ‘pheromone road’ to nearly optimal paths depending on the switching parameters. By introducing two types of order parameters characterizing the dependence of homing duration distributions on switching parameters, we discuss these global changes as phase transitions in ant path selections. We also report critical phenomena associated with the continuous phase transitions. 

This talk is based on our paper recently published in Physica A 643 (2024) 129798; https://doi.org/10.1016/j.physa.2024.129798. 

A preprint is also available at https://arxiv.org/abs/2311.01946.

Abstract:  Dynamically changing interpersonal relationships lie at the core of the emergence of in-group structures, such as polarity or conflict. We explore these dynamics in a simple continuous dynamical model based on Heider's balance theory. Previous theoretical findings include a rigorous proof of the emergence of in-group harmony or bipolar conflict (global minima) and the identification of local minima called jammed states, along with their corresponding energy spectrum in terms of structural complexity. However, some unrealistic scenarios constrain our capacity to contextualize and directly apply these results to real-world social behaviors. To address these challenges, we first introduce a unified dynamical model and find that group size can be critical to the onset of in-group bipolar conflict. Using random matrix statistics, we then characterize pathways to jammed states, which previously have not been discussed adequately. In a surprising twist, we also show that perturbing our dynamical model can elevate the likelihood of getting trapped in jammed states, giving rise to the novel notion of "noise-induced jammed states". We finally explore possible real-world implications as well as potential contributions to the existing literature in related fields such as anthropology and archaeology. This talk is based on a joint work with Masashi Shiraishi and Hiraku Nishimori (Meiji).

Abstract: This 1-day workshop is organized on the occasion of having two visitors to Chuo University from abroad, Prof. Z. Mouayn and Prof. L. Wei. The topics will include statistical mechanics, mathematical physics, probability theory, random matrix theory, quantum information theory, and our favorite research subject, complex systems.

Abstract: We consider a one-parameter family of non-Hermitian matrix-valued processes, which can be regarded as a dynamical extension of Girko's ensemble interpolating the GUE and the Ginibre ensemble of random matrices. In general, the eigenvalue processes of non-Hermitian matrix-valued processes should be treated together with their eigenvector-overlap processes because of the non-normality of matrices. In this talk, we give SDE representations for the eigenvalues and the eigenvector-overlap processes. We also discuss the time dependent weighted point processes associated with the eigenvalues and eigenvector-overlaps. This is based on the joint work with Syota Esaki (Oita University) and Makoto Katori (Chuo University). 

Abstract: In this talk, we consider stochastic processes associated with eigenvalues and eigenvectors of the non-Hermitian matrix-valued Brownian motion (nHBM). The nHBM is a dynamical N × N random matrix model whose entries are given by i.i.d. complex Brownian motions. For nHBM, the biorthogonality relation is imposed between the right and the left eigenvector processes, which allows for the scale-transformation invariance of the system. The eigenvalue processes associated with the nHBM are related to the Ginibre ensemble. It is difficult to give the SDE of the eigenvalue processes of the nHBM using only themselves. Each eigenvalue process is coupled with the eigenvector-overlap process, which is a Hermitian matrix-valued process with entries given by products of overlaps of the right and left eigenvectors. To analyze the dynamics of the eigenvalues, we think it is important to treat time-dependent point processes of eigenvalues and their variations weighted by the diagonal elements of the eigenvector-overlap processes. In this talk, we introduce tensor-valued processes associated with left and right eigenvectors and related time-dependent point processes.In addition, we discuss a relation to the time-dependent point process related to eigenvector-overlap processes. The present talk is based on the joint work with Makoto Katori (Chuo University), Jacek Malecki (Wroclaw University of Science and Technology) and Satoshi Yabuoku (National Institute of Technology, Kitakyushu College).

Abstract: We will discuss new methods to, in principle, obtain all cumulants of von Neumann entropy over different models of random states. The new methods uncover the structures of cumulants in terms of lower-order joint cumulants involving families of ancillary linear statistics that we have constructed. Importantly, the new methods avoid the tedious tasks of simplifying nested summations that prevent existing methods in the literature to obtain higher-order cumulants. This talk is based on an ongoing joint work with Youyi Huang. 

