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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

A very nice release by God Generator. This is recommended for everyone who likes weird bleeps, lo-fi goodness and effect fetishism. I also want to say sorry to the band itself because of the long time it took to get this online, but here it is finally!

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.

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.

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.

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!

Includes bibliographical references (p. 15-17)

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.

Original source:  Random animal walk cycle by Chibixi on DeviantArt Original description: a cat-ish thing walking 

We consider a model for random walks on random environments (RWRE) with random subset of the d-dimensional Euclidean lattice as the vertices, and uniform transition probabilities on 2d points (two "coordinate nearest points" in each of the d coordinate directions). We prove that the velocity of such random walks is almost surely 0, and give partial characterization of transience and recurrence for the different dimensions. Finally we prove Central Limit Theorem for such random walks, under a condition on the distance between nearest coordinate nearest points.

We study the effect of a single excluded site on the diffusion of a particle undergoing random walk in a d-dimensional lattice. The determination of the characteristic function allows to find explicitly the asymptotical behaviour of physical quantities such as the particle average position (drift) and the mean square deviation. Contrarily to the one-dimensional case, where the average coordinate diverges at infinite times as t**1/2 and where the diffusion constant D is changed due to the impurity, the effects of the latter are shown to be much less important in higher dimensions: for d=842, the position is simply shifted by a constant and the diffusion constant remains unaltered although dynamical corrections (logarithmic for d=2) still occur. Finally, the continuum space version of the model is analyzed; it is shown that d=1 is the lower dimensionality above which all the effects of the forbidden site are irrelevant.

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/

We propose to calculate bosonic and fermionic determinants with some general field background, and the corresponding 1-loop effective actions by evaluating random walk worldline loops generated statistically on the lattice. This is illustrated by some numerical calculations for constant gauge field backgrounds and then discussed for the general case.

Edge-reinforced random walk (ERRW), introduced by Coppersmith and Diaconis in 1986, is a random process, which takes values in the vertex set of a graph G, and is more likely to cross edges it has visited before. We show that it can be represented in terms of a Vertex-reinforced jump process (VRJP) with independent gamma conductances: the VRJP was conceived by Werner and first studied by Davis and Volkov (2002,2004), and is a continuous-time process favouring sites with more local time. We calculate, for any finite graph G, the limiting measure of the centred occupation time measure of VRJP, and interpret it as a supersymmetric hyperbolic sigma model in quantum field theory. This enables us to deduce that VRJP and ERRW are strongly recurrent in any dimension for large reinforcement, using a localisation result of Disertori and Spencer (2010).

Personalized recommender systems rely on personal usage data of each user in the system. However, privacy policies protecting users' rights prevent this data of being publicly available to a wider researcher audience. In this work, we propose a memory biased random walk model (MBRW) based on real clickstream graphs, as a generator of synthetic clickstreams that conform to statistical properties of the real clickstream data, while, at the same time, adhering to the privacy protection policies. We show that synthetic clickstreams can be used to learn recommender system models which achieve high recommender performance on real data and at the same time assuring that strong de-minimization guarantees are provided.

We consider the problem on nearly favorite points of simple random walk in $\mathbb{Z}^2$ which Dembo et al. suggested. We determine the power exponents for the numbers of pairs of $\alpha$-favorite points. These are conjectured to coincide with the exponents of the corresponding quantity for late points and the one for high points of the Gaussian free field for which their exact values are known. Our result verifies in almost sure sense. We also estimate this value in average and obtain the coincidence of it with the corresponding ones for the Gaussian free field.

In the information overloaded web, personalized recommender systems are essential tools to help users find most relevant information. The most heavily-used recommendation frameworks assume user interactions that are characterized by a single relation. However, for many tasks, such as recommendation in social networks, user-item interactions must be modeled as a complex network of multiple relations, not only a single relation. Recently research on multi-relational factorization and hybrid recommender models has shown that using extended meta-paths to capture additional information about both users and items in the network can enhance the accuracy of recommendations in such networks. Most of this work is focused on unweighted heterogeneous networks, and to apply these techniques, weighted relations must be simplified into binary ones. However, information associated with weighted edges, such as user ratings, which may be crucial for recommendation, are lost in such binarization. In this paper, we explore a random walk sampling method in which the frequency of edge sampling is a function of edge weight, and apply this generate extended meta-paths in weighted heterogeneous networks. With this sampling technique, we demonstrate improved performance on multiple data sets both in terms of recommendation accuracy and model generation efficiency.

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 …

We consider a class of inhomogeneous media known as composite media that is often encountered in experimental sciences and investigate the persistence probability of a random walker in such a system. Analytical and numerical results for the crossover time scales has been obtained for a composite system with two homogeneous components and three homogeneous components respectively.

