arXiv subjects

Makoto Katori

Publications and source records attributed to Makoto Katori.

At least 37 records · Page 2Linked to original sources

arXiv2021

Spreading and Suppression of Infection Clusters on the Ginibre Continuum Percolation Clusters

Off-lattice SIR models are constructed on continuum percolation clusters defined on the Ginibre point process (GPP) and on the Poisson point process (PPP). The static percolation transitions in the continuum percolation models as well as the infection-spreading transitions in the SIR models, which are regarded as the dynamic percolation transitions, are enhanced in the GPP-based model compared with the PPP-based model. This enhancement is caused by hyperuniformity of the GPP. On the other hand, in the extinction phase of infection on the phase diagram, a wide parameter region is determined in which formation of infection clusters is suppressed in the GPP-based model compared with the PPP-based model. We think that the PPP approximates a disordered configuration of individuals and the GPP does a configuration of citizens adopting a strategy to keep social distancing in a city in order to avoid contagion. The suppression of infection clusters observed in the GPP-based model will prove that such a strategy is effective when the infectivity is relatively small.

cond-mat.stat-mech↗

arXiv2021

Continuum Percolation and Stochastic Epidemic Models on Poisson and Ginibre Point Processes

The most studied continuum percolation model in two dimensions is the Boolean model consisting of disks with the same radius whose centers are randomly distributed on the Poisson point process (PPP). We also consider the Boolean percolation model on the Ginibre point process (GPP), which is a typical repelling point process realizing hyperuniformity. We think that the PPP approximates a disordered configuration of individuals, while the GPP does a configuration of citizens adopting a strategy to keep social distancing in a city in order to avoid contagion. We consider the SIR models with contagious infection on supercritical percolation clusters formed on the PPP and the GPP. By numerical simulations, we studied dependence of the percolation phenomena and the infection processes on the PPP- and the GPP-underlying graphs. We show that in a subcritical regime of infection rate the PPP-based models show emergence of infection clusters on clumping of points which is formed by fluctuation of uncorrelated Poissonian statistics. On the other hand, the cumulative numbers of infected individuals in processes are suppressed in the GPP-based models.

cond-mat.stat-mech↗

arXiv2021

Three phases of multiple SLE driven by non-colliding Dyson's Brownian motions

The present paper is concerned with properties of multiple Schramm--Loewner evolutions (SLEs) labelled by a parameter $κ\in (0,8]$. Specifically, we consider the solution of the multiple Loewner equation driven by a time change of Dyson's Brownian motions in the non-colliding regime. Although it is often considered that several properties of the solution can be studied by means of commutation relations of SLEs and the absolute continuity, this method is available only in the case that the curves generated by commuting SLEs are separated. Beyond this restriction, it is not even obvious that the solution of the multiple Loewner equation generates multiple curves. To overcome this difficulty, we employ the coupling of Gaussian free fields and multiple SLEs. Consequently, we prove the longstanding conjecture that the solution indeed generates multiple continuous curves. Furthermore, these multiple curves are (i) simple disjoint curves when $κ\in (0,4]$, (ii) intersecting curves when $κ\in (4,8)$, and (iii) space-filling curves when $κ=8$.

math.PR↗

arXiv2021

Local number variances and hyperuniformity of the Heisenberg family of determinantal point processes

The bulk scaling limit of eigenvalue distribution on the complex plane ${\mathbb{C}}$ of the complex Ginibre random matrices provides a determinantal point process (DPP). This point process is a typical example of disordered hyperuniform system characterized by an anomalous suppression of large-scale density fluctuations. As extensions of the Ginibre DPP, we consider a family of DPPs defined on the $D$-dimensional complex spaces ${\mathbb{C}}$, $D \in {\mathbb{N}}$, in which the Ginibre DPP is realized when $D=1$. This one-parameter family ($D \in {\mathbb{N}}$) of DPPs is called the Heisenberg family, since the correlation kernels are identified with the Szegő kernels for the reduced Heisenberg group. For each $D$, using the modified Bessel functions, an exact and useful expression is shown for the local number variance of points included in a ball with radius $R$ in ${\mathbb{R}}^{2D} \simeq {\mathbb{C}}^D$. We prove that any DPP in the Heisenberg family is in the hyperuniform state of Class I, in the sense that the number variance behaves as $R^{2D-1}$ as $R \to \infty$. Our exact results provide asymptotic expansions of the number variances in large $R$.

