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Kenkichi Tsunoda

Publications and source records attributed to Kenkichi Tsunoda.

17 recordsLinked to original sources
arXiv2026

Moderate deviations and laws of the iterated logarithm for geometric functionals in the sparse regime

We prove moderate deviation principles and laws of the iterated logarithm in the sparse regime for geometric and topological functionals associated with $k$-point connected components. These limit theorems are established at both the measure-valued and vector-valued levels, for Poisson and binomial point processes. As applications, we derive corresponding results for component counts in random geometric graphs and Čech complexes and for counts of Morse critical points.

math.PR↗
arXiv2026

Hydrodynamic limit of exclusion processes with killing on weighted Riemannian manifolds via a graph discretization

In the present paper, we consider an exclusion process with killing on a proximity graph constructed from a partition of a geodesically complete weighted Riemannian manifold. We rigorously derive its hydrodynamic equation, which is governed by a heat equation with a killing potential term on the manifold. Within our graph discretization framework, we define an empirical density field for the exclusion process and prove its convergence, under an appropriate space-time scaling, to the unique bounded weak solution of the hydrodynamic equation, provided that the weighted manifold is stochastically complete. Combining local regularity for the parabolic equation with techniques from stochastic analysis on manifolds, we prove the uniqueness of the bounded weak solution and give the explicit representation of the solution in terms of the minimal Schrödinger kernel. Our result requires neither a global lower Ricci curvature bound nor global boundedness of the weight function.

math.PR↗
arXiv2026

Critical stationary fluctuations in reaction--diffusion processes

We study stationary fluctuations at criticality for a one-dimensional reaction--diffusion process combining symmetric simple exclusion dynamics with Glauber-type spin flips. The strength of the Glauber interaction is tuned to the critical regime in which the quadratic term in the effective potential vanishes. Focusing on the stationary distribution, we show that the total magnetization scaled by $n^{3/4}$ exhibits non-Gaussian fluctuations. More precisely, we prove that under the invariant measure the rescaled magnetization converges in distribution to a random variable with density proportional to $\exp\{-2(θy^2 + y^4/2)\}$. In contrast with the previous result, we show that the density field acting on the faster modes, that is, those associated to zero-mean test functions, have much smaller Gaussian fluctuations. It follows from the previous two results that the rescaled density field projects onto the magnetization in the sense that its action on zero-mean test functions vanishes in the limit.

math.PR↗
arXiv2026

Hydrodynamic limit for Glauber-Kawasaki dynamics on the Sierpiński gasket

We prove the hydrodynamic limit for Glauber-Kawasaki dynamics on the Sierpiński gasket, a prototypical fractal graph that lacks translational invariance. The main novelty lies in incorporating Glauber dynamics, allowing for particle creation and annihilation with birth-death rates depending locally on the particle configuration. In the macroscopic limit, the particle density evolves according to a nonlinear reaction--diffusion equation, where the reaction term is explicitly determined by the microscopic rates. The key new ingredient is a replacement lemma adapted to the fractal geometry of the Sierpiński gasket. We establish this lemma by deriving 1-block and 2-blocks estimates on the Sierpiński gasket graph, which require new arguments due to the absence of classical lattice structures.

math.PR↗
arXiv2024

Sharp interface limit for a quasi-linear large deviation rate function

We discuss the sharp interface limit, leading to a mean curvature flow energy, for the rate function of the large deviation principle of a Glauber+Kawasaki process with speed change. We provide an explicit formula of the limiting functional given by the mobility and the transport coefficient.

math.AP↗
arXiv2024

Incompressible limit for weakly asymmetric simple exclusion processes coupled through collision

We establish the incompressible limit of weakly asymmetric simple exclusion processes coupled through particle collisions. The incompressible limit depends on various parameters in the particle system and is linked to fluid dynamics equations. Our main contributions to previous results are the extension of the parameter space and the focus on local particle jumps. Our proof uses the relative entropy method. The main novelties in the proof are a Boltzmann-Gibbs principle (a replacement lemma) and a spectral gap estimate.

math.PR↗
arXiv2022

Large deviation principle for persistence diagrams of random cubical filtrations

The objective of this article is to investigate the asymptotic behavior of the persistence diagrams of a random cubical filtration as the window size tends to infinity. Here, a random cubical filtration is an increasing family of random cubical sets, which are the union of randomly generated higher-dimensional unit cubes with integer coordinates in a Euclidean space. We first prove the strong law of large numbers for the persistence diagrams, inspired by the work of Hiraoka, Shirai, and Trinh, where the persistence diagram of a filtration of random geometric complexes is considered. As opposed to prior papers treating limit theorems for persistence diagrams, the present article aims to further study the large deviation behavior of persistence diagrams. We prove a large deviation principle for the persistence diagrams of a class of random cubical filtrations, and show that the rate function is given as the Fenchel--Legendre transform of the limiting logarithmic moment generating function. In the proof, we also establish a general method of lifting a large deviation principle for the tuples of persistent Betti numbers to persistence diagrams for broad application.

