arXiv subjects

Makoto Katori

Publications and source records attributed to Makoto Katori.

At least 19 recordsLinked to original sources

arXiv2026

Coupling of radial multiple SLE with Gaussian free field, and the hydrodynamic limit

Schramm--Loewner evolution (SLE) has been one of the central topics in the probabilistic study of two-dimensional critical systems. It is a random curve in two dimensions to which a cluster interface in a critical lattice system is supposed, or has been proved, to converge. The most archetypical setting for SLE is called chordal, where a random curve evolves in a simply-connected domain from a boundary point to another, whereas in its variant called radial, a random curve evolves from a boundary point to a distinguished interior point. Multiple SLE is a variant to another direction; it deals with multiple random curves, and it is a natural direction as there are certainly multiple cluster interfaces found in critical lattice systems. In this paper, we study multiple SLE in the radial setting, namely, radial multiple SLE. We report two main results. One is regarding the local set coupling between radial multiple SLE and Gaussian free field (GFF). This sort of coupling between SLE and GFF has been extensively studied in the chordal setting, and serves as a foundation for many recent developments. We show that the coupling between radial multiple SLE and GFF occurs if and only if the radial multiple SLE is driven by the circular Dyson Brownian motions. The circular Dyson Brownian motions are a typical example of stochastic log-gases. This fact motivates us to study the hydrodynamic limit of the corresponding radial multiple SLE, which refers to the dynamical law of large numbers, when the number of curves tends to infinity. In our other main result, we provide explicit description of the hydrodynamic limit.

math.PR↗

arXiv2026

Eigenvalues, eigenvector-overlaps, and regularized Fuglede-Kadison determinant of the non-Hermitian matrix-valued Brownian motion

The non-Hermitian matrix-valued Brownian motion is the stochastic process of a random matrix whose entries are given by independent complex Brownian motions. The bi-orthogonality relation is imposed between the right and the left eigenvector processes, which allows for their scale transformations with an invariant eigenvalue process. The eigenvector-overlap process is a Hermitian matrix-valued process, each element of which is given by a product of an overlap of right eigenvectors and that of left eigenvectors. We derive a set of stochastic differential equations (SDEs) for the coupled system of the eigenvalue process and the eigenvector-overlap process and prove the scale-transformation invariance of the obtained SDE system. The Fuglede--Kadison (FK) determinant associated with the present matrix-valued process is regularized by introducing an auxiliary complex variable. This variable is necessary to give the stochastic partial differential equations (SPDEs) for the time-dependent random field defined by the regularized FK determinant and for its squared and logarithmic variations. Time-dependent point process of eigenvalues and its variation weighted by the diagonal elements of the eigenvector-overlap process are related to the derivatives of the logarithmic regularized FK-determinant random-field. We also discuss the PDEs obtained by averaging the SPDEs.

math.PR↗

arXiv2025

Eigenvalue and pseudospectrum processes generated by nonnormal Toeplitz matrices with rank 1 perturbations

We introduce two kinds of matrix-valued dynamical processes generated by nonnormal Toeplitz matrices with the additive rank 1 perturbations $δJ$, where $δ\in {\mathbb{C}}$ and $J$ is the all-ones matrix. For each process, first we report the complicated motion of the numerically obtained eigenvalues. Then we derive the specific equation which determines the motion of non-zero simple eigenvalues and clarifies the time-dependence of degeneracy of the zero-eigenvalue $λ_0=0$. Comparison with the solutions of this equation, it is concluded that the numerically observed non-zero eigenvalues distributing around $λ_0$ are the exact eigenvalues not of the original system, but of the system perturbed by uncontrolled rounding errors of computer. The complex domain in which the eigenvalues of randomly perturbed system are distributed is identified with the pseudospectrum including $λ_0$ of the original system with $δJ$. We characterize the pseudospectrum processes using the symbol curves of the corresponding nonnormal Toeplitz operators without $δJ$. We report new phenomena in our second model such that at each time the outermost closed simple curve cut out from the symbol curve is realized as the exact eigenvalues, but the inner part of symbol curve is reduced in size and embedded in the pseudospectrum including $λ_0$. Such separation of exact simple eigenvalues and a degenerated eigenvalue associated with pseudospectrum will be meaningful for numerical analysis, since the former is stable and robust, but the latter is highly sensitive and unstable with respective to perturbations. The present study will be related to the pseudospectra approaches to non-Hermitian systems developed in quantum physics

math-ph↗

arXiv2025

Generalized Eigenspaces and Pseudospectra of Nonnormal and Defective Matrix-Valued Dynamical Systems

