One-dimensional Random Walk Research Articles

Markov processes of infinitely many nonintersecting random walks

Consider an N-dimensional Markov chain obtained from N one-dimensional random walks by Doob h-transform with the q-Vandermonde determinant. We prove that as N becomes large, these Markov chains converge to an infinite-dimensional Feller Markov process. The dynamical correlation functions of the limit process are determinantal with an explicit correlation kernel. The key idea is to identify random point processes on $${\mathbb Z}$$ with q-Gibbs measures on Gelfand–Tsetlin schemes and construct Markov processes on the latter space. Independently, we analyze the large time behavior of PushASEP with finitely many particles and particle-dependent jump rates (it arises as a marginal of our dynamics on Gelfand–Tsetlin schemes). The asymptotics is given by a product of a marginal of the GUE-minor process and geometric distributions.

Moderate Deviations for Stable Random Walks in Random Scenery

In this paper, a moderate deviation theorem for one-dimensional stable random walks in random scenery is proved. The proof relies on the analysis of maximum local times of stable random walks, and the comparison of moments between random walks in random scenery and self-intersection local times of the underlying random walks.

Moderate Deviations for Stable Random Walks in Random Scenery

In this paper, a moderate deviation theorem for one-dimensional stable random walks in random scenery is proved. The proof relies on the analysis of maximum local times of stable random walks, and the comparison of moments between random walks in random scenery and self-intersection local times of the underlying random walks.

Computational models for large-scale simulations of facilitated diffusion.

The binding of site-specific transcription factors to their genomic target sites is a key step in gene regulation. While the genome is huge, transcription factors belong to the least abundant protein classes in the cell. It is therefore fascinating how short the time frame is that they require to home in on their target sites. The underlying search mechanism is called facilitated diffusion and assumes a combination of three-dimensional diffusion in the space around the DNA combined with one-dimensional random walk on it. In this review, we present the current understanding of the facilitated diffusion mechanism and identify questions that lack a clear or detailed answer. One way to investigate these questions is through stochastic simulation and, in this manuscript, we support the idea that such simulations are able to address them. Finally, we review which biological parameters need to be included in such computational models in order to obtain a detailed representation of the actual process.

Large deviations and slowdown asymptotics for one-dimensional excited random walks

We study the large deviations of excited random walks on $\mathbb{Z}$. We prove a large deviation principle for both the hitting times and the position of the random walk and give a qualitative description of the respective rate functions. When the excited random walk is transient with positive speed $v_0$, then the large deviation rate function for the position of the excited random walk is zero on the interval $[0,v_0]$ and so probabilities such as $P(X_n 2$ is the expected total drift per site of the cookie environment.

Random walks with unbounded jumps among random conductances I: Uniform quenched CLT

We study a one-dimensional random walk among random conductances, with unbounded jumps. Assuming the ergodicity of the collection of conductances and a few other technical conditions (uniform ellipticity and polynomial bounds on the tails of the jumps) we prove a quenched uniform invariance principle for the random walk. This means that the rescaled trajectory of length n is (in a certain sense) close enough to the Brownian motion, uniformly with respect to the choice of the starting location in an interval of length $O(\sqrt{n})$ around the origin.

A combinatorial result with applications to self-interacting random walks

We give a series of combinatorial results that can be obtained from any two collections (both indexed by Z × N ) of left and right pointing arrows that satisfy some natural relationship. When applied to certain self-interacting random walk couplings, these allow us to reprove some known transience and recurrence results for some simple models. We also obtain new results for one-dimensional multi-excited random walks and for random walks in random environments in all dimensions.

Recurrence and Pólya Number of General One-Dimensional Random Walks

The recurrence properties of random walks can be characterized by Pólya number, i.e., the probability that the walker has returned to the origin at least once. In this paper, we consider recurrence properties for a general 1D random walk on a line, in which at each time step the walker can move to the left or right with probabilities l and r, or remain at the same position with probability o (l + r + o = 1). We calculate Pólya number P of this model and find a simple expression for P as, P = 1 − Δ, where Δ is the absolute difference of l and r (Δ = |l − r|). We prove this rigorous expression by the method of creative telescoping, and our result suggests that the walk is recurrent if and only if the left-moving probability l equals to the right-moving probability r.

