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Journal articles on the topic 'Symmetric random walk'

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Author: Grafiati

Published: 15 December 2023

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1

LI, KEQIN. "PERFORMANCE ANALYSIS AND EVALUATION OF RANDOM WALK ALGORITHMS ON WIRELESS NETWORKS." International Journal of Foundations of Computer Science 23, no. 04 (June 2012): 779–802. http://dx.doi.org/10.1142/s0129054112400369.

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We propose a model of dynamically evolving random networks and give an analytical result of the cover time of the simple random walk algorithm on a dynamic random symmetric planar point graph. Our dynamic network model considers random node distribution and random node mobility. We analyze the cover time of the parallel random walk algorithm on a complete network and show by numerical data that k parallel random walks reduce the cover time by almost a factor of k. We present simulation results for four random walk algorithms on random asymmetric planar point graphs. These algorithms include the simple random walk algorithm, the intelligent random walk algorithm, the parallel random walk algorithm, and the parallel intelligent random walk algorithm. Our random network model considers random node distribution and random battery transmission power. Performance measures include normalized cover time, probability distribution of the length of random walks, and load distribution.

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2

Zygmunt, Marcin J. "Non symmetric random walk on infinite graph." Opuscula Mathematica 31, no. 4 (2011): 669. http://dx.doi.org/10.7494/opmath.2011.31.4.669.

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3

Godrèche, Claude, and Jean-Marc Luck. "Survival probability of random walks and Lévy flights with stochastic resetting." Journal of Statistical Mechanics: Theory and Experiment 2022, no. 7 (July 1, 2022): 073201. http://dx.doi.org/10.1088/1742-5468/ac7a2a.

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Abstract We perform a thorough analysis of the survival probability of symmetric random walks with stochastic resetting, defined as the probability for the walker not to cross the origin up to time n. For continuous symmetric distributions of step lengths with either finite (random walks) or infinite variance (Lévy flights), this probability can be expressed in terms of the survival probability of the walk without resetting, given by Sparre Andersen theory. It is therefore universal, i.e. independent of the step length distribution. We analyze this survival probability at depth, deriving both exact results at finite times and asymptotic late-time results. We also investigate the case where the step length distribution is symmetric but not continuous, focusing our attention onto arithmetic distributions generating random walks on the lattice of integers. We investigate in detail the example of the simple Polya walk and propose an algebraic approach for lattice walks with a larger range.

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4

YANG, ZHIHUI. "LARGE DEVIATION ASYMPTOTICS FOR RANDOM-WALK TYPE PERTURBATIONS." Stochastics and Dynamics 07, no. 01 (March 2007): 75–89. http://dx.doi.org/10.1142/s0219493707001950.

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Symmetric random walks can be arranged to converge to a Wiener process in the area of normal deviation. However, random walks and Wiener processes have, in general, different asymptotics of the large deviation probabilities. The action functionals for random-walks and Wiener processes are compared in this paper. The correction term is calculated. Exit problem and stochastic resonance for random-walk-type perturbation are also considered and compared with the white-noise-type perturbation.

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5

Telcs, András, and Nicholas C. Wormald. "Branching and tree indexed random walks on fractals." Journal of Applied Probability 36, no. 4 (December 1999): 999–1011. http://dx.doi.org/10.1239/jap/1032374750.

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This paper deals with the recurrence of branching random walks on polynomially growing graphs. Amongst other things, we demonstrate the strong recurrence of tree indexed random walks determined by the resistance properties of spherically symmetric graphs. Several branching walk models are considered to show how the branching mechanism influences the recurrence behaviour.

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6

Telcs, András, and Nicholas C. Wormald. "Branching and tree indexed random walks on fractals." Journal of Applied Probability 36, no. 04 (December 1999): 999–1011. http://dx.doi.org/10.1017/s0021900200017812.

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This paper deals with the recurrence of branching random walks on polynomially growing graphs. Amongst other things, we demonstrate the strong recurrence of tree indexed random walks determined by the resistance properties of spherically symmetric graphs. Several branching walk models are considered to show how the branching mechanism influences the recurrence behaviour.

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7

Hilário, Marcelo R., Daniel Kious, and Augusto Teixeira. "Random Walk on the Simple Symmetric Exclusion Process." Communications in Mathematical Physics 379, no. 1 (August 26, 2020): 61–101. http://dx.doi.org/10.1007/s00220-020-03833-x.