We consider nonnormal matrix-valued dynamical systems with discrete time. For an eigenvalue of matrix, the number of times it appears as a root of the characteristic polynomial is called the algebraic multiplicity. On the other hand, the geometric multiplicity is the dimension of the linear space of eigenvectors associated with that eigenvalue. If the former exceeds the latter, then the eigenvalue is said to be defective and the matrix becomes nondiagonalizable by any similarity transformation. The discrete-time of our dynamics is identified with the geometric multiplicity of the zero eigenvalue $λ_0=0$. Its algebraic multiplicity takes about half of the matrix size at $t=1$ and increases stepwise in time, which keeps excess to the geometric multiplicity until their coincidence at the final time. Our model exhibits relaxation processes from far-from-normal to near-normal matrices, in which the defectivity of $λ_0$ is recovering in time. We show that such processes are realized as size reductions of pseudospectrum including $λ_0$. Here the pseudospectra are the domains on the complex plane which are not necessarily exact spectra but in which the resolvent of matrix takes extremely large values. The defective eigenvalue $λ_0$ is sensitive to perturbation and the eigenvalues of the perturbed systems are distributed densely in the pseudospectrum including $λ_0$. By constructing generalized eigenspace for $λ_0$, we give the Jordan block decomposition for the resolvent of matrix and characterize the pseudospectrum dynamics. Numerical study of the systems perturbed by Gaussian random matrices supports the validity of the present analysis.

math-ph↗

arXiv2025

On an identity of Chaundy and Bullard. III. Basic and elliptic extensions

The identity by Chaundy and Bullard expresses $1$ as a sum of two truncated binomial series in one variable where the truncations depend on two different non-negative integers. We present basic and elliptic extensions of the Chaundy--Bullard identity. The most general result, the elliptic extension, involves, in addition to the nome $p$ and the base $q$, four independent complex variables. Our proof uses a suitable weighted lattice path model. We also show how three of the basic extensions can be viewed as Bézout identities. Inspired by the lattice path model, we give a new elliptic extension of the binomial theorem, taking the form of an identity for elliptic commuting variables. We further present variants of the homogeneous form of the identity for $q$-commuting and for elliptic commuting variables.

math.CO↗

arXiv2025

Weighted Point Configurations with Hyperuniformity: An Ecological Example and Models

Random point configurations are said to be in hyperuniform states, if density fluctuations are anomalously suppressed in large-scale. Typical examples are found in Coulomb gas systems in two dimensions especially called log-gases in random matrix theory, in which points are repulsively correlated by long-range potentials. In infertile lands like deserts continuous survival competitions for water and nutrition will cause long-ranged repulsive interactions among plants. We have prepared digital data of spatial configurations of center-of-masses for bushes weighted by bush sizes which we call masses. Data analysis shows that such ecological point configurations do not show hyperuniformity as unmarked point processes, but are in hyperuniform states as marked point processes in which mass distributions are taken into account. We propose the non-equilibrium statistical-mechanics models to generate marked point processes having hyperuniformity, in which iterations of random thinning of points and coalescing of masses transform initial uncorrelated point processes into non-trivial point processes with hyperuniformity. Combination of data analysis and computer simulations shows the importance of strong correlations in probability law between spatial point configurations and mass distributions of individual points to realize hyperuniform marked point processes.

cond-mat.stat-mech↗

arXiv2025

Extinction and Metastability of Pheromone-Roads in Stochastic Models for Foraging Walks of Ants

Macroscopic changes of group behavior of eusocial insects are studied from the viewpoint of non-equilibrium phase transitions. Recent combined study of experiments and mathematical modeling by the group led by the third author suggests that a species of garden ant switches the individual foraging walk from pheromone-mediated to visual-cues-mediated depending on situation. If an initial pheromone-road between the nest and food sources is a detour, ants using visual cues can pioneer shorter paths. These shorter paths are reinforced by pheromone secreted by following ants, and then the detour ceases to exist. Once the old pheromone-road extincts, there will be almost no chance to reconstruct it. Hence the extinction of pheromone-road is expected to be regarded as a phase transition to an absorbing state. We propose a discrete-time model on a square lattice consisting of switching random walks interacting though time-dependent pheromone field. The numerical study shows that the critical phenomena of the present extinction transitions of pheromone-roads do not seem to belong to the directed percolation universality class associated with the usual absorbing-state transition. The new aspects are cased by the coexistence and competition with newly creating pheromone-roads. In a regime in the extinction phase, the annihilating road shows metastability and takes long time-period to be replaced by a new road.