math-ph↗

arXiv2020

Conformal welding problem, flow line problem, and multiple Schramm--Loewner evolution

A quantum surface (QS) is an equivalence class of pairs $(D,H)$ of simply connected domains $D\subsetneq\mathbb{C}$ and random distributions $H$ on $D$ induced by the conformal equivalence for random metric spaces. This distribution-valued random field is extended to a QS with $N+1$ marked boundary points (MBPs) with $N\in\mathbb{Z}_{\ge 0}$. We propose the conformal welding problem for it in the case of $N\in\mathbb{Z}_{\ge 1}$. If $N=1$, it is reduced to the problem introduced by Sheffield, who solved it by coupling the QS with the Schramm--Loewner evolution (SLE). When $N \ge 3$, there naturally appears room of making the configuration of MBPs random, and hence a new problem arises how to determine the probability law of the configuration. We report that the multiple SLE in $\mathbb{H}$ driven by the Dyson model on $\mathbb{R}$ helps us to fix the problems and makes them solvable for any $N \ge 3$. We also propose the flow line problem for an imaginary surface with boundary condition changing points (BCCPs). In the case when the number of BCCPs is two, this problem was solved by Miller and Sheffield. We address the general case with an arbitrary number of BCCPs in a similar manner to the conformal welding problem. We again find that the multiple SLE driven by the Dyson model plays a key role to solve the flow line problem.

math.PR↗

arXiv2020

Determinantal Point Processes, Stochastic Log-Gases, and Beyond

A determinantal point process (DPP) is an ensemble of random nonnegative-integer-valued Radon measures, whose correlation functions are all given by determinants specified by an integral kernel called the correlation kernel. First we show our new scheme of DPPs in which a notion of partial isometies between a pair of Hilbert spaces plays an important role. Many examples of DPPs in one-, two-, and higher-dimensional spaces are demonstrated, where several types of weak convergence from finite DPPs to infinite DPPs are given. Dynamical extensions of DPP are realized in one-dimensional systems of diffusive particles conditioned never to collide with each other. They are regarded as one-dimensional stochastic log-gases, or the two-dimensional Coulomb gases confined in one-dimensional spaces. In the second section, we consider such interacting particle systems in one dimension. We introduce a notion of determinantal martingale and prove that, if the system has determinantal martingale representation (DMR), then it is a determinantal stochastic process (DSP) in the sense that all spatio-temporal correlation function are expressed by a determinant. In the last section, we construct processes of Gaussian free fields (GFFs) on simply connected proper subdomains of $\mathbb{C}$ coupled with interacting particle systems defined on boundaries of the domains. There we use multiple Schramm--Loewner evolutions (SLEs) driven by the interacting particle systems. We prove that, if the driving processes are time-changes of the log-gases studied in the second section, then the obtained GFF with multiple SLEs are stationary. The stationarity defines an equivalence relation of GFFs, which will be regarded as a generalization of the imaginary surface studied by Miller and Sheffield.

math.PR↗

arXiv2020

Three-Parametric Marcenko-Pastur Density

The complex Wishart ensemble is the statistical ensemble of $M \times N$ complex random matrices with $M \geq N$ such that the real and imaginary parts of each element are given by independent standard normal variables. The Marcenko--Pastur (MP) density $ρ(x; r), x \geq 0$ describes the distribution for squares of the singular values of the random matrices in this ensemble in the scaling limit $N \to \infty$, $M \to \infty$ with a fixed rectangularity $r=N/M \in (0, 1]$. The dynamical extension of the squared-singular-value distribution is realized by the noncolliding squared Bessel process, and its hydrodynamic limit provides the two-parametric MP density $ρ(x; r, t)$ with time $t \geq 0$, whose initial distribution is $δ(x)$. Recently, Blaizot, Nowak, and Warchol studied the time-dependent complex Wishart ensemble with an external source and introduced the three-parametric MP density $ρ(x; r, t, a)$ by analyzing the hydrodynamic limit of the process starting from $δ(x-a), a > 0$. In the present paper, we give useful expressions for $ρ(x; r, t, a)$ and perform a systematic study of dynamic critical phenomena observed at the critical time $t_{\rm c}(a)=a$ when $r=1$. The universal behavior in the long-term limit $t \to \infty$ is also reported. It is expected that the present system having the three-parametric MP density provides a mean-field model for QCD showing spontaneous chiral symmetry breaking.