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Abstract We investigate the long-term behavior of a random walker evolving on top of the simple symmetric exclusion process (SSEP) at equilibrium, in dimension one. At each jump, the random walker is subject to a drift that depends on whether it is sitting on top of a particle or a hole, so that its asymptotic behavior is expected to depend on the density $$\rho \in [0, 1]$$ ρ ∈ [ 0 , 1 ] of the underlying SSEP. Our first result is a law of large numbers (LLN) for the random walker for all densities $$\rho $$ ρ except for at most two values $$\rho _-, \rho _+ \in [0, 1]$$ ρ - , ρ + ∈ [ 0 , 1 ] . The asymptotic speed we obtain in our LLN is a monotone function of $$\rho $$ ρ . Also, $$\rho _-$$ ρ - and $$\rho _+$$ ρ + are characterized as the two points at which the speed may jump to (or from) zero. Furthermore, for all the values of densities where the random walk experiences a non-zero speed, we can prove that it satisfies a functional central limit theorem (CLT). For the special case in which the density is 1/2 and the jump distribution on an empty site and on an occupied site are symmetric to each other, we prove a LLN with zero limiting speed. We also prove similar LLN and CLT results for a different environment, given by a family of independent simple symmetric random walks in equilibrium.

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8

Fujita, Takahiko. "A random walk analogue of Lévy’s Theorem." Studia Scientiarum Mathematicarum Hungarica 45, no. 2 (June 1, 2008): 223–33. http://dx.doi.org/10.1556/sscmath.45.2008.2.50.

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In this paper we will give a simple symmetric random walk analogue of Lévy’s Theorem. We will give a new definition of a local time of the simple symmetric random walk. We apply a discrete Itô formula to some absolute value like function to obtain a discrete Tanaka formula. Results in this paper rely upon a discrete Skorokhod reflection argument. This random walk analogue of Lévy’s theorem was already obtained by G. Simons ([14]) but it is still worth noting because we will use a discrete stochastic analysis to obtain it and this method is applicable to other research. We note some connection with previous results by Csáki, Révész, Csörgő and Szabados. Finally we observe that the discrete Lévy transformation in the present version is not ergodic. Lastly we give a Lévy-type theorem for simple nonsymmetric random walk using a discrete bang-bang process.

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9

ISHIMURA, N., and N. YOSHIDA. "ON THE CONVERGENCE OF DISCRETE PROCESSES WITH MULTIPLE INDEPENDENT VARIABLES." ANZIAM Journal 58, no. 3-4 (March 6, 2017): 379–85. http://dx.doi.org/10.1017/s1446181116000389.

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We discuss discrete stochastic processes with two independent variables: one is the standard symmetric random walk, and the other is the Poisson process. Convergence of discrete stochastic processes is analysed, such that the symmetric random walk tends to the standard Brownian motion. We show that a discrete analogue of Ito’s formula converges to the corresponding continuous formula.

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10

Fang, Xiao, Han L. Gan, Susan Holmes, Haiyan Huang, Erol Peköz, Adrian Röllin, and Wenpin Tang. "Arcsine laws for random walks generated from random permutations with applications to genomics." Journal of Applied Probability 58, no. 4 (November 22, 2021): 851–67. http://dx.doi.org/10.1017/jpr.2021.14.

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AbstractA classical result for the simple symmetric random walk with 2n steps is that the number of steps above the origin, the time of the last visit to the origin, and the time of the maximum height all have exactly the same distribution and converge when scaled to the arcsine law. Motivated by applications in genomics, we study the distributions of these statistics for the non-Markovian random walk generated from the ascents and descents of a uniform random permutation and a Mallows(q) permutation and show that they have the same asymptotic distributions as for the simple random walk. We also give an unexpected conjecture, along with numerical evidence and a partial proof in special cases, for the result that the number of steps above the origin by step 2n for the uniform permutation generated walk has exactly the same discrete arcsine distribution as for the simple random walk, even though the other statistics for these walks have very different laws. We also give explicit error bounds to the limit theorems using Stein’s method for the arcsine distribution, as well as functional central limit theorems and a strong embedding of the Mallows(q) permutation which is of independent interest.

