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Yuki Ueda

Publications and source records attributed to Yuki Ueda.

At least 19 recordsLinked to original sources
arXiv2026

Poisson operator on the interacting Fock space associated with a discrete-time quantum walk

We study the Poisson operator on the interacting Fock space associated with a discrete-time quantum walk, which we call the QW-Poisson operator. First, we investigate the spectral properties of the QW-Poisson distribution. In particular, we establish a relation between the spectral distributions of the Poisson operator and the reversed Poisson operator on a general interacting Fock space via a size-biased transform. Next, we study the edge behavior of the density of the QW-Poisson distribution. We show that a phase transition occurs at the left endpoint of the support: depending on the parameter, the density either decays to $0$ or blows up to $+\infty$. Moreover, this phase transition coincides with the transition in the number of atoms of the QW-Poisson distribution, equivalently, in the point spectrum of the QW-Poisson operator, and with whether $0$ belongs to the spectrum of the QW-Poisson operator. Finally, we study a connection between the interacting Fock space associated with a discrete-time quantum walk and noncommutative probability theory. More precisely, we compute the moment-generating function and moments of the QW-Poisson operator, and obtain a limit theorem for the Konno distribution via a Poisson approximation. We also investigate the Boolean self-decomposability of the Konno distribution and the shifted reversed QW-Poisson distribution.

math.FA↗
arXiv2026

Scaling limit theorem for mixed free and Boolean convolution powers

We prove a scaling limit theorem for a double sequence of probability measures involving additive free convolution $\boxplus$ and additive Boolean convolution $\uplus$. Let $\mu$ be a probability measure on $\mathbb{R}$ with mean zero and variance one, and let $M=M(N)>0$ satisfy $MN^{\alpha+1/2}\to t>0$. We study the weak limits, as $N\to \infty$, of the double arrays $D_{N^\alpha}((\mu^{\boxplus N})^{\uplus M})$. We show that the limit distribution is the Cauchy distribution with scale parameter $t$ if $\alpha>-1/2$, the $t$-fold Boolean convolution power of the standard semicircle law if $\alpha=-1/2$, and the point mass at the origin if $\alpha<-1/2$.

math.PR↗
arXiv2026

Convolution, cumulants and infinitesimal generators in the formal power series ring

We extend the notions of finite free convolution and finite free cumulants to the setting of formal power series by introducing their natural analogues, namely $t$-deformed convolution and $t$-deformed cumulants. In this framework, we establish $t$-deformed analogues of the law of large numbers and the central limit theorem, revealing structural parallels with classical, free, and finite free probability theories. We show that the case $t=-1$ recovers classical convolution at the level of moment generating functions, thereby connecting the theory directly to classical probability. We further investigate the infinitesimal generators associated with $\boxplus^t$-continuous semigroups, deriving explicit representation formulas that clarify how these generators describe the infinitesimal evolution of the semigroup. In the case $t = d$, our results yield explicit formulas for finite free infinitesimal generators. In the case $t = -1$, we relate these generators to those of one-dimensional L\'{e}vy processes by identifying the corresponding terms in their representations. This establishes a direct connection between $\boxplus^t$-convolution semigroups and classical L\'{e}vy-Khintchine-type generators.

math.PR↗
arXiv2026

Scale-free congestion clusters in large-scale traffic networks: a continuum modeling study

Recent empirical studies have reported that spatiotemporal congestion clusters in urban traffic exhibit scale-free statistics, with cluster size following a power-law distribution. In this study, we address whether macroscopic continuum descriptions of traffic flow are capable of generating such scale-free spatiotemporal congestion patterns. To this end, we analyze the second-order Aw-Rascle-Zhang model on directed networks under junction coupling. The governing equations are solved by a high-order discontinuous Galerkin scheme, and junction fluxes are determined by an optimization-based coupling procedure enforcing conservation and admissibility at intersections. Congestion is defined by thresholding the road-averaged density, and spatiotemporal clusters are extracted as connected components in space and time. Numerical experiments on lattice networks of varying sizes reveal that the cluster size follows a robust power-law distribution. Moreover, when rescaled by the linear system size inherent to the two-dimensional network geometry, the distribution collapses onto an approximately universal curve, indicating finite-size scaling governed by the linear system size. The observed power-law statistics and finite-size scaling are reminiscent of scale-invariant dynamics characteristic of self-organized criticality. These results demonstrate that macroscopic continuum traffic models can reproduce large-scale statistical features observed in real urban congestion dynamics.