Abstract: Propagation of correlation fronts and fluctuations around them in quantum many-body systems have attracted much attention in recent years. In this talk we discuss such propagating correlation fronts for a free fermionic system, starting from the alternating initial state. We find that all the moments of dynamical fluctuations around the correlation fronts are described by the universal GOE and GSE random matrix correlations in the long time limit. This may be comparable to the previous work of Eisler and Racz that all particle number correlations are described by the universal GUE random matrix correlations. The talk is based on a joint work with Kazuya Fujimoto. 

[1] K. Fujimoto, T. Sasamoto, Random Matrix Statistics in Propagating Correlation Fronts of Fermions, Phys. Rev. Lett. 132, 087101 (2024).

Abstract: TBAWe consider an orthogonal sequence of Gaussian random functions arising from the Gaussian entire function, by successive actions of the Landau levels raising operator. These functions are also closely related to the short-time Fourier transform of white noise with Hermite windows. We study inter-Landau levels interactions via the normalized pair correlation between the zeros of polyanalytic functions, which shows a two-dimensional analogue of interlacing property of orthogonal polynomials. This talk is based on a joint work with L. D. Abreu (University of Vienna).

Abstract: As the most basic mechanism to enable task allocation in ant colonies without a leader, the primary role of the caste system has been discussed [1] which assigns the specific task to individual workers according to their size, age, and other factors. Its plasticity is another characteristic feature of the task activity observed in ant colonies. Once a certain fraction of the specialists for a task is removed from the colony, the remaining part of the colony tends to cover the workload. With this process, stable maintenance of task fulfillment in the colony is realized [2]. As a mathematical model to numerically reproduce such workload compensation behavior in ant colonies, the response threshold model is widely known. To analytically treat the response threshold model [3], we introduce a corresponding master equation. In conjunction with the real data of ants' behavior obtained through long-term observation using tiny RFID tags attached to all individual workers in several colonies [4], we found that the master equation approach effectively describes various statistical features of the collective task engagement of ants. Specifically, the following two kinds of foraging-task activity statistics were indicated through the analysis of the master equation; i) Response threshold distribution for starting foraging task is not uniformly distributed but largely skewed among ants in each colony. ii) Foraging tasks in a colony are carried only by a small fraction of workers in the colony.

[1] E. O. Wilson, Behav. Ecol. Sociobio., 14, 47(1983). 

[2] D. Charbonneau et al., PLOS ONE, 12(9), e0184074(2017). 

[3] E. Bonabeau et al., Proc. R. Soc. Lond. B, 263, 1565(1996). 

[4] O. Yamanaka et al., Scientific Reports, 9, 1 (2019).

Abstract: Complex systems have recently attracted much attention, whether in natural sciences or in sociological sciences. Members constituting a complex system evolve through nonlinear interactions among each other. This means that in a complex system the multiplicative experience or, so to speak, history that any member has had produces its present characteristics. If attention is paid to any statistical property in any complex system, the lognormal distribution is the most natural and appropriate for the standard or “normal” statistics to look over the whole system. In fact, the lognormality emerges rather conspicuously when we examine various familiar examples in complex systems, such as GDP in the world, nursing-care period for the aged, populations of prefectures and municipalities, and our body height and weight.

Abstract: The talk consists of two parts. In the first part, delivered by Youyi Huang, we will discuss existing methods and results in computing exact cumulants of von Neumann entropy over different random state models. The existing methods rely on simplification of nested summations that becomes prohibitively tedious as the order of cumulant increases. Lu Wei will continue with the second part of the talk, where we will discuss new methods to obtain all cumulants of von Neumann entropy. The new methods uncover the structures of cumulants in terms of lower-order joint cumulants involving families of ancillary linear statistics while avoiding the tedious tasks of simplifying nested summations. The second part of the talk is based on an ongoing joint work with Youyi Huang.