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Naoki Kubota

Publications and source records attributed to Naoki Kubota.

16 recordsLinked to original sources
arXiv2026

Rate of divergence of time constant for frog model with vanishing initial density

The frog model with a Bernoulli initial configuration is an interacting particle system on the $d$-dimensional lattice ($d \geq 2$) with two types of particles: active and sleeping. Active particles perform independent simple random walks. In contrast, although the sleeping particles do not move at first, they become active and start moving once touched by the active particles. Initially, only the origin has a single active particle, and the other sites have sleeping particles according to a Bernoulli distribution. After the original active particle starts moving, further active particles are gradually generated under the above rule and propagate across the lattice. The time required for the propagation of active frogs is expected to increase as the parameter of the Bernoulli distribution decreases, since fewer frogs are available. The aim of this paper is to investigate this increase in the vanishing density limit. In particular, we observe that it diverges and the rate of divergence differs significantly between $d=2$ and $d \geq 3$.

math.PR↗
arXiv2024

Lipschitz-type estimate for the frog model with Bernoulli initial configuration

We consider the frog model with Bernoulli initial configuration, which is an interacting particle system on the multidimensional lattice consisting of two states of particles: active and sleeping. Active particles perform independent simple random walks. On the other hand, although sleeping particles do not move at first, they become active and can move around when touched by active particles. Initially, only the origin has one active particle, and the other sites have sleeping particles according to a Bernoulli distribution. Then, starting from the original active particle, active ones are gradually generated and propagate across the lattice, with time. It is of interest to know how the propagation of active particles behaves as the parameter of the Bernoulli distribution varies. In this paper, we treat the so-called time constant describing the speed of propagation, and prove that the absolute difference between the time constants for parameters $p,q \in (0,1]$ is bounded from above and below by multiples of $|p-q|$.

math.PR↗
arXiv2023

Upper tail large deviation for the one-dimensional frog model

In this paper, we study the upper tail large deviation for the one-dimensional frog model. In this model, sleeping and active frogs are assigned to vertices on $\mathbb Z$. While sleeping frogs do not move, the active ones move as independent simple random walks and activate any sleeping frogs. The main object of interest in this model is the asymptotic behavior of the first passage time ${\rm T}(0,n)$, which is the time needed to activate the frog at the vertex $n$, assuming there is only one active frog at $0$ at the beginning. While the law of large numbers and central limit theorems have been well established, the intricacies of large deviations remain elusive. Using renewal theory, Bérard and Ramírez have pointed out a slowdown phenomenon where the probability that the first passage time ${\rm T}(0,n)$ is significantly larger than its expectation decays sub-exponentially and lies between $\exp(-n^{1/2+o(1)})$ and $\exp(-n^{1/3+o(1)})$. In this article, using a novel covering process approach, we confirm that $1/2$ is the correct exponent, i.e., the rate of upper large deviations is given by $n^{1/2}$. Moreover, we obtain an explicit rate function that is characterized by properties of Brownian motion and is strictly concave.

math.PR↗
arXiv2022

Strict comparison for the Lyapunov exponents of the simple random walk in random potentials

We consider the simple random walk in i.i.d. nonnegative potentials on the $d$-dimensional cubic lattice $\mathbb{Z}^d$ ($d \geq 1$). In this model, the so-called Lyapunov exponent describes the cost of traveling for the simple random walk in the potential. The Lyapunov exponent depends on the distribution function of the potential, and the aim of this article is to prove that the Lyapunov exponent is strictly monotone in the distribution function of the potential with the order according to strict dominance. Furthermore, the comparison for the Lyapunov exponent also provides that for the rate function of this model.

math.PR↗
arXiv2022

Comparison of limit shapes for Bernoulli first-passage percolation

We consider Bernoulli first-passage percolation on the $d$-dimensional hypercubic lattice with $d \geq 2$. The passage time of edge $e$ is $0$ with probability $p$ and $1$ with probability $1-p$, independently of each other. Let $p_c$ be the critical probability for percolation of edges with passage time $0$. When $0\leq p<p_c$, there exists a nonrandom, nonempty compact convex set $\mathcal{B}_p$ such that the set of vertices to which the first-passage time from the origin is within $t$ is well-approximated by $t\mathcal{B}_p$ for all large $t$, with probability one. The aim of this paper is to prove that for $0\leq p<q<p_c$, the Hausdorff distance between $\mathcal{B}_p$ and $\mathcal{B}_q$ grows linearly in $q-p$. Moreover, we mention that the approach taken in the paper provides a lower bound for the expected size of the intersection of geodesics, that gives a nontrivial consequence for the \textit{critical} case.

math.PR↗
arXiv2022

Differences between Lyapunov exponents for the simple random walk in Bernoulli potentials

We consider the simple random walk on the $d$-dimensional lattice $\mathbb{Z}^d$ ($d \geq 1$), traveling in potentials which are Bernoulli distributed. The so-called Lyapunov exponent describes the cost of traveling for the simple random walk in the potential, and it is known that the Lyapunov exponent is strictly monotone in the parameter of the Bernoulli distribution. Hence, the aim of this paper is to investigate the effect of the potential on the Lyapunov exponent more precisely, and we derive some Lipschitz-type estimates for the difference between the Lyapunov exponents.