Optical implementation of quantum random walks using weak cross-Kerr media

Weak cross-Kerr media provides additional degrees of freedom of qubits in quantum information processing. In this paper, by exploiting weak cross-Kerr nonlinearity, we propose an optical implementation scheme of one-dimensional quantum random walks. The random walks are described by the interaction of single photons with cross-Kerr media. The proposed scheme can also be used to implement one-dimensional quantum random walks on an infinite line.

On the estimation of the potential of Sinai’s RWRE

We consider a one-dimensional random walk in random environment. We prove that the logarithm of the local time can be used as an estimator of the random environment. We give a constructive method allowing us to locally built, up to a translation, the random potential associated to the environment from a single trajectory of the random walk.

Asymptotic variance of the self-intersections of stable random walks using Darboux-Wiener theory

We present a Darboux-Wiener type lemma as a powerful alternative to the classical Tauberian theorem when monotonicity is not known a priori. We apply it to obtain the exact asymptotics of the variance of the self-intersections of a one-dimensional stable random walk. Finally we prove a functional central limit theorem for stable random walk in random scenery conjectured in [1].

Distribution of first-passage times to specific targets on compactly explored fractal structures

The distribution of the first passage times (FPT) of a one-dimensional random walker to a target site follows a power law F(t)~t(-3/2). We generalize this result to another situation pertinent to compact exploration and consider the FPT of a random walker with specific source and target points on an infinite fractal structure with spectral dimension d(s)<2. We show that the probability density of the first return to the origin has the form F(t)~t(d(s)/2-2), and the FPT to a specific target at distance r follows the law F(r,t)~r(d(w)-d(f)) t(d(s)/2-2), where d(w) and d(f) are the walk dimension and the fractal dimension of the structure, respectively. The distance dependence of F(r,t) reproduces the one of the mean FPT of a random walk in a confined domain.

Power-law analysis of surface-plasmon-enhanced electromagnetic field dependence of blinking SERS of thiacyanine or thiacarbocyanine adsorbed on single silver nanoaggregates

Blinking statistics in surface-enhanced Raman scattering (SERS) of thiacyanine or thiacarbocyanine adsorbed on single Ag nanoaggregates were analyzed by a power law. A power law reproduces the probability distributions of both the bright and dark SERS occurrences against their duration times. As the localized surface plasmon resonance (LSPR) wavelength of a single Ag nanoaggregate approached the excitation wavelength or the excitation laser intensity increases, the power-law exponents were close to -1.5, a value derived from a one-dimensional random walk model. When the LSPR wavelength left the excitation wavelength or the excitation laser intensity decreases, the power-law exponents deviated from -1.5. The decrease in the power-law exponents in the bright SERS, which indicates a decrease in the probabilities of the long-lived bright SERS, and the increase in the power-law exponents in the dark SERS coincide with the increasing shallowness and narrowing of a optical trapping potential well due to a surface-plasmon-enhanced electromagnetic field around a junction of the Ag nanoaggregates excited at a wavelength apart from the LSPR wavelength or under the low laser intensity, i.e., the low original electromagnetic field, respectively.

Correlation property and electronic transfer propertyof Cantor-styled artificial DNA sequence

For the Cantor-styled artificial DNA sequence constructed by following the inflation rule A→ABA and B→BBB, using the one-dimensional random walk model and Hurst analysis, we calculate numerically the displacement and its standard deviation, the rescaled range function and the Hurst exponent. The results are compared with those of one-dimensional random binary sequence. We show that Cantor-styled DNA sequence presents correlated behavior, scaling invariability, and self-similarity. In addition, according to the renormalization-group method, we study the charge transfer properties of Cantor-styled DNA sequence, including localization length, charge transmission coefficient and Lyapunov exponent. We find that some resonant peaks can survive in relatively long Cantor-styled DNA sequences, which also implies that there are long-range correlations in Cantor-styled DNA sequences.

