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The Starving Time - Random Walk Jamendo Album #003533 Tracklisting: 01 - The Schoolchildren's Blizzard 02 - Bin Yamin 03 - Fader 04 - Odessa Antiplague Station 05 - CI,LI 06 - Disquiet 07 - Malware 08 - Tachycardia 09 - Mean Streak (Live) 10 - Soft Loud Formula Please read the Readme.txt and License.txt files for important origin and licensing information.

The Starving Time - Random Walk Jamendo Album #003533 Tracklisting: 01 - The Schoolchildren's Blizzard 02 - Bin Yamin 03 - Fader 04 - Odessa Antiplague Station 05 - CI,LI 06 - Disquiet 07 - Malware 08 - Tachycardia 09 - Mean Streak (Live) 10 - Soft Loud Formula Please read the Readme.txt and License.txt files for important origin and licensing information.

This article is from Proteome Science , volume 9 . Abstract Background: Protein complexes are important entities to organize various biological processes in the cell, like signal transduction, gene expression, and molecular transmission. In most cases, proteins perform their intrinsic tasks in association with their specific interacting partners, forming protein complexes. Therefore, an enriched catalog of protein complexes in a cell could accelerate further research to elucidate the mechanisms underlying many biological processes. However, known complexes are still limited. Thus, it is a challenging problem to computationally predict protein complexes from protein-protein interaction networks, and other genome-wide data sets. Methods: Macropol et al. proposed a protein complex prediction algorithm, called RRW, which repeatedly expands a current cluster of proteins according to the stationary vector of a random walk with restarts with the cluster whose proteins are equally weighted. In the cluster expansion, all the proteins within the cluster have equal influences on determination of newly added protein to the cluster. In this paper, we extend the RRW algorithm by introducing a random walk with restarts with a cluster of proteins, each of which is weighted by the sum of the strengths of supporting evidence for the direct physical interactions involving the protein. The resulting algorithm is called NWE (Node-Weighted Expansion of clusters of proteins). Those interaction data are obtained from the WI-PHI database. Results: We have validated the biological significance of the results using curated complexes in the CYC2008 database, and compared our method to RRW and MCL (Markov Clustering), a popular clustering-based method, and found that our algorithm outperforms the other algorithms. Conclusions: It turned out that it is an effective approach in protein complex prediction to expand a cluster of proteins, each of which is weighted by the sum of the strengths of supporting evidence for the direct physical interactions involving the protein.

In this lecture we will design a method to randomly sample from a convex body, and this method will be a subroutine in approximately computing volume.

A suggestion by Meek et al. (1985) that the gravity wave field be viewed as stochastic, with significant nonlinearities, is applied to calculate diffusivities. The purpose here is to calculate the diffusivity for stochastic wave model and compare it with previous diffusivity estimates. The researchers do this for an idealized case in which the wind velocity changes but slowly, and for which saturation is the principal mechanism by which wave energy is lost. A related calculation was given in a very brief way (Weinstock, 1976), but the approximations were not fully justified, nor were the physical pre-suppositions clearly explained. The observations of Meek et al. (1985) have clarified the pre-suppositions for the researchers and provided a rationalization and improvement of the approximations employed.

In the latest episode of ECE Tech Talk, Dr.Talty describes his random walk of life. He provides detailed memories of his time at GM and Ford and how he ended up there. Dr.Talty also shares how he finds working at a university the closes thing to a fountain of youth. Mayank and Dhwan also promise Dr.Talty that after the pandemic they will gather and play darts. Enjoy!