cond-mat.stat-mech↗

arXiv2025

Interacting Particle Systems Modeling Self-Propelled Motions

In non-equilibrium statistical physics, active matters in both living and non-living systems have been extensively studied. In particular, self-propelled particle systems provide challenging research subjects in experimental and theoretical physics, since individual and collective behaviors of units performing persistent motions can not be described by usual fluctuation theory for equilibrium systems. A typical example of man-made self-propelled systems which can be easily handled in small-sized experiments is a system of camphor floats put on the surface of water. Based on the experimental and theoretical studied by Nishimori et al. (J. Phys. Soc. Jpn. 86 (2017) 101012), we propose a new type of mathematical models for complex motions of camphor disks on the surface of water. In the previous mathematical models introduced by Nishimori et al. are coupled systems of the equations of motion for camphor disks described by ordinary differential equations and the partial differential equation for the concentration field of camphor molecules in water. Here we consider coupled systems of equation of motions of camphor disks and random walks representing individual camphor molecules in water. In other words, we take into account non-equilibrium fluctuations by introducing stochastic processes into the deterministic models. Numerical simulation shows that our models can represent self-propelled motions of individual camphor disk as well as repulsive interactions among them. We focus on the one-dimensional models in which viscosity is dominant, and derive a dynamical system of a camphor disk by taking the average of random variables of our stochastic system. By studying both of stochastic models and dynamical systems, we clarify the transitions between three phases of motions for a camphor disk depending on parameters.

cond-mat.stat-mech↗

arXiv2024

Switching particle systems for foraging ants showing phase transitions in path selections

Switching interacting particle systems studied in probability theory are the stochastic processes of hopping particles on a lattice made up of slow and fast particles, where the switching between these types of particles occurs randomly at a given transition rate. This paper explores how such stochastic processes involving multiple particles can model group behaviors of ants. Recent experimental research by the last author's group 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 study critical phenomena associated with continuous phase transitions.

cond-mat.stat-mech↗

arXiv2024

Accumulated spectrograms for hyperuniform determinantal point processes

We define the accumulated spectrogram associated to a locally trace class orthogonal projection operator and to a bounded set using the polar decomposition of its restriction on that set and prove a convergence theorem for accumulated spectrograms along an exhaustion in the case when the corresponding determinantal point process is hyperuniform. We prove that a radial determinantal point process on Rd is always hyperuniform along the exhaustion formed by the dilations of a bounded open set, and as a consequence, we obtain that dilations of the corresponding accumulated spectrogram converge to the indicator function of the considered set, establishing thus a universal phenomenon. Our result is a generalisation of a theorem by Abreu-Gröchenig-Romero in [1] concerning time-frequency localization operators.

math.PR↗

arXiv2022

Point Processes and Multiple SLE/GFF Coupling

In the series of lectures, we will discuss probability laws of random points, curves, and surfaces. Starting from a brief review of the notion of martingales, one-dimensional Brownian motion (BM), and the $D$-dimensional Bessel processes, BES$_{D}$, $D \geq 1$, first we study Dyson's Brownian motion model with parameter $β>0$, DYS$_β$, which is regarded as multivariate extensions of BES$_D$ with the relation $β=D-1$. Next, using the reproducing kernels of Hilbert function spaces, the Gaussian analytic functions (GAFs) are defined on a unit disk and an annulus. As zeros of the GAFs, determinantal point processes and permanental-determinantal point processes are obtained. Then, the Schramm--Loewner evolution with parameter $κ>0$, SLE$_κ$, is introduced, which is driven by a BM on ${\mathbb{R}}$ and generates a family of conformally invariant probability laws of random curves on the upper half complex plane ${\mathbb{H}}$. We regard SLE$_κ$ as a complexification of BES$_D$ with the relation $κ=4/(D-1)$. The last topic of lectures is the construction of the multiple SLE$_κ$, which is driven by the $N$-particle process on ${\mathbb{R}}$ and generates $N$ interacting random curves in ${\mathbb{H}}$. We prove that the multiple SLE/GFF coupling is established, if and only if the driving $N$-particle process on ${\mathbb{R}}$ is identified with DYS$_β$ with the relation $β=8/κ$.