physics.soc-ph↗
arXiv2025

Monotone max-convolution and subordination functions for free max-convolution

We show that the distribution of the spectral maximum of monotonically independent self-adjoint operators coincides with the classical max-convolution of their distributions. In free probability, it was proven that for any probability measures $\sigma,\mu$ on $\mathbb{R}$ there is a unique probability measure $\mathbb{A}_\sigma(\mu)$ satisfying $\sigma\boxplus \mu = \sigma \triangleright \mathbb{A}_\sigma(\mu)$, where $\boxplus$ and $\triangleright$ are free and monotone additive convolutions, respectively. We recall that the reciprocal Cauchy transform of $\mathbb{A}_\sigma(\mu)$ is the subordination function for free additive convolution. Motivated by this analogy, we introduce subordination functions for free max-convolution and prove their existence and structural properties.

math.OA↗
arXiv2025

Generalized Meixner-type free gamma distributions:convolution formulas and potential correspondence

We introduce and study a class of generalized Meixner-type free gamma distributions $\mu_{t,\theta,\lambda}$ ($t,\theta>0$ and $\lambda\ge 1$), which includes both the free gamma distributions introduced by Anshelevich and certain scaled free beta prime distributions introduced by Yoshida. We investigate fundamental properties and mixture structures of these distributions. In particular, we consider the Gibbs distribution $\frac{1}{\mathcal{Z}_{t,\theta,\lambda}} \exp\{-V_{t,\theta,\lambda}(x)\}$ associated with a family of potentials $V_{t,\theta,\lambda}$, and show that $\mu_{t,\theta,\lambda}$ maximizes Voiculescu's free entropy with potential $V_{t,\theta,\lambda}$ for parameters $t,\theta>0$ and $1\le \lambda<1+t/\theta$. This result substantially extends the range of classcal-free correspondences obtained the potential function, differing from those arising from the Bercovici-Pata bijection. Moreover, we identify algebraic relations involving noncommutative random variables distributed as free gamma distributions.

math.PR↗
arXiv2025

Higher-order asymptotic expansion with error estimate for the multidimensional Laplace-type integral under perturbations

We consider the asymptotic behavior of the multidimensional Laplace-type integral with a perturbed phase function. Under suitable assumptions, we derive a higher-order asymptotic expansion with an error estimate, generalizing some previous results including Laplace's method. The key points of the proof are a precise asymptotic analysis based on a lot of detailed Taylor expansions, and a careful consideration of the effects of the perturbations on the Hessian matrix of the phase function.

math.CA↗
arXiv2024

$S$-transform in Finite Free Probability

We present a simplified explanation of why free fractional convolution corresponds to the differentiation of polynomials, by finding how the finite free cumulants of a polynomial behave under differentiation. This approach allows us to understand the limiting behaviour of the coefficients $\widetilde{\mathsf{e}}_k(p_d)$ of $p_d$ when the degree $d$ tends to infinity and the empirical root distribution of $p_d$ has a limiting distribution $\mu$ on $[0,\infty)$. Specifically, we relate the asymptotic behaviour of the ratio of consecutive coefficients to Voiculescu's $S$-transform of $\mu$. This prompts us to define a new notion of finite $S$-transform, which converges to Voiculescu's $S$-transform in the large $d$ limit. It also satisfies several analogous properties to those of the $S$-transform in free probability, including multiplicativity and monotonicity. This new insight has several applications that strengthen the connection between free and finite free probability. Most notably, we generalize the approximation of $\boxtimes_d$ to $\boxtimes$ and prove a finite approximation of the Tucci--Haagerup--M\"oller limit theorem in free probability, conjectured by two of the authors. We also provide finite analogues of the free multiplicative Poisson law, the free max-convolution powers and some free stable laws.

