On a random walk with memory and its relation with Markovian processes

Turban, L.

doi:10.1088/1751-8113/43/28/285006

Cited by 4 publications

(10 citation statements)

References 31 publications

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“…In the appendix (see appendix D), we discuss how one can rewrite the chaos game equation in terms of a random walk with variable size steps. Our work directly links to the work of Krapisvsky and Turban [32,33] in which results on the density distribution are presented in this context. Interestingly, this distribution is known as Bernoulli convolutions [34].…”

Section: Point Densitymentioning

confidence: 61%

“…In the statistical physics community, random walks with variable size steps have been considered in [30,31]. To make the link between the chaos game and those studies more explicit, we start from the chaos game equation (1) (using x 0 = 0) and define σ j = 2c n−j−1 − 1 as spin variables (taking values in {−1, 1}).…”

Section: The Average Number Of Fermions In the System: N Nmentioning

confidence: 99%

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Quantum repeated interactions and the chaos game

Platini¹,

Low²

2018

J. Phys. A: Math. Theor.

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Inspired by the algorithm of Barnsley's chaos game, we construct an open quantum system model based on the repeated interaction process. We shown that the quantum dynamics of the appropriate fermionic/bosonic system (in interaction with an environment) provides a physical model of the chaos game. When considering fermionic operators, we follow the system's evolution by focusing on its reduced density matrix. The system is shown to be in a Gaussian state (at all time t) and the average number of particles is shown to obey the chaos game equation. Considering bosonic operators, with a system initially prepared in coherent states, the evolution of the system can be tracked by investigating the dynamics of the eigenvalues of the annihilation operator. This quantity is governed by a chaos game-like equation from which different scenarios emerge.

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Section: Point Densitymentioning

confidence: 61%

Section: The Average Number Of Fermions In the System: N Nmentioning

confidence: 99%

Quantum repeated interactions and the chaos game

Platini¹,

Low²

2018

J. Phys. A: Math. Theor.

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“…To close, it is worth mentioning that the potentially spectacular effects of longranged memory on random walks have been scrutinized in a long series of recent works [26,27,28,29,30,31,32].…”

Section: Discussionmentioning

confidence: 99%

On stochastic differential equations with random delay

2011

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We consider stochastic dynamical systems defined by differential equations with a uniform random time delay. The latter equations are shown to be equivalent to deterministic higher-order differential equations: for an n-th order equation with random delay, the corresponding deterministic equation has order n + 1. We analyze various examples of dynamical systems of this kind, and find a number of unusual behaviors. For instance, for the harmonic oscillator with random delay, the energy grows as exp((3/2) t 2/3 ) in reduced units. We then investigate the effect of introducing a discrete time step ε. At variance with the continuous situation, the discrete random recursion relations thus obtained have intrinsic fluctuations. The crossover between the fluctuating discrete problem and the deterministic continuous one as ε goes to zero is studied in detail on the example of a first-order linear differential equation.

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“…One of the most common types of a discrete random process is a random walk, first introduced by Pearson in 1905, see [7]. There exist many variations of a random walk with various applications to real-life problems [9], [10]. Yet there are still new possibilities and options regarding how to alter and improve the classical random walk and present yet another model representing different real-life events.…”

Section: Introductionmentioning

confidence: 99%

“…Yet there are still new possibilities and options regarding how to alter and improve the classical random walk and present yet another model representing different real-life events. One such modification is the random walk with varying step size introduced in 2010 by Turban [10] which, together with the idea of self-exciting point processes [3] and the perspective of model applications in reliability analysis and also in sports statistics, served as an inspiration for the random walk with varying transition probabilities…”

Section: Introductionmentioning

confidence: 99%

Discrete random processes with memory: Models and applications

Kouřim¹,

Volf²

2020

Appl.Math.

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The contribution focuses on Bernoulli-like random walks, where the past events significantly affect the walk's future development. The main concern of the paper is therefore the formulation of models describing the dependence of transition probabilities on the process history. Such an impact can be incorporated explicitly and transition probabilities modulated using a few parameters reflecting the current state of the walk as well as the information about the past path. The behavior of proposed random walks, as well as the task of their parameter estimation, are studied both theoretically and with the aid of simulations.

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On a random walk with memory and its relation with Markovian processes

Cited by 4 publications

References 31 publications

Quantum repeated interactions and the chaos game

Quantum repeated interactions and the chaos game

On stochastic differential equations with random delay

Discrete random processes with memory: Models and applications

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