We study the entropy of the set traced by an $n$-step random walk on $\Z^d$. We show that for $d \geq 3$, the entropy is of order $n$. For $d = 2$, the entropy is of order $n/\log^2 n$. These values are essentially governed by the size of the boundary of the trace.

We apply results from Baryshnikov, Brady, Bressler and Pemantle (2008) to compute limiting probability profiles for various quantum random walks in one and two dimensions. Using analytic machinery we show some features of the limit distribution that are not evident in an empirical intensity plot of the time 10,000 distribution. Some conjectures are stated and computational techniques are discussed as well.

Some asymptotic properties of a Brownian motion in multifractal time, also called multifractal random walk, are established. We show the almost sure and $L^1$ convergence of its structure function. This is an issue directly connected to the scale invariance and multifractal property of the sample paths. We place ourselves in a mixed asymptotic setting where both the observation length and the sampling frequency may go together to infinity at different rates. The results we obtain are similar to the ones that were given by Ossiander and Waymire and Bacry \emph{et al.} in the simpler framework of Mandelbrot cascades.

In this paper, we deal with the asymptotic distribution of the maximum increment of a random walk with a regularly varying jump size distribution. This problem is motivated by a long-standing problem on change point detection for epidemic alternatives. It turns out that the limit distribution of the maximum increment of the random walk is one of the classical extreme value distributions, the Fr\'{e}chet distribution. We prove the results in the general framework of point processes and for jump sizes taking values in a separable Banach space.

We study the trajectory of a simple random walk on a d-regular graph with d>2 and locally tree-like structure as the number n of vertices grows. Examples of such graphs include random d-regular graphs and large girth expanders. For these graphs, we investigate percolative properties of the set of vertices not visited by the walk until time un, where u>0 is a fixed positive parameter. We show that this so-called vacant set exhibits a phase transition in u in the following sense: there exists an explicitly computable threshold u* such that, with high probability as n grows, if u u*, then it has a volume of order log(n). The critical value u* coincides with the critical intensity of a random interlacement process (introduced by Sznitman [arXiv:0704.2560]) on a d-regular tree. We also show that the random interlacement model describes the structure of the vacant set in local neighbourhoods.

Let a simple random walk run inside a torus of dimension three or higher for a number of steps which is a constant proportion of the volume. We examine geometric properties of the range, the random subgraph induced by the set of vertices visited by the walk. Distance and mixing bounds for the typical range are proven that are a $k$-iterated log factor from those on the full torus for arbitrary $k$. The proof uses hierarchical renormalization and techniques that can possibly be applied to other random processes in the Euclidean lattice. We use the same technique to bound the heat kernel of a random walk on the trace of random interlacements.

We find explicit eigenvectors for the transition matrix of a random walk due to Bidegare, Hanlon and Rockmore. This is accomplished by using Brown and Diaconis' analysis of its stationary distribution, together with some combinatorics of functions on the face lattice of a hyperplane arrangement, due to Gelfand and Varchenko.

We study the behavior of the Riemann zeta function on the critical line when the imaginary part of the argument is sampled by the Cauchy random walk. We develop a complete second order theory for the corresponding system of random variables and show that it behaves almost like a system of non-correlated variables. Exploiting this fact in relation with known criteria for almost sure convergence allows to investigate its almost sure asymptotic behavior.

We focus on the parametric estimation of the distribution of a Markov environment from the observation of a single trajectory of a one-dimensional nearest-neighbor path evolving in this random environment. In the ballistic case, as the length of the path increases, we prove consistency, asymptotic normality and efficiency of the maximum likelihood estimator. Our contribution is two-fold: we cast the problem into the one of parameter estimation in a hidden Markov model (HMM) and establish that the bivariate Markov chain underlying this HMM is positive Harris recurrent. We provide different examples of setups in which our results apply, in particular that of DNA unzipping model, and we give a simple synthetic experiment to illustrate those results.

We consider a Branching Random Walk on $\R$ whose step size decreases by a fixed factor, $0 1/2$ the limit measure is almost surely (a.s.) absolutely continuous with respect to the Lebesgue measure, but for Pisot $1/b$ it is a.s. singular; (2) for all $b > (\sqrt{5}-1)/2$ the support of the measure is a.s. the closure of its interior; (3) for Pisot $1/b$ the support of the measure is ``fractured'': it is a.s. disconnected and the components of the complement are not isolated on both sides.

We consider the model of the one-dimensional cookie random walk when the initial cookie distribution is spatially uniform and the number of cookies per site is finite. We give a criterion to decide whether the limiting speed of the walk is non-zero. In particular, we show that a positive speed may be obtained for just 3 cookies per site. We also prove a result on the continuity of the speed with respect to the initial cookie distribution.