Random quantum walk is a quantum thought experiment that I have generated and translated into a 3D model. See more explanation here: https://prezi.com/p/edit/k0xltuvk3wnm/

Random quantum walk is a quantum thought experiment that I have generated and translated into a 3D model. See more explanation here: https://prezi.com/p/edit/k0xltuvk3wnm/

Random walks are a fundamental tool used widely across several areas of computer science - theory, web algorithms, distributed networks, as well as mathematics and statistical physics. On the web and in distributed graphs, random walks are used for several algorithmic applications such as sampling, ranking, mining similarity, estimating connectivity, and graph partitioning. In the first part of the talk, I will present our work on performing random walks on large graphs presented as edge streams. The naive technique for performing a random walk of length ℓ requires O(ℓ) passes over the input stream. We present the first nontrivial technique that shows how to perform such walks in O(√ℓ) passes. We show how this technique can be used to estimate PageRank vectors in O(n) space and O(√M) passes, where n is the number of nodes in the graph, and M is the mixing time. In comparison, a standard implementation of PageRank requires O(n) space and O(M) passes. In subsequent work, we use this technique to obtain sub-linear time algorithms for computing random walks on distributed networks. These techniques have shown how to break past the longstanding linear-time barrier and perform random walks much more efficiently. In the latter part of the talk, I will briefly present work on understanding user feedback based ranking mechanisms employed on the web. Such ranking mechanisms are ubiquitous, for e.g., ranking on youtube, forums, social networks, digg, question-answering sites etc. The main metric used to evaluate a mechanism is the ranking accuracy vs. the cost of reviews. We show that for many reasonable probability models, the widely used thumbs (or stars) based mechanisms cannot produce approximately accurate rankings with bounded reviews per item. We provide a ranking mechanism based on pairwise comparisons which achieves approximate rankings with bounded cost. We have a system Shout Velocity (http://shoutvelocity.com), which is a twitter-like forum, and implements a comparison based ranking mechanism. ©2010 Microsoft Corporation. All rights reserved.

A local Random Walk Method (RWM) for potential problems governed by Lapalace's and Paragon's equations is developed for two- and three-dimensional problems. The RWM is implemented and demonstrated in a multiprocessor parallel environment on a Beowulf cluster of computers. A speed gain of 16 is achieved as the number of processors is increased from 1 to 23.

Randomness is a fundamental part of natural physical phenomena. Yet, it is often unappreciated how these stochastic processes affect our daily lives. On this program, Dr. Leonard Mlodinow discussed the random walk.

The damped random walk (DRW) model is increasingly used to model the variability in quasar optical light curves, but it is still uncertain whether the DRW model provides an adequate description of quasar optical variability across all time scales. Using a sample of OGLE quasar light curves, we consider four modifications to the DRW model by introducing additional parameters into the covariance function to search for deviations from the DRW model on both short and long time scales. We find good agreement with the DRW model on time scales that are well sampled by the data (from a month to a few years), possibly with some intrinsic scatter in the additional parameters, but this conclusion depends on the statistical test employed and is sensitive to whether the estimates of the photometric errors are correct to within ~10%. On very short time scales (below a few months), we see some evidence of the existence of a cutoff below which the correlation is stronger than the DRW model, echoing the recent finding of Mushotzky et al. (2011) using quasar light curves from Kepler. On very long time scales (> a few years), the light curves do not constrain models well, but are consistent with the DRW model.

The spatial search problem on regular lattice structures in integer number of dimensions $d\geq2$ has been studied extensively, using both coined and coinless quantum walks. The relativistic Dirac operator has been a crucial ingredient in these studies. Here we investigate the spatial search problem on fractals of non-integer dimensions. Although the Dirac operator cannot be defined on a fractal, we construct the quantum walk on a fractal using the flip-flop operator that incorporates a Klein-Gordon mode. We find that the scaling behavior of the spatial search is determined by the spectral (and not the fractal) dimension. Our numerical results have been obtained on the well-known Sierpinski gaskets in two and three dimensions.

We consider a one dimensional sub-ballistic random walk evolving in a parametric i.i.d. random environment. We study the asymptotic properties of the maximum likelihood estimator (MLE) of the parameter based on a single observation of the path till the time it reaches a distant site. In that purpose, we adapt the method developed in the ballistic case by Comets et al (2014) and Falconnet, Loukianova and Matias (2014). Using a supplementary assumption due to the specificity of the sub-ballistic regime, we prove consistency and asymptotic normality as the distant site tends to infinity. To emphazis the role of the additional assumption, we investigate the Temkin model with unknown support, and it turns out that the MLE is consistent but, unlike in the ballistic regime, the Fisher information is infinite. We also explore the numerical performance of our estimation procedure.

