Related papers: Elementary potential theory on the hypercube

In this paper, we present a quantum Bernoulli noises approach to quantum walks on hypercubes. We first obtain an alternative description of a general hypercube and then, based on the alternative description, we find that the operators…

For an $n$-element subset $U$ of $\mathbb{Z}^2$, select $x$ from $U$ according to harmonic measure from infinity, remove $x$ from $U$, and start a random walk from $x$. If the walk leaves from $y$ when it first enters $U$, add $y$ to $U$.…

We consider one-dimensional random walks in random environment which are transient to the right. Our main interest is in the study of the sub-ballistic regime, where at time $n$ the particle is typically at a distance of order $O(n^\kappa)$…

We consider random walks on the surface of the sphere $S_{n-1}$ ($n \geq 2$) of the $n$-dimensional Euclidean space $E_n$, in short a hypersphere. By solving the diffusion equation in $S_{n-1}$ we show that the usual law $<r^2 > \varpropto…

One-dimensional problem for quantum harmonic oscillator with "regular+random" frequency subjected to the external "regular+random" force is considered. Averaged transition probabilities are found.

Let $(X_t, t \geq 0)$ be an $\alpha$-stable random walk with values in $\Z^d$. Let $l_t(x) = \int_0^t \delta_x(X_s) ds$ be its local time. For $p>1$, not necessarily integer, $I_t = \sum_x l_t^p(x)$ is the so-called $p$-fold self-…

A central feature of quantum metrology is the possibility of Heisenberg scaling, a quadratic improvement over the limits of classical statistics. This scaling, however, is notoriously fragile to noise. While for some noise types it can be…

We present an effective potential that allows quantum thermal expectation values of a position-dependent observable to be estimated as a classical ensemble average of the corresponding function. We follow the approach of Feynman and Hibbs,…

The separability and Runge-Lenz-type dynamical symmetry of the internal dynamics of certain two-electron Quantum Dots, found by Simonovi\'c et al. [1], is traced back to that of the perturbed Kepler problem. A large class of axially…

Let $S_n$ be a random walk with i.i.d. increments which have zero mean and finite variance. For every $x\ge0$ we define the stopping time $\tau_x:=\inf\{n\ge1:x+S_n\le0\}$ and consider the probabilities $\mathbb{P}(x+S_n\ge y,\tau_x>n)$. We…

We show exact dimensionality of harmonic measures associated with random walks on groups acting on a hyperbolic space under finite first moment condition, and establish the dimension formula by the entropy over the drift. We also treat the…

We give two algorithms that allow to get arbitrary precision asymptotics for the harmonic potential of a random walk.

In this paper we extend the construction of random walks with a prescribed Poisson boundary to the case of measures in the class of a generalized Gibbs state. The price for dropping the $\alpha$-quasiconformal assumptions is that we must…

We consider two dimensional random walks conditioned to stay in the positive quadrant. Assuming that the increments of the walk have finite second moments and that the drift vector is co-oriented with one of two axes, we construct positive…

We study the asymptotic probability that a random walk with heavy-tailed increments crosses a high boundary on a random time interval. We use new techniques to extend results of Asmussen [Ann. Appl. Probab. 8 (1998) 354-374] to completely…

We propose to view hermitian metrics on trivial holomorphic vector bundles $E\to\Omega$ as noncommutative analogs of functions defined on the base $\Omega$, and curvature as the notion corresponding to the Laplace operator or…

As an undergraduate exercise, in an article (2012 Am. J. Phys. $\bf{80}$ 780-14), quantum and classical uncertainties for dimensionless variables of position and momentum were evaluated in three potentials: infinite well, bouncing ball, and…

We look at random walks in Dirichlet environment. It was known that in dimension $d\geq 3$, if the walk is sub-ballistic, the displacement of the walk is polynomial of order $\kappa$ for some explicit $\kappa$. We show that the walk, after…

Numerical studies are presented to assess error estimates for a separable (Hartree) approximation for dynamically evolving composite quantum systems which exhibit distinct scales defined by their mass and frequency ratios. The relevant…

Let $\Gamma$ be a nonelementary hyperbolic group with a word metric $d$ and $\partial\Gamma$ its hyperbolic boundary equipped with a visual metric $d_a$ for some parameter $a>1$. Fix a superexponential symmetric probability $\mu$ on…