math.PR↗
arXiv2019

Gaussian fluctuation for superdiffusive elephant random walks

Elephant random walk is a kind of one-dimensional discrete-time random walk with infinite memory: For each step, with probability $α$ the walker adopts one of his/her previous steps uniformly chosen at random, and otherwise he/she performs like a simple random walk (possibly with bias). It admits phase transition from diffusive to superdiffusive behavior at the critical value $α_c=1/2$. For $α\in (α_c, 1)$, there is a scaling factor $a_n$ of order $n^α$ such that the position $S_n$ of the walker at time $n$ scaled by $a_n$ converges to a nondegenerate random variable $W$, whose distribution is not Gaussian. Our main result shows that the fluctuation of $S_n$ around $W \cdot a_n$ is still Gaussian. We also give a description of phase transition induced by bias decaying polynomially in time.

math.PR↗
arXiv2019

Continuity result for the rate function of the simple random walk on supercritical percolation clusters

We consider the simple random walk on supercritical percolation clusters in the multidimensional cubic lattice. In this model, a quenched large deviation principle holds for the position of the random walk. Its rate function depends on the law of the percolation configuration, and the aim of this paper is to study the continuity of the rate function in the law. To do this, it is useful that the rate function is expressed by the so-called Lyapunov exponent, which is the asymptotic cost paid by the random walk for traveling in a landscape of percolation configurations. In this context, we first observe the continuity of the Lyapunov exponent in the law of the percolation configuration, and then lift it to the rate function.

math.PR↗
arXiv2016

Deviation bounds for the first passage time in the frog model

We consider the so-called frog model with random initial configurations. The dynamics of this model is described as follows: Some particles are randomly assigned on any site of the multidimensional cubic lattice. Initially, only particles at the origin are active and these independently perform simple random walks. The other particles are sleeping and do not move at first. When sleeping particles are hit by an active particle, they become active and start moving in a similar fashion. The aim of this paper is to derive large deviation and concentration bounds for the first passage time at which an active particle reaches a target site.

math.PR↗
arXiv2016

Rate of convergence in first-passage percolation under low moments

We consider first-passage percolation on the $d$ dimensional cubic lattice for $d \geq 2$; that is, we assign independently to each edge $e$ a nonnegative random weight $t_e$ with a common distribution and consider the induced random graph distance (the passage time), $T(x,y)$. It is known that for each $x \in \mathbb{Z}^d$, $μ(x) = \lim_n T(0,nx)/n$ exists and that $0 \leq \mathbb{E}T(0,x) - μ(x) \leq C\|x\|_1^{1/2}\log \|x\|_1$ under the condition $\mathbb{E}e^{αt_e}<\infty$ for some $α>0$. By combining tools from concentration of measure with Alexander's methods, we show how such bounds can be extended to $t_e$'s with distributions that have only low moments. For such edge-weights, we obtain an improved bound $C (\|x\|_1 \log \|x\|_1)^{1/2}$ and bounds on the rate of convergence to the limit shape.

math.PR↗
arXiv2016

Concentrations for the simple random walk in unbounded nonnegative potentials

We consider the simple random walk in i.i.d. nonnegative potentials on the multidimensional cubic lattice. Our goal is to investigate the cost paid by the simple random walk for traveling from the origin to a remote location in a landscape of potentials. In particular, we obtain concentration inequalities for the travel cost in unbounded nonnegative potentials.

math.PR↗
arXiv2015

Upper bounds on the non-random fluctuations in first passage percolation with low moment conditions

We consider first passage percolation with i.i.d. weights on edges of the d-dimensional cubic lattice. Under the assumptions that a weight is equal to zero with probability smaller than the critical probability of bond percolation in the d-dimensional cubic lattice, and has moments bigger than 1, we investigate upper bounds on the so-called non-random fluctuations of the model. In addition, we give an application of our result to a lower bound for variance of the first passage percolation in the case where the limit shape has flat edges.

math.PR↗
arXiv2013

Quenched large deviations for multidimensional random walk in random environment with holding times

We consider a random walk in random environment with random holding times, that is, the random walk jumping to one of its nearest neighbors with some transition probability after a random holding time. Both the transition probabilities and the laws of the holding times are randomly distributed over the integer lattice. Our main result is a quenched large deviation principle for the position of the random walk. The rate function is given by the Legendre transform of the so-called Lyapunov exponents for the Laplace transform of the first passage time. By using this representation, we derive some asymptotics of the rate function in some special cases.

math.PR↗
arXiv2013

Quenched invariance principle for simple random walk on discrete point processes

We consider the simple random walk on random graphs generated by discrete point processes. This random graph has a random subset of a cubic lattice as the vertices and lines between any consecutive vertices on lines parallel to each coordinate axis as the edges. Under the assumption that discrete point processes are finitely dependent and stationary, we prove that the quenched invariance principle holds, that is, for almost every configuration of a point process, the path distribution of the walk converges weakly to that of a Brownian motion.

math.PR↗