We consider a one dimensional sub-ballistic random walk evolving in a parametric i.i.d. random environment. We study the asymptotic properties of the maximum likelihood estimator (MLE) of the parameter based on a single observation of the path till the time it reaches a distant site. In that purpose, we adapt the method developed in the ballistic case by Comets et al (2014) and Falconnet, Loukianova and Matias (2014). Using a supplementary assumption due to the specificity of the sub-ballistic regime, we prove consistency and asymptotic normality as the distant site tends to infinity. To emphazis the role of the additional assumption, we investigate the Temkin model with unknown support, and it turns out that the MLE is consistent but, unlike in the ballistic regime, the Fisher information is infinite. We also explore the numerical performance of our estimation procedure.

Stefan van der Walt's talk at PyConZA 2013

Stefan van der Walt's talk at PyConZA 2013

For one-dimensional simple random walk in a general i.i.d. scenery and its limiting process we construct a coupling with explicit rate of approximation extending a recent result for Gaussian sceneries due to Khoshnevisan and Lewis. Furthermore we explicity identify the constant in the law of iterated logarithm.

Random walk on the set of irreducible representations of a finite group is investigated. For the symmetric and general linear groups, a sharp convergence rate bound is obtained and a cutoff phenomenon is proved. As related results, an asymptotic description of Plancherel measure of the finite general linear groups is given, and a connection of these random walks with quantum computing is noted.

We consider Sinai's walk in i.i.d. random scenery and focus our attention on a conjecture of R\'ev\'esz \cite{r05} concerning the upper limits of Sinai's walk in random scenery when the scenery is bounded from above. A close study of the competition between the concentration property for Sinai's walk and negative values for the scenery enables us to prove that the conjecture is true if the scenery has "thin" negative tails and is false otherwise.

Using a probabilistic interpretation of the Burau representation of the braid group offered by Vaughan Jones, we generalize the Burau representation to a representation of the semigroup of string links. This representation is determined by a linear system, and is dominated by finite type string link invariants. For positive string links, the representation matrix can be interpreted as the transition matrix of a Markov process. For positive non-separable links, we show that all states are persistent. (This is an extensively revised version of the first author's original paper titled "Burau representation and random walk on knots".)

In this paper, we reveal the branching structure for a non-homogeneous random walk with bounded jumps. The ladder time $T_1,$ the first hitting time of $[1,\infty)$ by the walk starting from $0,$ could be expressed in terms of a non-homogeneous multitype branching process. As an application of the branching structure, we prove a law of large numbers of random walk in random environment with bounded jumps and specify the explicit invariant density for the Markov chain of ``the environment viewed from the particle" .The invariant density and the limit velocity could be expressed explicitly in terms of the environment.

For a symmetric, homogeneous and irreducible random walk on d-dimensional integer lattice Z^d, having zero mean and a finite variance of jumps, we study the passage times (with possible infinite values) determined by the starting point x, the hitting state y and the taboo state z. We find the probability that these passages times are finite and analyze the tails of their cumulative distribution functions. In particular, it turns out that for the random walk on Z^d, except for a simple (nearest neighbor) random walk on Z, the order of the tail decrease is specified by dimension d only. In contrast, for a simple random walk on Z, the asymptotic properties of hitting times with taboo essentially depend on the mutual location of the points x, y and z. These problems originated in our recent study of branching random walk on Z^d with a single source of branching.

We define an analog of Plancherel measure for the set of rooted unlabeled trees on n vertices, and a Markov chain which has this measure as its stationary distribution. Using the combinatorics of commutation relations, we show that order n^2 steps are necessary and suffice for convergence to the stationary distribution.

