Related papers: Weak limit theorem for a one-dimensional split-ste…

In this paper we present a new and flexible method to show that, in one dimension, various self-repellent random walks converge to self-repellent Brownian motion in the limit of weak interaction after appropriate space-time scaling. Our…

We implement the proof of principle for the quantum walk of one ion in a linear ion trap. With a single-step fidelity exceeding 0.99, we perform three steps of an asymmetric walk on the line. We clearly reveal the differences to its…

The behaviors of one-dimensional quantum random walks are strikingly different from those of classical ones. However, when decoherence is involved, the limiting distributions take on many classical features over time. In this paper, we…

In this paper, we prove convergence in distribution of Langevin processes in the overdamped asymptotics. The proof relies on the classical perturbed test function (or corrector) method, which is used both to show tightness in path space,…

Let $S=(S_k)_{k\geq 0}$ be a random walk on $\mathbb{Z}$ and $\xi=(\xi_{i})_{i\in\mathbb{Z}}$ a stationary random sequence of centered random variables, independent of $S$. We consider a random walk in random scenery that is the sequence of…

We consider a discrete-time 2-state quantum walk on the line. The state of the quantum walker evolves according to a rule which is determined by a coin-flip operator and a position-shift operator. In this paper we take a 3-periodic time…

We prove a weak law of large numbers for the capacity of the range of simple random walks on $\mathbb{Z}^{4}$. On $\mathbb{Z}^{3}$, we show that the capacity, properly scaled, converges in distribution towards the corresponding quantity for…

We consider 2-state quantum walks (QWs) on the line, which are defined by two matrices. One of the matrices operates the walk at only half-time. In the usual QWs, localization does not occur at all. However, our walk can be localized around…

We present an Ito's formula for the one-dimensional discrete-time quantum walk and give some examples including a Tanaka's formula by using the formula. Moreover we discuss integrals for the quantum walk.

We prove a conjecture of Toth and Veto about the weak convergence of the self repelling random walk with directed edges under diffusive scaling to a uniform distribution.

Consider the boundary case in a one-dimensional super-critical branching random walk. It is known that upon the survival of the system, the minimal position after $n$ steps behaves in probability like ${3\over 2} \log n$ when $n\to \infty$.…

Using a central limit theorem for arrays of interacting quantum systems, we give analytical expressions for the density of states and the partition function at finite temperature of such a system, which are valid in the limit of infinite…

We investigate the two-component quantum walk in one-dimensional lattice. We show that the inter-component interaction strength together with the hopping imbalance between the components exhibit distinct features in the quantum walk for…

We prove a functional limit theorem for Markov chains that, in each step, move up or down by a possibly state dependent constant with probability $1/2$, respectively. The theorem entails that the law of every one-dimensional regular…

We introduce the concept of a quantum walk with two particles and study it for the case of a discrete time walk on a line. A quantum walk with more than one particle may contain entanglement, thus offering a resource unavailable in the…

There has been much recent work on quantum inequalities to constrain negative energy. These are uncertainty principle-type restrictions on the magnitude and duration of negative energy densities or fluxes. We consider several examples of…

Quantum walks are quantum counterparts of random walks and their probability distributions are different from each other. A quantum walker distributes on a Hilbert space and it is observed at a location with a probability. The finding…

Photonics provide an efficient way to implement quantum walks, the quantum analogue of classical random walk that demonstrates rich physics with potential applications. However, most photonic quantum walks do not involve photon…

Some limit theorems of the type $\int_{\Omega}f_n dm_n -- --> \int_{\Omega}f dm$ are presented for scalar, (vector), (multi)-valued sequences of m_n-integrable functions f_n. The convergences obtained, in the vector and multivalued…

We continue the line of research of random walks with barrier initiated by Iksanov and M{\"o}hle (2008). Assuming that the tail of the step of the underlying random walk has a power-like behavior at infinity with exponent $-\alpha$,…