math.PR↗

arXiv2018

Macdonald denominators for affine root systems, orthogonal theta functions, and elliptic determinantal point processes

Rosengren and Schlosser introduced notions of ${\it R}_N$-theta functions for the seven types of irreducible reduced affine root systems, ${\it R}_N={\it A}_{N-1}$, ${\it B}_{N}$, ${\it B}^{\vee}_N$, ${\it C}_{N}$, ${\it C}^{\vee}_N$, ${\it BC}_{N}$, ${\it D}_{N}$, $N \in \mathbb{N}$, and gave the Macdonald denominator formulas. We prove that, if the variables of the ${\it R}_N$-theta functions are properly scaled with $N$, they construct seven sets of biorthogonal functions, each of which has a continuous parameter $t \in (0, t_{\ast})$ with given $0< t_{\ast} < \infty$. Following the standard method in random matrix theory, we introduce seven types of one-parameter ($t \in (0, t_{\ast})$) families of determinantal point processes in one dimension, in which the correlation kernels are expressed by the biorthogonal theta functions. We demonstrate that they are elliptic extensions of the classical determinantal point processes whose correlation kernels are expressed by trigonometric and rational functions. In the scaling limits associated with $N \to \infty$, we obtain four types of elliptic determinantal point processes with an infinite number of points and parameter $t \in (0, t_{\ast})$. We give new expressions for the Macdonald denominators using the Karlin--McGregor--Lindström--Gessel--Viennot determinants for noncolliding Brownian paths, and show the realization of the associated elliptic determinantal point processes as noncolliding Brownian brides with a time duration $t_{\ast}$, which are specified by the pinned configurations at time $t=0$ and $t=t_{\ast}$.

math.PR↗

arXiv2018

Two-Dimensional Elliptic Determinantal Point Processes and Related Systems

We introduce new families of determinantal point processes (DPPs) on a complex plane ${\mathbb{C}}$, which are classified into seven types following the irreducible reduced affine root systems, $R_N=A_{N-1}$, $B_N$, $B^{\vee}_N$, $C_N$, $C^{\vee}_N$, $BC_N$, $D_N$, $N \in {\mathbb{N}}$. Their multivariate probability densities are doubly periodic with periods $(L, iW)$, $0 < L, W < \infty$, $i=\sqrt{-1}$. The construction is based on the orthogonality relations with respect to the double integrals over the fundamental domain, $[0, L) \times i [0, W)$, which are proved in this paper for the $R_N$-theta functions introduced by Rosengren and Schlosser. In the scaling limit $N \to \infty, L \to \infty$ with constant density $ρ=N/(LW)$ and constant $W$, we obtain four types of DPPs with an infinite number of points on ${\mathbb{C}}$, which have periodicity with period $i W$. In the further limit $W \to \infty$ with constant $ρ$, they are degenerated into three infinite-dimensional DPPs. One of them is uniform on ${\mathbb{C}}$ and equivalent with the Ginibre point process studied in random matrix theory, while other two systems are rotationally symmetric around the origin, but non-uniform on ${\mathbb{C}}$. We show that the elliptic DPP of type $A_{N-1}$ is identified with the particle section, obtained by subtracting the background effect, of the two-dimensional exactly solvable model for one-component plasma studied by Forrester. Other two exactly solvable models of one-component plasma are constructed associated with the elliptic DPPs of types $C_N$ and $D_N$. Relationship to the Gaussian free field on a torus is discussed for these three exactly solvable plasma models.