Let $(B_t)_{0\leq t\leq T}$ be either a Bernoulli random walk or a Brownian motion with drift, and let $M_t:=\max\{B_s: 0\leq s\leq t\}$, $0\leq t\leq T$. This paper solves the general optimal prediction problem \sup_{0\leq\tau\leq T}\sE[f(M_T-B_\tau)], where the supremum is over all stopping times $\tau$ adapted to the natural filtration of $(B_t)$, and $f$ is a nonincreasing convex function. The optimal stopping time $\tau^*$ is shown to be of "bang-bang" type: $\tau^*\equiv 0$ if the drift of the underlying process $(B_t)$ is negative, and $\tau^*\equiv T$ is the drift is positive. This result generalizes recent findings by S. Yam, S. Yung and W. Zhou [{\em J. Appl. Probab.} {\bf 46} (2009), 651--668] and J. Du Toit and G. Peskir [{\em Ann. Appl. Probab.} {\bf 19} (2009), 983--1014], and provides additional mathematical justification for the dictum in finance that one should sell bad stocks immediately, but keep good ones as long as possible.

We introduce a new method for proving central limit theorems for random walk on nilpotent groups. The method is illustrated in a local central limit theorem on the Heisenberg group, weakening the necessary conditions on the driving measure. As a second illustration, the method is used to study walks on the $n\times n$ uni-upper triangular group with entries taken modulo $p$. The method allows sharp answers to the behavior of individual coordinates: coordinates immediately above the diagonal require order $p^2$ steps for randomness, coordinates on the second diagonal require order $p$ steps; coordinates on the $k$th diagonal require order $p^{\frac{2}{k}}$ steps.

We investigate the dynamic scaling properties of stochastic particle systems on a non-deterministic scale-free network. It has been known that the dynamic scaling behavior depends on the degree distribution exponent of the underlying scale-free network. Our study shows that it also depends on the global structure of the underlying network. In random walks on the tree structure scale-free network, we find that the relaxation time follows a power-law scaling $\tau\sim N$ with the network size $N$. And the random walker return probability decays algebraically with the decay exponent which varies from node to node. On the other hand, in random walks on the looped scale-free network, they do not show the power-law scaling. We also study a pair-annihilation process on the scale-free network with the tree and the looped structure, respectively. We find that the particle density decays algebraically in time both cases, but with the different exponent.

We show that p-adic analysis provides a quite natural basis for the description of relaxation in hierarchical glassy systems. For our purposes, we specify the Markov stochastic process considered by S. Albeverio and W. Karwowski. As a result we have obtained a random walk on p-adic integer numbers, which provide the generalization of Cayley tree proposed by Ogielski and Stein. The temperature-dependent power-law decay and the Kohlrausch law are derived.

We consider a model of loop-erased random walks on the finite pre-Sierpinski gasket which permits rigorous analysis. We prove the existence of the scaling limit and show that the path of the limiting process is almost surely self-avoiding, while having Hausdorff dimension strictly greater than 1. This result means that the path has infinitely fine creases, while having no self-intersection. Our loop-erasing procedure is formulated by a `larger-scale-loops-first' rule. It enables us to obtain exact recursion relations, making use of `self-similarity' of a fractal structure.

Neural network models offer a theoretical testbed for the study of learning at the cellular level. The only experimentally verified learning rule, Hebb's rule, is extremely limited in its ability to train networks to perform complex tasks. An identified cellular mechanism responsible for Hebbian-type long-term potentiation, the NMDA receptor, is highly versatile. Its function and efficacy are modulated by a wide variety of compounds and conditions and are likely to be directed by non-local phenomena. Furthermore, it has been demonstrated that NMDA receptors are not essential for some types of learning. We have shown that another neural network learning rule, the chemotaxis algorithm, is theoretically much more powerful than Hebb's rule and is consistent with experimental data. A biased random-walk in synaptic weight space is a learning rule immanent in nervous activity and may account for some types of learning -- notably the acquisition of skilled movement.