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11

Georgiou, Nicholas, Mikhail V. Menshikov, Aleksandar Mijatović, and Andrew R. Wade. "Anomalous recurrence properties of many-dimensional zero-drift random walks." Advances in Applied Probability 48, A (July 2016): 99–118. http://dx.doi.org/10.1017/apr.2016.44.

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AbstractFamously, a d-dimensional, spatially homogeneous random walk whose increments are nondegenerate, have finite second moments, and have zero mean is recurrent if d∈{1,2}, but transient if d≥3. Once spatial homogeneity is relaxed, this is no longer true. We study a family of zero-drift spatially nonhomogeneous random walks (Markov processes) whose increment covariance matrix is asymptotically constant along rays from the origin, and which, in any ambient dimension d≥2, can be adjusted so that the walk is either transient or recurrent. Natural examples are provided by random walks whose increments are supported on ellipsoids that are symmetric about the ray from the origin through the walk's current position; these elliptic random walks generalize the classical homogeneous Pearson‒Rayleigh walk (the spherical case). Our proof of the recurrence classification is based on fundamental work of Lamperti.

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12

Bérard, Jean. "The almost sure central limit theorem for one-dimensional nearest-neighbour random walks in a space-time random environment." Journal of Applied Probability 41, no. 01 (March 2004): 83–92. http://dx.doi.org/10.1017/s0021900200014054.

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The central limit theorem for random walks on ℤ in an i.i.d. space-time random environment was proved by Bernabeiet al.for almost all realization of the environment, under a small randomness assumption. In this paper, we prove that, in the nearest-neighbour case, when the averaged random walk is symmetric, the almost sure central limit theorem holds for anarbitrarylevel of randomness.

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13

Fort, G., E. Moulines, G. O. Roberts, and J. S. Rosenthal. "On the geometric ergodicity of hybrid samplers." Journal of Applied Probability 40, no. 1 (March 2003): 123–46. http://dx.doi.org/10.1239/jap/1044476831.

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In this paper, we consider the random-scan symmetric random walk Metropolis algorithm (RSM) on ℝd. This algorithm performs a Metropolis step on just one coordinate at a time (as opposed to the full-dimensional symmetric random walk Metropolis algorithm, which proposes a transition on all coordinates at once). We present various sufficient conditions implying V-uniform ergodicity of the RSM when the target density decreases either subexponentially or exponentially in the tails.

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14

Fort, G., E. Moulines, G. O. Roberts, and J. S. Rosenthal. "On the geometric ergodicity of hybrid samplers." Journal of Applied Probability 40, no. 01 (March 2003): 123–46. http://dx.doi.org/10.1017/s0021900200022300.

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In this paper, we consider the random-scan symmetric random walk Metropolis algorithm (RSM) on ℝ d . This algorithm performs a Metropolis step on just one coordinate at a time (as opposed to the full-dimensional symmetric random walk Metropolis algorithm, which proposes a transition on all coordinates at once). We present various sufficient conditions implying V-uniform ergodicity of the RSM when the target density decreases either subexponentially or exponentially in the tails.

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15

Bookstein, Fred L. "Random walk and the existence of evolutionary rates." Paleobiology 13, no. 4 (1987): 446–64. http://dx.doi.org/10.1017/s0094837300009039.

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Before one can study evolutionary rates one must reject the null model of symmetric random walk. for which the requisite quantity does not exist. As random walks reliably simulate all the features we find so compelling in the fossil record—jumps, trends, and irregular cycles—rejection of this irritating hypothesis is much more difficult than one might hope. This paper reviews principal theorems from the mathematical literature of random walk and shows how they may be applied to empirical data by scaling net changes according to the square root of elapsed time. The notorious pair of “opposite” findings, equilibrium and anagenesis, may be construed as deviations from random walk in opposite directions. Malmgren's data on Globorotalia tumida, previously interpreted as an example of punctuated anagenesis, are consistent with a random walk showing neither punctuation nor anagenesis, but instead varying in speed over four subsequences.

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16

Zodage, Aniket, Rosalind J. Allen, Martin R. Evans, and Satya N. Majumdar. "A sluggish random walk with subdiffusive spread." Journal of Statistical Mechanics: Theory and Experiment 2023, no. 3 (March 1, 2023): 033211. http://dx.doi.org/10.1088/1742-5468/acc4b1.