We use a reflection argument, introduced by Gessel and Zeilberger, to count the number of k-step walks between two points which stay within a chamber of a Weyl group. We apply this technique to walks in the alcoves of the classical affine Weyl groups. In all cases, we get determinant formulas for the number of k-step walks. One important example is the region m>x_1>x_2>...>x_n>0, which is a rescaled alcove of the affine Weyl group C_n. If each coordinate is considered to be an independent particle, this models n non-colliding random walks on the interval (0,m). Another case models n non-colliding random walks on the circle.

We consider a d-dimensional random walk in random scenery X(n), where the scenery consists of i.i.d. with exponential moments but a tail decay of the form exp(-c t^a) with any}. We show that this probability is of order exp(-(ny)^b) with b=a/(a+1).

We study space-time fluctuations around a characteristic line for a one-dimensional interacting system known as the random average process. The state of this system is a real-valued function on the integers. New values of the function are created by averaging previous values with random weights. The fluctuations analyzed occur on the scale n^{1/4} where n is the ratio of macroscopic and microscopic scales in the system. The limits of the fluctuations are described by a family of Gaussian processes. In cases of known product-form equilibria, this limit is a two-parameter process whose time marginals are fractional Brownian motions with Hurst parameter 1/4. Along the way we study the limits of quenched mean processes for a random walk in a space-time random environment. These limits also happen at scale n^{1/4} and are described by certain Gaussian processes that we identify. In particular, when we look at a backward quenched mean process, the limit process is the solution of a stochastic heat equation.

We consider simple random walk on the incipient infinite cluster for the spread-out model of oriented percolation on $Z^d \times Z_+$. In dimensions $d>6$, we obtain bounds on exit times, transition probabilities, and the range of the random walk, which establish that the spectral dimension of the incipient infinite cluster is 4/3, and thereby prove a version of the Alexander--Orbach conjecture in this setting. The proof divides into two parts. One part establishes general estimates for simple random walk on an arbitrary infinite random graph, given suitable bounds on volume and effective resistance for the random graph. A second part then provides these bounds on volume and effective resistance for the incipient infinite cluster in dimensions $d>6$, by extending results about critical oriented percolation obtained previously via the lace expansion.

We give a simple non-analytic proof of Biggins' theorem on martingale convergence for branching random walks.

We outline basic properties of a symmetric random walk in one dimension, in which the length of the nth step equals lambda^n, with lambda oo, the probability that the endpoint is at x, P_{lambda}(x;N), approaches a limiting distribution P_{lambda}(x) that has many beautiful features. For lambda

The paper gives some properties of hitting times and an analogue of the Wiener-Hopf factorization for the Kendall random walk. We show also that the Williamson transform is the best tool for problems connected with the Kendall generalized convolution.

We use maximal entropy random walk (MERW) to study the trapping problem in dendrimers modeled by Cayley trees with a deep trap fixed at the central node. We derive an explicit expression for the mean first passage time from any node to the trap, as well as an exact formula for the average trapping time (ATT), which is the average of the source-to-trap mean first passage time over all non-trap starting nodes. Based on the obtained closed-form solution for ATT, we further deduce an upper bound for the leading behavior of ATT, which is the fourth power of $\ln N$, where $N$ is the system size. This upper bound is much smaller than the ATT of trapping depicted by unbiased random walk in Cayley trees, the leading scaling of which is a linear function of $N$. These results show that MERW can substantially enhance the efficiency of trapping performed in dendrimers.

This paper considers the question: how many times does a simple random walk revisit the most frequently visited site among the inner boundary points? It is known that in ${\mathbb{Z}}^2$, the number of visits to the most frequently visited site among all of the points of the random walk range up to time $n$ is asymptotic to $\pi^{-1}(\log n)^2$, while in ${\mathbb{Z}}^d$ $(d\ge3)$, it is of order $\log n$. We prove that the corresponding number for the inner boundary is asymptotic to $\beta_d\log n$ for any $d\ge2$, where $\beta_d$ is a certain constant having a simple probabilistic expression.

The epidemic-type aftershock sequence model (ETAS) is a simple stochastic process modeling seismicity, based on the two best-established empirical laws, the Omori law (power law decay ~1/t^{1+\theta} of seismicity after an earthquake) and Gutenberg-Richter law (power law distribution of earthquake energies). In order to describe also the space distribution of seismicity, we use in addition a power law distribution ~1/r^{1+\mu} of distances between triggered and triggering earthquakes. We present an exact mapping between the ETAS model and a class of CTRW (continuous time random walk) models, based on the identification of their corresponding Master equations. This mapping allows us to use the wealth of results previously obtained on anomalous diffusion of CTRW. We provide a classification of the different regimes of diffusion of seismic activity triggered by a mainshock. Specifically, we derive the relation between the average distance between aftershocks and the mainshock as a function of the time from the mainshock and of the joint probability distribution of the times and locations of the aftershocks. Our predictions are checked by careful numerical simulations. We stress the distinction between the ``bare'' Omori law describing the seismic rate activated directly by a mainshock and the ``renormalized'' Omori law taking into account all possible cascades from mainshocks to aftershocks of aftershock of aftershock, and so on. In particular, we predict that seismic diffusion or sub-diffusion occurs and should be observable only when the observed Omori exponent is less than 1, because this signals the operation of the renormalization of the bare Omori law, also at the origin of seismic diffusion in the ETAS model.