math.OA↗
arXiv2024

A note on convergence of densities to free extreme value distributions

The concept of free extreme value distributions as universal limit laws for the spectral maximum of free noncommutative real random variables was discovered by Ben Arous and Voiculescu in 2006. This paper contributes to study the convergence of densities towards free extreme value distributions under the von Mises condition for sample distributions.

math.PR↗
arXiv2023

On a diffusion equation with rupture

We propose a model to describe an evolution of a bubble cluster with rupture. In a special case, the equation is reduced to a single parabolic equation with evaporation for the thickness of a liquid layer covering bubbles. We postulate that a bubble collapses if this liquid layer becomes thin. We call this collapse a rupture. We prove for our model that there is a periodic-in-time solution if the place of rupture occurs only in the largest bubble. Numerical tests indicate that there may not exist a periodic solution if such an assumption is violated.

math.AP↗
arXiv2023

On the free L\'{e}vy measure of the normal distribution

Belinschi et al. [Adv. Math., 226 (2011), 3677--3698] proved that the normal distribution is freely infinitely divisible. This paper establishes a certain monotonicity, real analyticity and asymptotic behavior of the density of the free L\'{e}vy measure. The monotonicity property strengthens the result in Hasebe et al. [Int. Math. Res. Not. (2019), 1758--1787] that the normal distribution is freely selfdecomposable.

math.PR↗
arXiv2023

New combinatorial identity for the set of partitions and limit theorems in finite free probability theory

We provide a refined combinatorial identity for the set of partitions of $\{1,\dots, n\}$, which plays an important role in investigating several limit theorems related to finite free convolutions. Firstly, we present the finite free analogue of Sakuma and Yoshida's limit theorem. That is, we provide the limit of $\{D_{1/m}((p_d^{\boxtimes_d m})^{\boxplus_d m})\}_{m\in \mathbb{N}}$ as $m\rightarrow\infty$ in two cases: (i) $m/d\rightarrow t$ for some $t>0$, or (ii) $m/d\rightarrow0$. The second application presents a central limit theorem for finite free multiplicative convolution. We establish a connection between this theorem and the multiplicative free semicircular distributions through combinatorial identities. Our last result gives alternative proofs for Kabluchko's limit theorems concerning the unitary Hermite and the Laguerre polynomials.

math.PR↗
arXiv2022

On Boolean selfdecomposable distributions

This paper introduces the class of selfdecomposable distributions concerning Boolean convolution. A general regularity property of Boolean selfdecomposable distributions is established; in particular the number of atoms is at most two and the singular continuous part is zero. We then analyze how shifting probability measures changes Boolean selfdecomposability. Several examples are presented to supplement the above results. Finally, we prove that the standard normal distribution $N(0,1)$ is Boolean selfdecomposable but the shifted one $N(m,1)$ is not for sufficiently large $|m|$.

math.PR↗
arXiv2021

On freely quasi-infinitely divisible distributions

Inspired by the notion of quasi-infinite divisibility (QID), we introduce and study the class of freely quasi-infinitely divisible (FQID) distributions on $\mathbb{R}$, i.e. distributions which admit the free L\'{e}vy-Khintchine-type representation with signed L\'{e}vy measure. We prove several properties of the FQID class, some of them in contrast to those of the QID class. For example, a FQID distribution may have negative Gaussian part, and the total mass of its signed L\'{e}vy measure may be negative. Finally, we extend the Bercovici-Pata bijection, providing a characteristic triplet, with the L\'{e}vy measure having nonzero negative part, which is at the same time classical and free characteristic triplet.

math.PR↗
arXiv2021

Rates of convergence for laws of the spectral maximum of free random variables

Let $\{X_n\}_n$ be a sequence of freely independent, identically distributed non-commutative random variables. Consider a sequence $\{W_n\}_n$ of the renormalized spectral maximum of random variables $X_1,\cdots, X_n$. It is known that the renormalized spectral maximum $W_n$ converges to the free extreme value distribution under certain conditions on the distribution function. In this paper, we provide a rate of convergence in the Kolmogorov distance between a distribution function of $W_n$ and the free extreme value distribution.

math.PR↗