math-ph↗

arXiv2018

Excursion Processes Associated with Elliptic Combinatorics

Researching elliptic analogues for equalities and formulas is a new trend in enumerative combinatorics which has followed the previous trend of studying $q$-analogues. Recently Schlosser proposed a lattice path model in the square lattice with a family of totally elliptic weight-functions including several complex parameters and discussed an elliptic extension of the binomial theorem. In the present paper, we introduce a family of discrete-time excursion processes on $\mathbb{Z}$ starting from the origin and returning to the origin in a given time duration $2T$ associated with Schlosser's elliptic combinatorics. The processes are inhomogeneous both in space and time and hence expected to provide new models in non-equilibrium statistical mechanics. By numerical calculation we show that the maximum likelihood trajectories on the spatio-temporal plane of the elliptic excursion processes and of their reduced trigonometric versions are not straight lines in general but are nontrivially curved depending on parameters. We analyze asymptotic probability laws in the long-term limit $T \to \infty$ for a simplified trigonometric version of excursion process. Emergence of nontrivial curves of trajectories in a large scale of space and time from the elementary elliptic weight-functions exhibits a new aspect of elliptic combinatorics.

math-ph↗

arXiv2018

Hydrodynamic Limit of Multiple SLE

Recently del Monaco and Schleißinger addressed an interesting problem whether one can take the limit of multiple Schramm--Loewner evolution (SLE) as the number of slits $N$ goes to infinity. When the $N$ slits grow from points on the real line ${\mathbb{R}}$ in a simultaneous way and go to infinity within the upper half plane ${\mathbb{H}}$, an ordinary differential equation describing time evolution of the conformal map $g_t(z)$ was derived in the $N \to \infty$ limit, which is coupled with a complex Burgers equation in the inviscid limit. It is well known that the complex Burgers equation governs the hydrodynamic limit of the Dyson model defined on ${\mathbb{R}}$ studied in random matrix theory, and when all particles start from the origin, the solution of this Burgers equation is given by the Stieltjes transformation of the measure which follows a time-dependent version of Wigner's semicircle law. In the present paper, first we study the hydrodynamic limit of the multiple SLE in the case that all slits start from the origin. We show that the time-dependent version of Wigner's semicircle law determines the time evolution of the SLE hull, $K_t \subset {\mathbb{H}} \cup {\mathbb{R}}$ , in this hydrodynamic limit. Next we consider the situation such that a half number of the slits start from $a>0$ and another half of slits start from $-a < 0$, and determine the multiple SLE in the hydrodynamic limit. After reporting these exact solutions, we will discuss the universal long-term behavior of the multiple SLE and its hull $K_t$ in the hydrodynamic limit.

math-ph↗

arXiv2017

Elliptic Determinantal Processes and Elliptic Dyson Models

We introduce seven families of stochastic systems of interacting particles in one-dimension corresponding to the seven families of irreducible reduced affine root systems. We prove that they are determinantal in the sense that all spatio-temporal correlation functions are given by determinants controlled by a single function called the spatio-temporal correlation kernel. For the four families ${A}_{N-1}$, ${B}_N$, ${C}_N$ and ${D}_N$, we identify the systems of stochastic differential equations solved by these determinantal processes, which will be regarded as the elliptic extensions of the Dyson model. Here we use the notion of martingales in probability theory and the elliptic determinant evaluations of the Macdonald denominators of irreducible reduced affine root systems given by Rosengren and Schlosser.

math.PR↗

arXiv2016

Elliptic Bessel processes and elliptic Dyson models realized as temporally inhomogeneous processes

The Bessel process with parameter $D>1$ and the Dyson model of interacting Brownian motions with coupling constant $β>0$ are extended to the processes in which the drift term and the interaction terms are given by the logarithmic derivatives of Jacobi's theta functions. They are called the elliptic Bessel process, eBES$^{(D)}$, and the elliptic Dyson model, eDYS$^{(β)}$, respectively. Both are realized on the circumference of a circle $[0, 2 πr)$ with radius $r >0$ as temporally inhomogeneous processes defined in a finite time interval $[0, t_*), t_* < \infty$. Transformations of them to Schrödinger-type equations with time-dependent potentials lead us to proving that eBES$^{(D)}$ and eDYS$^{(β)}$ can be constructed as the time-dependent Girsanov transformations of Brownian motions. In the special cases where $D=3$ and $β=2$, observables of the processes are defined and the processes are represented for them using the Brownian paths winding round a circle and pinned at time $t_*$. We show that eDYS$^{(2)}$ has the determinantal martingale representation for any observable. Then it is proved that eDYS$^{(2)}$ is determinantal for all observables for any finite initial configuration without multiple points. Determinantal processes are stochastic integrable systems in the sense that all spatio-temporal correlation functions are given by determinants controlled by a single continuous function called the spatio-temporal correlation kernel.