math.PR↗

arXiv2022

Functional Equations Solving Initial-Value Problems of Complex Burgers-Type Equations for One-Dimensional Log-Gases

We study the hydrodynamic limits of three kinds of one-dimensional stochastic log-gases known as Dyson's Brownian motion model, its chiral version, and the Bru-Wishart process studied in dynamical random matrix theory. We define the measure-valued processes so that their Cauchy transforms solve the complex Burgers-type equations. We show that applications of the method of characteristic curves to these partial differential equations provide the functional equations relating the Cauchy transforms of measures at an arbitrary time with those at the initial time. We transform the functional equations for the Cauchy transforms to those for the $R$-transforms and the $S$-transforms of the measures, which play central roles in free probability theory. The obtained functional equations for the $R$-transforms and the $S$-transforms are simpler than those for the Cauchy transforms and useful for explicit calculations including the computation of free cumulant sequences. Some of the results are argued using the notion of free convolutions.

math.PR↗

arXiv2022

Hyperuniformity of the determinantal point processes associated with the Heisenberg group

The Ginibre point process is given by the eigenvalue distribution of a non-hermitian complex Gaussian matrix in the infinite matrix-size limit. This is a determinantal point process (DPP) on the complex plane ${\mathbb{C}}$ in the sense that all correlation functions are given by determinants specified by an integral kernel called the correlation kernel. Shirai introduced the one-parameter ($m \in {\mathbb{N}}_0$) extensions of the Ginibre DPP and called them the Ginibre-type point processes. In the present paper we consider a generalization of the Ginibre and the Ginibre-type point processes on ${\mathbb{C}}$ to the DPPs in the higher-dimensional spaces, ${\mathbb{C}}^D, D=2,3, \dots$, in which they are parameterized by a multivariate level $m \in {\mathbb{N}}_0^D$. We call the obtained point processes the extended Heisenberg family of DPPs, since the correlation kernels are generally identified with the correlations of two points in the space of Heisenberg group expressed by the Schrödinger representations. We prove that all DPPs in this large family are in Class I of hyperuniformity.

math.PR↗

arXiv2022

Scaling limit for determinantal point processes on spheres

The unitary group with the Haar probability measure is called Circular Unitary Ensemble. All the eigenvalues lie on the unit circle in the complex plane and they can be regarded as a determinantal point process on $\mathbb{S}^1$. It is also known that the scaled point processes converge weakly to the determinantal point process associated with the so-called sine kernel as the size of matrices tends to $\infty$. We extend this result to the case of high-dimensional spheres and show that the scaling limit processes are determinantal point processes associated with the kernels expressed by the Bessel functions of the first kind.

math.PR↗

arXiv2022

Local universality of determinantal point processes on Riemannian manifolds

We consider the Laplace-Beltrami operator $Δ_g$ on a smooth, compact Riemannian manifold $(M,g)$ and the determinantal point process $\mathcal{X}_λ$ on $M$ associated with the spectral projection of $-Δ_g$ onto the subspace corresponding to the eigenvalues up to $λ^2$. We show that the pull-back of $\mathcal{X}_λ$ by the exponential map $\exp_p : T_p^*M \to M$ under a suitable scaling converges weakly to the universal determinantal point process on $T_p^* M$ as $λ\to \infty$.

math.PR↗

arXiv2022

Zeros of the i.i.d. Gaussian Laurent series on an annulus: weighted Szegő kernels and permanental-determinantal point processes