Neural network models offer a theoretical testbed for the study of learning at the cellular level. The only experimentally verified learning rule, Hebb's rule, is extremely limited in its ability to train networks to perform complex tasks. An identified cellular mechanism responsible for Hebbian-type long-term potentiation, the NMDA receptor, is highly versatile. Its function and efficacy are modulated by a wide variety of compounds and conditions and are likely to be directed by non-local phenomena. Furthermore, it has been demonstrated that NMDA receptors are not essential for some types of learning. We have shown that another neural network learning rule, the chemotaxis algorithm, is theoretically much more powerful than Hebb's rule and is consistent with experimental data. A biased random-walk in synaptic weight space is a learning rule immanent in nervous activity and may account for some types of learning -- notably the acquisition of skilled movement.

Neural network models offer a theoretical testbed for the study of learning at the cellular level. The only experimentally verified learning rule, Hebb's rule, is extremely limited in its ability to train networks to perform complex tasks. An identified cellular mechanism responsible for Hebbian-type long-term potentiation, the NMDA receptor, is highly versatile. Its function and efficacy are modulated by a wide variety of compounds and conditions and are likely to be directed by non-local phenomena. Furthermore, it has been demonstrated that NMDA receptors are not essential for some types of learning. We have shown that another neural network learning rule, the chemotaxis algorithm, is theoretically much more powerful than Hebb's rule and is consistent with experimental data. A biased random-walk in synaptic weight space is a learning rule immanent in nervous activity and may account for some types of learning -- notably the acquisition of skilled movement.

Testing out the video on my Canon SD110

We consider the biased random walk on a critical Galton-Watson tree conditioned to survive, and confirm that this model with trapping belongs to the same universality class as certain one-dimensional trapping models with slowly-varying tails. Indeed, in each of these two settings, we establish closely-related functional limit theorems involving an extremal process and also demonstrate extremal aging occurs.

We consider a transient excited random walk on $Z$ and study the asymptotic behavior of the occupation time of a currently most visited site. In particular, our results imply that, in contrast to the random walks in random environment, a transient excited random walk does not spend an asymptotically positive fraction of time at its favorite (most visited up to a date) sites.

One can view quantum mechanics as a generalization of classical probability theory that provides for pairwise interference among alternatives. Adopting this perspective, we ``quantize'' the classical random walk by finding, subject to a certain condition of ``strong positivity'', the most general Markovian, translationally invariant ``decoherence functional'' with nearest neighbor transitions.

Is a random walk algorithm and the code in openscad to generate it. The pictures include copies made in shapeways and printed at home every iteration is different. I added 5 renditions in stl format. Note: I sell this pieces in my shapeways store.

A simple random walk (SRW) on a graph is a Markov chain whose state space is the vertex set and the next state distribution is uniform among the neighbors of the current state. A graph is called recurrent if a SRW on it returns to the starting vertex with probability 1, and called transient otherwise. The path of a walk on a graph is simply the set of edges this walk has traversed. Our main result is that the path of a SRW on any graph is a recurrent graph. The proof uses the electrical network interpretation of random walks. We will give a sketch of the proof, including the necessary background, and discuss related questions and conjectures. Based on joint works with Itai Benjamini, Russell Lyons and Oded Schramm. ©2008 Microsoft Corporation. All rights reserved.

A simple random walk (SRW) on a graph is a Markov chain whose state space is the vertex set and the next state distribution is uniform among the neighbors of the current state. A graph is called recurrent if a SRW on it returns to the starting vertex with probability 1, and called transient otherwise. The path of a walk on a graph is simply the set of edges this walk has traversed. Our main result is that the path of a SRW on any graph is a recurrent graph. The proof uses the electrical network interpretation of random walks. We will give a sketch of the proof, including the necessary background, and discuss related questions and conjectures. Based on joint works with Itai Benjamini, Russell Lyons and Oded Schramm. ©2008 Microsoft Corporation. All rights reserved.