math.PR↗

arXiv2016

Percolation of optical excitation mediated by near-field interactions

Optical excitation transfer in nanostructured matter has been intensively studied in various material systems for versatile applications. Herein, we discuss the percolation of optical excitations in randomly organized nanostructures caused by optical near-field interactions governed by Yukawa potential in a two-dimensional stochastic model. The model results demonstrate the appearance of two phases of percolation of optical excitation as a function of the localization degree of near-field interaction. Moreover, it indicates sublinear scaling with percolation distance when the light localization is strong. The results provide fundamental insights into optical excitation transfer and will facilitate the design and analysis of nanoscale signal-transfer characteristics.

cond-mat.dis-nn↗

arXiv2016

Characterizations of the hydrodynamic limit of the Dyson model

Under appropriate conditions for the initial configuration, the empirical measure of the $N$-particle Dyson model with parameter $β\geq 1$ converges to a unique measure-valued process as $N$ goes to infinity, which is independent of $β$. The limit process is characterized by its Stieltjes transform called the Green's function. Since the Green's function satisfies the complex Burgers equation in the inviscid limit, this is called the hydrodynamic limit of the Dyson model. We review the relations among the hydrodynamic equation of the Green's function, the continuity equation of the probability density function, and the functional equation of the Green's function. The basic tools to prove the relations are the Hilbert transform, a special case of the Sokhotski-Plemelj theorem, and the method of characteristics for solving partial differential equations. For two special initial configurations, we demonstrate how to characterize the limit processes using these relations.

math.PR↗

arXiv2015

Characteristic Polynomials of Random Matrices and Noncolliding Diffusion Processes

We consider the noncolliding Brownian motion (BM) with $N$ particles starting from the eigenvalue distribution of Gaussian unitary ensemble (GUE) of $N \times N$ Hermitian random matrices with variance $σ^2$. We prove that this process is equivalent with the time shift $t \to t+σ^2$ of the noncolliding BM starting from the configuration in which all $N$ particles are put at the origin. In order to demonstrate nontriviality of such equivalence for determinantal processes, we show that, even from its special consequence, determinantal expressions are derived for the ensemble averages of products of characteristic polynomials of random matrices in GUE. Another determinantal process, noncolliding squared Bessel process with index $ν>-1$, is also studied in parallel with the noncolliding BM and corresponding results for characteristic polynomials are given for random matrices in the chiral GUE as well as in the Gaussian ensembles of class C and class D.

math.PR↗

arXiv2015

Phase Diagram of Collective Motion of Bacterial Cells in a Shallow Circular Pool

The collective motion of bacterial cells in a shallow circular pool is systematically studied using the bacterial species $Bacillus$ $subtilis$. The ratio of cell length to pool diameter (i.e., the reduced cell length) ranges from 0.06 to 0.43 in our experiments. Bacterial cells in a circular pool show various types of collective motion depending on the cell density in the pool and the reduced cell length. The motion is classified into six types, which we call random motion, turbulent motion, one-way rotational motion, two-way rotational motion, random oscillatory motion, and ordered oscillatory motion. Two critical values of reduced cell lengths are evaluated, at which drastic changes in collective motion are induced. A phase diagram is proposed in which the six phases are arranged.

physics.bio-ph↗

arXiv2015

Self-Elongation with Sequential Folding of a Filament of Bacterial Cells

Under hard-agar and nutrient-rich conditions, a cell of $Bacillus$ $subtilis$ grows as a single filament owing to the failure of cell separation after each growth and division cycle. The self-elongating filament of cells shows sequential folding processes, and multifold structures extend over an agar plate. We report that the growth process from the exponential phase to the stationary phase is well described by the time evolution of fractal dimensions of the filament configuration. We propose a method of characterizing filament configurations using a set of lengths of multifold parts of a filament. Systems of differential equations are introduced to describe the folding processes that create multifold structures in the early stage of the growth process. We show that the fitting of experimental data to the solutions of equations is excellent, and the parameters involved in our model systems are determined.,

cond-mat.soft↗