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Abstract We study a one-dimensional sluggish random walk with space-dependent transition probabilities between nearest-neighbour lattice sites. Motivated by trap models of slow dynamics, we consider a model in which the trap depth increases logarithmically with distance from the origin. This leads to a random walk which has symmetric transition probabilities that decrease with distance | k | from the origin as 1 / | k | for large | k | . We show that the typical position after time t scales as t 1 / 3 with a nontrivial scaling function for the position distribution which has a trough (a cusp singularity) at the origin. Therefore an effective central bias away from the origin emerges even though the transition probabilities are symmetric. We also compute the survival probability of the walker in the presence of a sink at the origin and show that it decays as t − 1 / 3 at late times. Furthermore we compute the distribution of the maximum position, M(t), to the right of the origin up to time t, and show that it has a nontrivial scaling function. Finally we provide a generalisation of this model where the transition probabilities decay as 1 / | k | α with α > 0.

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17

Bérard, Jean. "The almost sure central limit theorem for one-dimensional nearest-neighbour random walks in a space-time random environment." Journal of Applied Probability 41, no. 1 (March 2004): 83–92. http://dx.doi.org/10.1239/jap/1077134669.

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The central limit theorem for random walks on ℤ in an i.i.d. space-time random environment was proved by Bernabei et al. for almost all realization of the environment, under a small randomness assumption. In this paper, we prove that, in the nearest-neighbour case, when the averaged random walk is symmetric, the almost sure central limit theorem holds for an arbitrary level of randomness.

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18

Prigent, Martin, and Matthew I. Roberts. "Noise sensitivity and exceptional times of transience for a simple symmetric random walk in one dimension." Probability Theory and Related Fields 178, no. 1-2 (June 18, 2020): 327–67. http://dx.doi.org/10.1007/s00440-020-00978-7.

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Abstract We define a dynamical simple symmetric random walk in one dimension, and show that there almost surely exist exceptional times at which the walk tends to infinity. This is in contrast to the usual dynamical simple symmetric random walk in one dimension, for which such exceptional times are known not to exist. In fact we show that the set of exceptional times has Hausdorff dimension 1/2 almost surely, and give bounds on the rate at which the walk diverges at such times. We also show noise sensitivity of the event that our random walk is positive after n steps. In fact this event is maximally noise sensitive, in the sense that it is quantitatively noise sensitive for any sequence $$\varepsilon _n$$ ε n such that $$n\varepsilon _n\rightarrow \infty $$ n ε n → ∞ . This is again in contrast to the usual random walk, for which the corresponding event is known to be noise stable.

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19

Mathieu, P., and A. Piatnitski. "Quenched invariance principles for random walks on percolation clusters." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 463, no. 2085 (July 3, 2007): 2287–307. http://dx.doi.org/10.1098/rspa.2007.1876.

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We consider a supercritical Bernoulli percolation model in , d ≥2, and study the simple symmetric random walk on the infinite percolation cluster. The aim of this paper is to prove the almost sure (quenched) invariance principle for this random walk.

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20

BARBÉ, ANDRÉ. "NECESSITY FROM CHANCE: SELF-ORGANIZED REPLICATION OF SYMMETRIC PATTERNS THROUGH SYMMETRIC RANDOM INTERACTIONS." International Journal of Bifurcation and Chaos 19, no. 04 (April 2009): 1185–225. http://dx.doi.org/10.1142/s0218127409023585.

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We present an algorithm, completely random in nature, that, without invoking a fitness function or purposeful design, produces symmetric replicas in a population of so-called cellicules. Cellicules consist of "cells" arranged in a structure with geometric symmetry S. Each cell has one of two possible states, thus defining a state-configuration pattern on a cellicule. The algorithm acts recurrently on a population of cellicules, possibly randomly initialized, through a random "copying interaction" between two randomly selected cellicules that first undergo a random reorientation in accordance with the symmetry S. The dynamics of the algorithm is analyzed in detail for several symmetries. This shows that it is a random walk with absorbing states which correspond to a population in which all cellicules have an identical S-symmetric configuration pattern. We discuss some aspects concerning the evolution of cellicule-populations under mixing and mutation, and some variations on the basic algorithm.

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21

KOIRALA, ROBERT. "On Simple Symmetric Random Walk in 􀢊 -Dimensional Integer Lattice." Journal of Ultra Scientist of Physical Sciences Section A 29, no. 10 (October 2, 2017): 410–17. http://dx.doi.org/10.22147/jusps-a/291001.