On an annulus ${\mathbb{A}}_q :=\{z \in {\mathbb{C}}: q < |z| < 1\}$ with a fixed $q \in (0, 1)$, we study a Gaussian analytic function (GAF) and its zero set which defines a point process on ${\mathbb{A}}_q$ called the zero point process of the GAF. The GAF is defined by the i.i.d.~Gaussian Laurent series such that the covariance kernel parameterized by $r >0$ is identified with the weighted Szegő kernel of ${\mathbb{A}}_q$ with the weight parameter $r$ studied by Mccullough and Shen. The GAF and the zero point process are rotationally invariant and have a symmetry associated with the $q$-inversion of coordinate $z \leftrightarrow q/z$ and the parameter change $r \leftrightarrow q^2/r$. When $r=q$ they are invariant under conformal transformations which preserve ${\mathbb{A}}_q$. Conditioning the GAF by adding zeros, new GAFs are induced such that the covariance kernels are also given by the weighted Szegő kernel of Mccullough and Shen but the weight parameter $r$ is changed depending on the added zeros. We also prove that the zero point process of the GAF provides a permanental-determinantal point process (PDPP) in which each correlation function is expressed by a permanent multiplied by a determinant. Dependence on $r$ of the unfolded 2-correlation function of the PDPP is studied. If we take the limit $q \to 0$, a simpler but still non-trivial PDPP is obtained on the unit disk ${\mathbb{D}}$. We observe that the limit PDPP indexed by $r \in (0, \infty)$ can be regarded as an interpolation between the determinantal point process (DPP) on ${\mathbb{D}}$ studied by Peres and Virág ($r \to 0$) and that DPP of Peres and Virág with a deterministic zero added at the origin ($r \to \infty$).

math.PR↗

arXiv2021

Partial Isometries, Duality, and Determinantal Point Processes

A determinantal point process (DPP) is an ensemble of random nonnegative-integer-valued Radon measures $Ξ$ on a space $S$ with measure $λ$, whose correlation functions are all given by determinants specified by an integral kernel $K$ called the correlation kernel. We consider a pair of Hilbert spaces, $H_{\ell}, \ell=1,2$, which are assumed to be realized as $L^2$-spaces, $L^2(S_{\ell}, λ_{\ell})$, $\ell=1,2$, and introduce a bounded linear operator ${\cal W} : H_1 \to H_2$ and its adjoint ${\cal W}^{\ast} : H_2 \to H_1$. We show that if ${\cal W}$ is a partial isometry of locally Hilbert--Schmidt class, then we have a unique DPP on $(Ξ_1, K_1, λ_1)$ associated with ${\cal W}^* {\cal W}$. In addition, if ${\cal W}^*$ is also of locally Hilbert--Schmidt class, then we have a unique pair of DPPs, $(Ξ_{\ell}, K_{\ell}, λ_{\ell})$, $\ell=1,2$. We also give a practical framework which makes ${\cal W}$ and ${\cal W}^{\ast}$ satisfy the above conditions. Our framework to construct pairs of DPPs implies useful duality relations between DPPs making pairs. For a correlation kernel of a given DPP our formula can provide plural different expressions, which reveal different aspects of the DPP. In order to demonstrate these advantages of our framework as well as to show that the class of DPPs obtained by this method is large enough to study universal structures in a variety of DPPs, we report plenty of examples of DPPs in one-, two-, and higher-dimensional spaces $S$, where several types of weak convergence from finite DPPs to infinite DPPs are given. One-parameter ($d \in \mathbb{N}$) series of infinite DPPs on $S=\mathbb{R}^d$ and $\mathbb{C}^d$ are discussed, which we call the Euclidean and the Heisenberg families of DPPs, respectively, following the terminologies of Zelditch.

math.PR↗

arXiv2021

Gaussian free fields coupled with multiple SLEs driven by stochastic log-gases

Miller and Sheffield introduced the notion of an imaginary surface as an equivalence class of pairs of simply connected proper subdomains of $\mathbb{C}$ and Gaussian free fields (GFFs) on them under the conformal equivalence. They considered the situation in which the conformal maps are given by a chordal Schramm--Loewner evolution (SLE). In the present paper, we construct GFF-valued processes on $\mathbb{H}$ (the upper half-plane) and $\mathbb{O}$ (the first orthant of $\mathbb{C}$) by coupling a GFF with a multiple SLE evolving in time on each domain. We prove that a GFF on $\mathbb{H}$ and $\mathbb{O}$ is locally coupled with a multiple SLE if the multiple SLE is driven by the stochastic log-gas called the Dyson model defined on $\mathbb{R}$ and the Bru--Wishart process defined on $\mathbb{R}_+$, respectively. We obtain pairs of time-evolutionary domains and GFF-valued processes.

math.PR↗