We consider the time dependent dynamics of an atom in a two-color pumped cavity, longitudinally through a side mirror and transversally via direct driving of the atomic dipole. The beating of the two driving frequencies leads to a time dependent effective optical potential that forces the atom into a non-trivial motion, strongly resembling a discrete random walk behavior between lattice sites. We provide both numerical and analytical analysis of such a quasi-random walk behavior.

In this paper, we study the probability of visiting a distant point $a\in \mathbb{Z}^4$ by critical branching random walk starting from the origin. We prove that this probability is bounded by $1/(|a|^2\log |a|)$ up to a constant.

Given a permutation sigma of the integers {-n,-n+1,...,n} we consider the Markov chain X_{sigma}, which jumps from k to sigma (k\pm 1) equally likely if k\neq -n,n. We prove that the expected hitting time of {-n,n} starting from any point is Theta(n) with high probability when sigma is a uniformly chosen permutation. We prove this by showing that with high probability, the digraph of allowed transitions is an Eulerian expander; we then utilize general estimates of hitting times in directed Eulerian expanders.

We study a class of tridiagonal matrix models, the "q-roots of unity" models, which includes the sign ($q=2$) and the clock ($q=\infty$) models by Feinberg and Zee. We find that the eigenvalue densities are bounded by and have the symmetries of the regular polygon with $2 q$ sides, in the complex plane. Furthermore the averaged traces of $M^k$ are integers that count closed random walks on the line, such that each site is visited a number of times multiple of $q$. We obtain an explicit evaluation for them.

Random walks are used for modeling various dynamics in, for example, physical, biological, and social contexts. Furthermore, their characteristics provide us with useful information on the phase transition and critical phenomena of even broader classes of related stochastic models. Abundant results are obtained for random walk on simple graphs such as the regular lattices and the Cayley trees. However, random walks and related processes on more complex networks, which are often more relevant in the real world, are still open issues, possibly yielding different characteristics. In this paper, we investigate the return times of random walks on random graphs with arbitrary vertex degree distributions. We analytically derive the distributions of the return times. The results are applied to some types of networks and compared with numerical data.

We establish the almost sure validity of the multifractal formalism for R^d-valued branching random walks on the whole relative interior of the natural convex domain of study.

We consider a simple random walk W_i in 1 or 2 dimensions, in which the walker may choose to stand still for a limited time. The time horizon is n, the maximum consecutive time steps which can be spent standing still is m_n and the goal is to maximize P(W_n=0). We show that for dimension 1, if m_n grows faster than (\log n)^{2+\gamma} for some \gamma>0, there is a strategy for each n such that P(W_n = 0) approaches 1. For dimension 2, if m_n grows faster than a positive power of n then there are strategies keeping P(W_n=0) bounded away from 0.

Let $\sigma$ be a permutation of $\{0,\ldots,n\}$. We consider the Markov chain $X$ which jumps from $k\neq 0,n$ to $\sigma(k+1)$ or $\sigma(k-1)$, equally likely. When $X$ is at 0 it jumps to either $\sigma(0)$ or $\sigma(1)$ equally likely, and when $X$ is at $n$ it jumps to either $\sigma(n)$ or $\sigma(n-1)$, equally likely. We show that the identity permutation maximizes the expected hitting time of n, when the walk starts at 0. More generally, we prove that the hitting time of a random walk on a strongly connected $d$-directed graph is maximized when the graph is the line $[0,n]\cap\Z$ with $d-2$ self-loops at every vertex and $d-1$ self-loops at 0 and $n$.

We investigated the discrete-time quantum random walks on a line in periodic potential. The probability distribution with periodic potential is more complex compared to the normal quantum walks, and the standard deviation $\sigma$ has interesting behaviors for different period $q$ and parameter $\theta$. We studied the behavior of standard deviation with variation in walk steps, period, and $\theta$. The standard deviation increases approximately linearly with $\theta$ and decreases with $1/q$ for $\theta\in(0,\pi/4)$, and increases approximately linearly with $1/q$ for $\theta\in[\pi/4,\pi/2)$. When $q=2$, the standard deviation is lazy for $\theta\in[\pi/4+n\pi,3\pi/4+n\pi],n\in Z$.