This talk will describe a general recipe for replacing discrete stochastic processes by deterministic analogues that satisfy the same first-order limit laws but have smaller fluctuations. The recipe will be applied to several illustrative problems in the study of random walk and random aggregation. In particular, a derandomized version of the internal diffusion-limited aggregation model in two dimensions gives rise to a growing blob that is remarkably close to circular and also displays intriguing internal structures (see http://www.math.wisc.edu/~propp/million.gif). This is joint work with Ander Holroyd and Lionel Levine. An early write-up of derandomized aggregation: www.math.wisc.edu/~propp/hidden/rotor Email-log of some messages I sent out about derandomized walk: www.math.wisc.edu/~propp/hidden/test/rotorwalk.to Lionel Levine's undergraduate thesis: www.math.berkeley.edu/~levine/rotorrouter.pdf Slides from a talk given by Lionel Levine: www.math.berkeley.edu/~levine/slides/ Lionel Levine and Adam Kampff's picture of the rotor-router aggregation blob after 270,000 particles have aggregated: www.math.berkeley.edu/~levine/private/rotorrouter/bigblob.bmp Two close-ups of that same picture: www.math.berkeley.edu/~levine/private/rotorrouter/closeup.bmp Ed Pegg's picture of the rotor-router blob after 750,000 particles have aggregated: www.math.wisc.edu/~propp/proppcircle.gif Ander Holroyd's picture of the rotor-router blob after 1,000,000 particles have aggregated: www.math.wisc.edu/~propp/million.gif Vishal Sanwalani's picture of the state achieved by the abelian sandpile model when sixty thousand grains have been added: www.math.wisc.edu/~propp/hidden/501.gif Hal Canary's applets for demonstrating derandomized walk and aggregation: http://ups.physics.wisc.edu/~hal/SSL/2003/ ©2003 Microsoft Corporation. All rights reserved.

Knowledge of the position of a mobile target has often to be ascertained from knowledge of the position at an earlier time and a model of the target motion. The models considered in this research contribution are based on random walk theory and are applicable where the motion is of a complex, time- varying character. Differences in the initial conditions assumed and minor variations in the model account for the different sections. In each case, density and distribution functions are derived for the coordinates of the target.

Financial time series are volatile, non-stationary and non-linear data that are affected by external economic factors. There is several performant predictive approaches such as univariate ARIMA model and more recently Recurrent Neural Network. The accurate forecasting of budget data is a strategic and challenging task for an optimal management of resources, it requires the use of the most accurate model. We propose a predictive approach that uses and compares the Machine Learning ARIMA model and Deep Learning Recurrent LSTM model. The application and the comparative analysis show that the LSTM model outperforms the ARIMA model, mainly thanks to the LSTMs ability to learn non-linear relationship from data.

This article is from Sensors (Basel, Switzerland) , volume 13 . Abstract An actual account of the angle random walk (ARW) coefficients of gyros in the constant rate biased rate ring laser gyro (RLG) inertial navigation system (INS) is very important in practical engineering applications. However, no reported experimental work has dealt with the issue of characterizing the ARW of the constant rate biased RLG in the INS. To avoid the need for high cost precise calibration tables and complex measuring set-ups, the objective of this study is to present a cost-effective experimental approach to characterize the ARW of the gyros in the constant rate biased RLG INS. In the system, turntable dynamics and other external noises would inevitably contaminate the measured RLG data, leading to the question of isolation of such disturbances. A practical observation model of the gyros in the constant rate biased RLG INS was discussed, and an experimental method based on the fast orthogonal search (FOS) for the practical observation model to separate ARW error from the RLG measured data was proposed. Validity of the FOS-based method was checked by estimating the ARW coefficients of the mechanically dithered RLG under stationary and turntable rotation conditions. By utilizing the FOS-based method, the average ARW coefficient of the constant rate biased RLG in the postulate system is estimated. The experimental results show that the FOS-based method can achieve high denoising ability. This method estimate the ARW coefficients of the constant rate biased RLG in the postulate system accurately. The FOS-based method does not need precise calibration table with high cost and complex measuring set-up, and Statistical results of the tests will provide us references in engineering application of the constant rate biased RLG INS.