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22

He, Xue Dong, Sang Hu, Jan Obłój, and Xun Yu Zhou. "Two explicit Skorokhod embeddings for simple symmetric random walk." Stochastic Processes and their Applications 129, no. 9 (September 2019): 3431–45. http://dx.doi.org/10.1016/j.spa.2018.09.013.

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23

Hough, Bob. "The random k cycle walk on the symmetric group." Probability Theory and Related Fields 165, no. 1-2 (July 3, 2015): 447–82. http://dx.doi.org/10.1007/s00440-015-0636-6.

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24

Connor, Stephen. "Optimal Coadapted Coupling for a Random Walk on the Hyper-Complete Graph." Journal of Applied Probability 50, no. 04 (December 2013): 1117–30. http://dx.doi.org/10.1017/s0021900200013838.

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The problem of constructing an optimal coadapted coupling for a pair of symmetric random walks onZ2dwas considered by Connor and Jacka (2008), and the existence of a coupling which is stochastically fastest in the class of all such coadapted couplings was demonstrated. In this paper we show how to generalise this construction to an optimal coadapted coupling for the continuous-time symmetric random walk onKnd, whereKnis the complete graph withnvertices. Moreover, we show that although this coupling is not maximal for anyn(i.e. it does not achieve equality in the coupling inequality), it does tend to a maximal coupling asn→ ∞.

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Connor, Stephen. "Optimal Coadapted Coupling for a Random Walk on the Hyper-Complete Graph." Journal of Applied Probability 50, no. 4 (December 2013): 1117–30. http://dx.doi.org/10.1239/jap/1389370103.

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The problem of constructing an optimal coadapted coupling for a pair of symmetric random walks on Z2d was considered by Connor and Jacka (2008), and the existence of a coupling which is stochastically fastest in the class of all such coadapted couplings was demonstrated. In this paper we show how to generalise this construction to an optimal coadapted coupling for the continuous-time symmetric random walk on Knd, where Kn is the complete graph with n vertices. Moreover, we show that although this coupling is not maximal for any n (i.e. it does not achieve equality in the coupling inequality), it does tend to a maximal coupling as n → ∞.

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26

Roerdink, J. B. T. M. "A Markov chain identity and monotonicity of the diffusion constants for a random walk in a heterogeneous environment." Mathematical Proceedings of the Cambridge Philosophical Society 108, no. 1 (July 1990): 111–26. http://dx.doi.org/10.1017/s0305004100069000.

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AbstractWe consider a 2-dimensional square lattice which is partitioned into a periodic array of rectangular cells, on which a nearest neighbour random walk with symmetric increments is defined whose transition probabilities only depend on the relative position within a cell. On the basis of a determinantal identity proved in this paper, we obtain a result for finite Markov chains which shows that the diffusion constants for the random walk are monotonic functions of the individual transition probabilities. We point out the similarity of this monotonicity property to Rayleigh's Monotonicity Law for electric networks or, equivalently, reversible random walks.

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27

Sarkar, Jyotirmoy, and Saran Ishika Maiti. "Symmetric Random Walks on Regular Tetrahedra, Octahedra, and Hexahedra." Calcutta Statistical Association Bulletin 69, no. 1 (May 2017): 110–28. http://dx.doi.org/10.1177/0008068317695974.

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We study a symmetric random walk on the vertices of three regular polyhedra. Starting from the origin, at each step the random walk moves, independently of all previous moves, to one of the vertices adjacent to the current vertex with equal probability. We find the distributions, or at least the means and the standard deviations, of the number of steps needed (a) to return to origin, (b) to visit all vertices, and (c) to return to origin after visiting all vertices. We also find the distributions of (i) the number of vertices visited before return to origin, (ii) the last vertex visited, and (iii) the number of vertices visited during return to origin after visiting all vertices.

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28

Zadarazhniuk, H. A. "An analogue of Aldous’s theorem on mixing times of a random walk for complex reflection groups." Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series 59, no. 1 (April 3, 2023): 51–61. http://dx.doi.org/10.29235/1561-2430-2023-59-1-51-61.