math.PR↗
arXiv2022

Motion by mean curvature from Glauber-Kawasaki dynamics with speed change

We derive a continuum mean-curvature flow as a certain hydrodynamic scaling limit of Glauber-Kawasaki dynamics with speed change. The Kawasaki part describes the movement of particles through particle interactions. It is speeded up in a diffusive space-time scaling. The Glauber part governs the creation and annihilation of particles. The Glauber part is set to favor two levels of particle density. It is also speeded up in time, but at a lesser rate than the Kawasaki part. Under this scaling, a mean-curvature interface flow emerges, with a homogenized `surface tension-mobility' parameter reflecting microscopic rates. The interface separates the two levels of particle density. Similar hydrodynamic limits have been derived in two recent papers; one where the Kawasaki part describes simple nearest neighbor interactions, and one where the Kawasaki part is replaced by a zero-range process. We extend the main results of these two papers beyond nearest-neighbor interactions. The main novelty of our proof is the derivation of a `Boltzmann-Gibbs' principle which covers a class of local particle interactions.

math.PR↗
arXiv2022

Glauber-Exclusion dynamics : rapid mixing regime

We show that for any attractive Glauber-Exclusion process on the one-dimensional lattice of size $N$ with periodic boundary condition, if the corresponding hydrodynamic limit equation has a reaction term with a strictly convex potential, then the total-variation mixing time is of order $O(\log N)$. In particular, the result covers the full high-temperature regime in the original model introduced by De Masi, Ferrari and Lebowitz (1985).

math.PR↗
arXiv2022

Constant-speed interface flow from unbalanced Glauber-Kawasaki dynamics

We derive the hydrodynamic limit of Glauber-Kawasaki dynamics. The Kawasaki part is simple and describes independent movement of the particles with hard core exclusive interactions. It is speeded up in a diffusive space-time scaling. The Glauber part describes the birth and death of particles. It is set to favor two levels of particle density with a preference for one of the two. It is also speeded up in time, but at a lesser rate than the Kawasaki part. Under this scaling, the limiting particle density instantly takes either of the two favored density values. The interface which separates these two values evolves with constant speed (Huygens' principle). Similar hydrodynamic limits have been derived in four recent papers. The crucial difference with these papers is that we consider Glauber dynamics which has a preferences for one of the two favored density values. As a result, we observe limiting dynamics on a shorter time scale, and the evolution is different from the mean curvature flow obtained in the four previous papers. While several steps in our proof can be adopted from these papers, the proof for the propagation of the interface is new.

math.PR↗
arXiv2021

Exponentially slow mixing and hitting times of rare events for a reaction--diffusion model

We consider the superposition of symmetric simple exclusion dynamics speeded-up in time, with spin-flip dynamics in a one-dimensional interval with periodic boundary conditions. We show that the mixing time has an exponential lower bound in the system size if the potential of the hydrodynamic equation has more than two local minima. We also apply our estimates to show that the normalized hitting times of rare events converge to a mean one exponential random variable if the potential has a unique minimum.

math.PR↗
arXiv2019

Motion by mean curvature from Glauber-Kawasaki dynamics

We study the hydrodynamic scaling limit for the Glauber-Kawasaki dynamics. It is known that, if the Kawasaki part is speeded up in a diffusive space-time scaling, one can derive the Allen-Cahn equation which is a kind of the reaction-diffusion equation in the limit. This paper concerns the scaling that the Glauber part, which governs the creation and annihilation of particles, is also speeded up but slower than the Kawasaki part. Under such scaling, we derive directly from the particle system the motion by mean curvature for the interfaces separating sparse and dense regions of particles as a combination of the hydrodynamic and sharp interface limits.

math.PR↗
arXiv2019

Hydrostatic limit for exclusion process with slow boundary revisited

We revisit in this short article the hydrostatic limit for the exclusion process with slow boundary. The original proof of this result relies on estimates of the correlation functions. We achieve the same result based on analysis of two different time scales, which do not need any information about the correlation functions.

math.PR↗
arXiv2018

Strong law of large numbers for Betti numbers in the thermodynamic regime

We establish the strong law of large numbers for Betti numbers of random Čech complexes built on $\mathbb R^N$-valued binomial point processes and related Poisson point processes in the thermodynamic regime. Here we consider both the case where the underlying distribution of the point processes is absolutely continuous with respect to the Lebesgue measure on $\mathbb R^N$ and the case where it is supported on a $C^1$ compact manifold of dimension strictly less than $N$. The strong law is proved under very mild assumption which only requires that the common probability density function belongs to $L^p$ spaces, for all $1\leq p < \infty$.

math.PR↗
arXiv2018

Static large deviations for a reaction-diffusion model

We consider the superposition of a symmetric simple exclusion dynamics, speeded-up in time, with a spin-flip dynamics in a one-dimensional interval with periodic boundary conditions. We prove the large deviations principle for the empirical measure under the stationary state. We deduce from this result that the stationary state is concentrated on the stationary solutions of the hydrodynamic equation which are stable.

math.PR↗
arXiv2018

Limit theorems for random cubical homology

This paper studies random cubical sets in $\mathbb{R}^d$. Given a cubical set $X\subset \mathbb{R}^d$, a random variable $ω_Q\in[0,1]$ is assigned for each elementary cube $Q$ in $X$, and a random cubical set $X(t)$ is defined by the sublevel set of $X$ consisting of elementary cubes with $ω_Q\leq t$ for each $t\in[0,1]$. Under this setting, the main results of this paper show the limit theorems (law of large numbers and central limit theorem) for Betti numbers and lifetime sums of random cubical sets and filtrations. In addition to the limit theorems, the positivity of the limiting Betti numbers is also shown.

math.PR↗