Black and white photograph from the Photo Collection from Patricia Williams MacVeagh. This photograph features Tina Guignon and horse, Random Walk. Photo taken September 3, 1972 at the B & B (Busch & Baskowitz) Horse Show, St. Louis, MO. The Photo Collection of Patricia Williams MacVeagh was donated to the  National Sporting Library & Museum . File #: 0000BP-0036

We give a characterization of the line digraph of a regular digraph. We make use of the characterization, to show that the underlying digraph of a coined quantum random walk is a line digraph. We remark the connection between line digraphs and in-split graphs in symbolic dynamics.

We connect this question to a problem of estimating the probability that the image of certain random matrices does not intersect with a subset of the unit sphere $\mathbb{S}^{n-1}$. In this way, the case of a discretized Brownian motion is related to Gordon's escape theorem dealing with standard Gaussian matrices. The approach allows us to prove that with high probability, the $\pi/2$-covering time of certain random walks on $\mathbb{S}^{n-1}$ is of order $n$. For certain spherical simplices on $\mathbb{S}^{n-1}$, we extend the "escape" phenomenon to a broad class of random matrices; as an application, we show that $e^{Cn}$ steps are sufficient for the standard walk on $\mathbb{Z}^n$ to absorb the origin into its convex hull with a high probability.

We consider a one dimensional random walk in a random environment (RWRE) with a positive speed $\lim_{n\to\infty}\frac{X_n}{n}=v_\alpha>0$. Gantert and Zeitouni showed that if the environment has both positive and negative local drifts then the quenched slowdown probabilities $P_\omega(X_n < xn)$ with $x \in (0,v_\alpha)$ decay approximately like $\exp\{-n^{1-1/s}\}$ for a deterministic $s > 1$. More precisely, they showed that $n^{-\gamma} \log P_\omega( X_n < x n)$ converges to $0$ or $-\infty$ depending on whether $\gamma > 1-1/s$ or $\gamma < 1-1/s$. In this paper, we improve on this by showing that $n^{-1+1/s} \log P_\omega( X_n < x n)$ oscillates between $0$ and $-\infty$, almost surely. This had previously been shown by Gantert only in a very special case of random environments.

In this article we present an effective Hamiltonian approach for Discrete Time Quantum Random Walk. A form of the Hamiltonian for one dimensional quantum walk has been prescribed, utilizing the fact that Hamiltonians are the generators of time translations. Then an attempt has been made to generalize the techniques to higher dimensions. We find that the Hamiltonian can be written as the sum of a Weyl Hamiltonian and a Dirac comb potential. The time evolution operator obtained from this prescribed Hamiltonian is in complete agreement with that of the standard approach. But in higher dimension we find that the time evolution operator is additive, instead of being multiplicative \cite{Chandrasekhar:2013SREP08229}. We showed that in case of two-step walk, effectively the time evolution operator can have multiplicative form. In case of a square lattice, quantum walk has been studied computationally for different coins and the results for both the additive and the multiplicative approaches have been compared. Using the Graphene Hamiltonian the walk has been studied on a Graphene lattice and we conclude the preference of additive approach over the multiplicative one.

We introduce history-dependent discrete-time quantum random walk models by adding uncorrelated memory terms and also by modifying Hamiltonian of the walker to include couplings with memory-keeping agents. We next numerically study the correlation effects in these models. We also propose a correlation exponent as a relevant and promising tool for investigation of correlation or memory (hence non-Markovian) effects. Our analysis can easily be applied to more realistic models in which different regimes may emerge because of competition between different underlying physical mechanisms.

Let $(W_n(\theta))_{n\in\mathbb N_0}$ be the Biggins martingale associated with a supercritical branching random walk and denote by $W_\infty(\theta)$ its limit. Assuming essentially that the martingale $(W_n(2\theta))_{n\in\mathbb N_0}$ is uniformly integrable and that $\text{Var} W_1(\theta)$ is finite, we prove a functional central limit theorem for the tail process $(W_\infty(\theta) - W_{n+r}(\theta))_{r\in\mathbb N_0}$ and a law of the iterated logarithm for $W_\infty(\theta)-W_n(\theta)$, as $n\to\infty$.