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The subject of this paper is the mixing time of random walks on minimal Cayley graphs of complex reflection groups G(m,1,n). The key role in estimating it is played by the coupling of distributions, which has been used before for the same task on symmetric groups. The difficulty with its adaptation for the current case is that there are now two components in a walk, which are to be coupled, and they influence each other’s behaviour. To solve this problem, random walks are split into several blocks for each of which the time needed for their states to match is estimated separately. The result is upper and lower bounds on mixing times of random walks on complex reflection groups, analogous to those obtained by Aldous for a symmetric group.

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29

Grey, D. R. "Persistent random walks may have arbitrarily large tails." Advances in Applied Probability 21, no. 1 (March 1989): 229–30. http://dx.doi.org/10.2307/1427206.

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Grey, D. R. "Persistent random walks may have arbitrarily large tails." Advances in Applied Probability 21, no. 01 (March 1989): 229–30. http://dx.doi.org/10.1017/s0001867800017286.

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Le Gall, Jean-François, and Shen Lin. "THE RANGE OF TREE-INDEXED RANDOM WALK." Journal of the Institute of Mathematics of Jussieu 15, no. 2 (September 10, 2014): 271–317. http://dx.doi.org/10.1017/s1474748014000280.

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We provide asymptotics for the range $R_{n}$ of a random walk on the $d$-dimensional lattice indexed by a random tree with $n$ vertices. Using Kingman’s subadditive ergodic theorem, we prove under general assumptions that $n^{-1}R_{n}$ converges to a constant, and we give conditions ensuring that the limiting constant is strictly positive. On the other hand, in dimension $4$, and in the case of a symmetric random walk with exponential moments, we prove that $R_{n}$ grows like $n/\!\log n$. We apply our results to asymptotics for the range of a branching random walk when the initial size of the population tends to infinity.

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32

Uchaikin, Vladimir V., Renat T. Sibatov, and Dmitry N. Bezbatko. "On a Generalization of One-Dimensional Kinetics." Mathematics 9, no. 11 (May 31, 2021): 1264. http://dx.doi.org/10.3390/math9111264.

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One-dimensional random walks with a constant velocity between scattering are considered. The exact solution is expressed in terms of multiple convolutions of path-distributions assumed to be different for positive and negative directions of the walk axis. Several special cases are considered when the convolutions are expressed in explicit form. As a particular case, the solution of A. S. Monin for a symmetric random walk with exponential path distribution and its generalization to the asymmetric case are obtained. Solution of fractional telegraph equation with the fractional material derivative is presented. Asymptotic behavior of its solution for an asymmetric case is provided.

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Deligiannidis, George, and Sergey Utev. "Optimal Bounds for the Variance of Self-Intersection Local Times." International Journal of Stochastic Analysis 2016 (July 20, 2016): 1–10. http://dx.doi.org/10.1155/2016/5370627.

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For a Zd-valued random walk (Sn)n∈N0, let l(n,x) be its local time at the site x∈Zd. For α∈N, define the α-fold self-intersection local time as Ln(α)≔∑xl(n,x)α. Also let LnSRW(α) be the corresponding quantities for the simple random walk in Zd. Without imposing any moment conditions, we show that the variance of the self-intersection local time of any genuinely d-dimensional random walk is bounded above by the corresponding quantity for the simple symmetric random walk; that is, var(Ln(α))=O(var⁡(LnSRW(α))). In particular, for any genuinely d-dimensional random walk, with d≥4, we have var⁡(Ln(α))=O(n). On the other hand, in dimensions d≤3 we show that if the behaviour resembles that of simple random walk, in the sense that lim infn→∞var⁡Lnα/var⁡(LnSRW(α))>0, then the increments of the random walk must have zero mean and finite second moment.

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34

Sama-ae, Al-ameen, Nattakarn Chaidee, and Kritsana Neammanee. "Half-normal approximation for statistics of symmetric simple random walk." Communications in Statistics - Theory and Methods 47, no. 4 (January 2, 2018): 779–92. http://dx.doi.org/10.1080/03610926.2016.1139129.

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35

Palacios, José Luis. "On the Simple Symmetric Random Walk and its Maximal Function." American Statistician 62, no. 2 (May 2008): 138–40. http://dx.doi.org/10.1198/000313008x304846.

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36

Gorenflo, Rudolf, Gianni De Fabritiis, and Francesco Mainardi. "Discrete random walk models for symmetric Lévy–Feller diffusion processes." Physica A: Statistical Mechanics and its Applications 269, no. 1 (July 1999): 79–89. http://dx.doi.org/10.1016/s0378-4371(99)00082-5.

