arXiv subjects

Yuta Arai

Publications and source records attributed to Yuta Arai.

4 recordsLinked to original sources

arXiv2026

Basins of Attraction to Multiple Fixed Points in Discrete-time Hysteresis Neural Networks

This paper studies multiple fixed points in a discrete-time hysteresis neural network. The network consists of binary hysteresis neurons characterized by the threshold parameter. Depending on the parameter, the network can have a variety of multiple binary fixed points. Stability of each fixed point is characterized by basin of attraction (BOA): the set of initial points falling into the fixed point. In order to evaluate the distribution of BOA sizes, we present entropy. In order to escape from the curse of dimensionality, we introduce a simple problem: classification of binary data set. In the classification, BOAs correspond to classes. In the problem, we clarify that the threshold parameter can control the entropy, especially, can maximize the entropy: the distribution approaches to uniform. As a concrete example, we consider an item response data set in education. Using two fundamental metrics in the item response theory, the classification results are evaluated.

cs.NE↗

arXiv2023

On the KPZ scaling and the KPZ fixed point for TASEP

We consider all totally asymmetric simple exclusion processes (TASEPs) whose transition probabilities are given in the Schütz-type formulas and which jump with homogeneous rates. We show that the multi-point distribution of particle positions and the coefficient of KPZ scaling are described using the probability generating function of the distribution followed when the rightmost particle jumps. For all TASEPs satisfying certain assumptions, We also prove the pointwise convergence of the kernels appearing in the joint distribution of particle positions to those appearing in the KPZ fixed point formula. Our result generalizes the result of Matetski, Quastel, and Remenik [Acta Math., 227, 115-203, (2021)].

math.PR↗

arXiv2020

The KPZ fixed point for discrete time TASEPs

We consider two versions of discrete time totally asymmetric simple exclusion processes (TASEPs) with geometric and Bernoulli random hopping probabilities. For the process mixed with these and continuous time dynamics, we obtain a single Fredholm determinant representation for the joint distribution function of particle positions with arbitrary initial data. This formula is a generalization of the recent result by Mateski, Quastel and Remenik and allows us to take the KPZ scaling limit. For both the discrete time geometric and Bernoulli TASEPs, we show that the distribution function converges to the one describing the KPZ fixed point.

math-ph↗

arXiv2019

Rigidity and non-rigidity for uniform perturbed lattice

A point process on the topological space S is at most countable subset without a random accumulation point in S. In studies of the point processes, there is a problem of seeing the properties of rigidity and tolerance, and this problem is studied actively in recent years. When let $\displaystyle\mathbb{Z}(\mathbf{X}):=(z+X_z)_{z\in\mathbb{Z}^d}$ be the perturbed lattice that is the lattice $\mathbb{Z}^d$ perturbed by independent and identically random variables $(X_z)_{z\in\mathbb{Z}^d}$ taking values in $\mathbb{R}^d$, regarding the Gaussian perturbed lattice, Peres and Sly showed that there exist the phase transitions with respect to the rigidity and the tolerance when $d\geq 3$ in recent paper. In this paper, when random variables $(X_z)_{z\in\mathbb{Z}^d}$ follow uniform distribution, we show the mutually absolute continuity of the measure without one point and the original measure on a restricted set of spaces of the point process in $d\geq 4$. Also, as a consequence of the above, we show that when random variables $(X_z)_{z\in\mathbb{Z}^d}$ follow the uniform distribution, phase transitions related to the tolerance can be seen in $d\geq 4$.

math.PR↗