Diffusion between evolving interfaces

Diffusion in an evolving environment is studied by continuous-time Monte Carlosimulations. Diffusion is modeled by continuous-time random walkers on a lattice, in adynamic environment provided by bubbles between two one-dimensional interfaces drivensymmetrically towards each other. For one-dimensional random walkers constrained by theinterfaces, the bubble size distribution dominates diffusion. For two-dimensional randomwalkers, it is also controlled by the topography and dynamics of the interfaces. The resultsof the one-dimensional case are recovered in the limit where the interfaces are stronglydriven. Even with simple hard-core repulsion between the interfaces and the particles,diffusion is found to depend strongly on the details of the dynamical rules of particles closeto the interfaces.

Transience/recurrence and the speed of a one-dimensional random walk in a “have your cookie and eat it” environment

Considérons une variante de la marche aléatoire simple et symétrique sur les entiers, avec le mécanisme de transition suivant: A chaque site x, la probabilité de sauter à droite est ω(x)∈[½, 1), jusqu’à la première fois que le processus saute à gauche du site x, après laquelle la probabilité de sauter à droite est ½. Nous examinons les propriétés de transience/recurrence pour ce processus, dans les environnements déterministes et aussi dans les environnements stationnaires et ergodiques {ω(x)}x∈Z. Dans les environnements déterministes, nous étudions aussi la vitesse du processus.

Current fluctuations of a system of one-dimensional random walks in random environment

We study the current of particles that move independently in a common static random environment on the one-dimensional integer lattice. A two-level fluctuation picture appears. On the central limit scale the quenched mean of the current process converges to a Brownian motion. On a smaller scale the current process centered at its quenched mean converges to a mixture of Gaussian processes. These Gaussian processes are similar to those arising from classical random walks, but the environment makes itself felt through an additional Brownian random shift in the spatial argument of the limiting current process.

Loops, tails and trains: A simple model for structural transformations of grafted adsorbing neutral polymer brushes

The structure of grafted adsorbing polymers on surfaces is described as a statistical ensemble of loops generated by an one-dimensional random walk perpendicular to the surface. The configuration of each chain is considered as a succession of closed loops ended by an open loop (a tail). The probability of formation of each individual loop is the product between the probability of first return to the surface and a Boltzmann factor containing the free energy of the Flory-Huggins kind, which is approximated by the minimum free energy of all possible configurations of that loop. At high grafting densities, the attractive interactions between monomers and surface control the fraction of polymer belonging to either closed loops or tails, hence the formation of a stretched grafted brush. At low grafting densities, the increase of that interaction above a critical value generates an abrupt collapse of the brush on the surface. Whereas for long polymers (with more than about 100 Kuhn segments), the structure of the brush can be determined, in general, only via Monte-Carlo sampling, it is argued that the two structural transitions indicated above can be well predicted by simple approximations.

Grain-Boundary Fluctuations in Two-Dimensional Colloidal Crystals

We study grain-boundary fluctuations in two-dimensional colloidal crystals in real space and time using video microscopy. The experimentally obtained static and dynamic correlation functions are very well described by expressions obtained using capillary wave theory. This directly leads to values for the interfacial stiffness and the interface mobility, the key parameters in curvature-driven grain-boundary migration. Furthermore, we show that the average grain-boundary position exhibits a one-dimensional random walk as recently suggested by computer simulations [Z. T. Trautt, M. Upmanyu, and A. Karma, Science 314, 632 (2006)]. The interface mobility determined from the mean-square displacement of the average grain-boundary position is in good agreement with values inferred from grain-boundary fluctuations.

Weak convergence for the minimal position in a branching random walk: A simple proof

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 $$ \frac{3} {2} $$ log n when n → ∞. We give a simple and self-contained proof of this result, based exclusively on elementary properties of sums of i.i.d. real-valued random variables.