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37

Csáki, Endre. "Some joint distributions in Bernoulli excursions." Journal of Applied Probability 31, A (1994): 239–50. http://dx.doi.org/10.2307/3214959.

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Some exact and asymptotic joint distributions are given for certain random variables defined on the excursions of a simple symmetric random walk. We derive appropriate recursion formulas and apply them to get certain expressions for the joint generating or characteristic functions of the random variables.

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Csáki, Endre. "Some joint distributions in Bernoulli excursions." Journal of Applied Probability 31, A (1994): 239–50. http://dx.doi.org/10.1017/s0021900200107090.

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Some exact and asymptotic joint distributions are given for certain random variables defined on the excursions of a simple symmetric random walk. We derive appropriate recursion formulas and apply them to get certain expressions for the joint generating or characteristic functions of the random variables.

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39

Piatnitski, A., and E. Zhizhina. "Scaling Limit of Symmetric Random Walk in High-Contrast Periodic Environment." Journal of Statistical Physics 169, no. 3 (September 23, 2017): 595–613. http://dx.doi.org/10.1007/s10955-017-1883-y.

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40

Aguech, Rafik, and Mohamed Abdelkader. "Two-Dimensional Moran Model: Final Altitude and Number of Resets." Mathematics 11, no. 17 (September 2, 2023): 3774. http://dx.doi.org/10.3390/math11173774.

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In this paper, we consider a two-dimension symmetric random walk with reset. We give, in the first part, some results about the distribution of every component. In the second part, we give some results about the final altitude Zn. Finally, we analyse the statistical properties of NnX, the number of resets (the number of returns to state 1 after n steps) of the first component of the random walk. As a principal tool in these studies, we use the probability generating function.

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41

Apers, Simon, and Alain Scarlet. "Quantum fast-forwarding: Markov chains and graph property testing." Quantum Information and Computation 19, no. 3&4 (March 2019): 181–213. http://dx.doi.org/10.26421/qic19.3-4-1.

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We introduce a new tool for quantum algorithms called quantum fast-forwarding (QFF). The tool uses quantum walks as a means to quadratically fast-forward a reversible Markov chain. More specifically, with P the Markov chain transition matrix and D = \sqrt{P\circ P^T} its discriminant matrix (D=P if P is symmetric), we construct a quantum walk algorithm that for any quantum state |v> and integer t returns a quantum state \epsilon-close to the state D^t|v>/\|D^t|v>. The algorithm uses O(|D^t|v>|^{-1}\sqrt{t\log(\epsilon\|D^t|v>})^{-1}}) expected quantum walk steps and O(\|D^t|v>|^{-1}) expected reflections around |v>. This shows that quantum walks can accelerate the transient dynamics of Markov chains, complementing the line of results that proves the acceleration of their limit behavior. We show that this tool leads to speedups on random walk algorithms in a very natural way. Specifically we consider random walk algorithms for testing the graph expansion and clusterability, and show that we can quadratically improve the dependency of the classical property testers on the random walk runtime. Moreover, our quantum algorithm exponentially improves the space complexity of the classical tester to logarithmic. As a subroutine of independent interest, we use QFF for determining whether a given pair of nodes lies in the same cluster or in separate clusters. This solves a robust version of s-t connectivity, relevant in a learning context for classifying objects among a set of examples. The different algorithms crucially rely on the quantum speedup of the transient behavior of random walks.

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42

Connor, Stephen, and Saul Jacka. "Optimal Co-Adapted Coupling for the Symmetric Random Walk on the Hypercube." Journal of Applied Probability 45, no. 3 (September 2008): 703–13. http://dx.doi.org/10.1239/jap/1222441824.

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Let X and Y be two simple symmetric continuous-time random walks on the vertices of the n-dimensional hypercube, Z2n. We consider the class of co-adapted couplings of these processes, and describe an intuitive coupling which is shown to be the fastest in this class.

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Connor, Stephen, and Saul Jacka. "Optimal Co-Adapted Coupling for the Symmetric Random Walk on the Hypercube." Journal of Applied Probability 45, no. 03 (September 2008): 703–13. http://dx.doi.org/10.1017/s0021900200004654.

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Let X and Y be two simple symmetric continuous-time random walks on the vertices of the n-dimensional hypercube, Z 2 n . We consider the class of co-adapted couplings of these processes, and describe an intuitive coupling which is shown to be the fastest in this class.

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44

Sherlock, Chris, and Gareth Roberts. "Optimal scaling of the random walk Metropolis on elliptically symmetric unimodal targets." Bernoulli 15, no. 3 (August 2009): 774–98. http://dx.doi.org/10.3150/08-bej176.

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45

Molchanov, S. A., and E. B. Yarovaya. "Large deviations for a symmetric branching random walk on a multidimensional lattice." Proceedings of the Steklov Institute of Mathematics 282, no. 1 (October 2013): 186–201. http://dx.doi.org/10.1134/s0081543813060163.

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46

Butov, A. A., and E. V. Krichagina. "A functional limit theorem for a symmetric walk in a random environment." Russian Mathematical Surveys 43, no. 2 (April 30, 1988): 163–64. http://dx.doi.org/10.1070/rm1988v043n02abeh001710.

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47

Vallois, Pierre, and Charles S. Tapiero. "The range inter-event process in a symmetric birth-death random walk." Applied Stochastic Models in Business and Industry 17, no. 3 (2001): 293–306. http://dx.doi.org/10.1002/asmb.440.

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48

Hayashi, Masafumi, So Oshiro, and Masato Takei. "Rate of moment convergence in the central limit theorem for the elephant random walk." Journal of Statistical Mechanics: Theory and Experiment 2023, no. 2 (February 1, 2023): 023202. http://dx.doi.org/10.1088/1742-5468/acb265.

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Abstract The one-dimensional elephant random walk is a typical model of discrete-time random walk with step-reinforcement, and is introduced by Schütz and Trimper (2004 Phys. Rev. E 70 045101). It has a parameter α ∈ ( − 1 , 1 ) : the case α = 0 corresponds to the simple symmetric random walk, and when α > 0 (resp. α < 0), the mean displacement of the walker at time n grows (resp. vanishes) like n α . The walk admits a phase transition at α = 1 / 2 from the diffusive behavior to the superdiffusive behavior. In this paper, we study the rate of the moment convergence in the central limit theorem for the position of the walker when − 1 < α ⩽ 1 / 2 . We find a crossover phenomenon in the rate of convergence of the 2mth moments with m = 2 , 3 , … inside the diffusive regime − 1 < α < 1 / 2 .

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Dombry, C. "A weighted random walk model, with application to a genetic algorithm." Advances in Applied Probability 39, no. 2 (June 2007): 550–68. http://dx.doi.org/10.1239/aap/1183667623.

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We consider a weighted random walk model defined as follows. An n-step random walk on the integers with distribution Pn is weighted by giving the path S=(S0,…,Sn) a probability proportional to where the function f is the so-called fitness function. In the case of power-type fitness, we prove the convergence of the renormalized path to a deterministic function with exponential speed. This function is a solution to a variational problem. In the case of the simple symmetric random walk, explicit computations are done. Our result relies on large deviations techniques and Varadhan's integral lemma. We then study an application of this model to mutation-selection dynamics on the integers where a random walk operates the mutation. This dynamics is the infinite-population limit of that of mutation-selection genetic algorithms. We prove that the population grows to ∞ and make explicit its growth speed. This is a toy model for modelling the effect of stronger selection at ∞ for genetic algorithms taking place in a noncompact space.

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50

Dombry, C. "A weighted random walk model, with application to a genetic algorithm." Advances in Applied Probability 39, no. 02 (June 2007): 550–68. http://dx.doi.org/10.1017/s0001867800001889.

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Abstract:

We consider a weighted random walk model defined as follows. An n-step random walk on the integers with distribution P n is weighted by giving the path S=(S 0,…,S n ) a probability proportional to where the function f is the so-called fitness function. In the case of power-type fitness, we prove the convergence of the renormalized path to a deterministic function with exponential speed. This function is a solution to a variational problem. In the case of the simple symmetric random walk, explicit computations are done. Our result relies on large deviations techniques and Varadhan's integral lemma. We then study an application of this model to mutation-selection dynamics on the integers where a random walk operates the mutation. This dynamics is the infinite-population limit of that of mutation-selection genetic algorithms. We prove that the population grows to ∞ and make explicit its growth speed. This is a toy model for modelling the effect of stronger selection at ∞ for genetic algorithms taking place in a noncompact space.

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