Variable survival exponents in history-dependent random walks: hard movable reflector

Dickman, Ronald; Araujo, Francisco Fontenele; ben‐Avraham, Daniel

doi:10.1590/s0103-97332003000300006

Cited by 6 publications

(12 citation statements)

References 21 publications

(46 reference statements)

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“…1 , which follows from (3.6) and(3.8). Thus the scaled and centered random variable (ℓ t − 1)/(1 − λ) has the same symmetric probability density as X t−1 .…”

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confidence: 84%

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

Turban¹

2010

J. Phys. A: Math. Theor.

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We study a one-dimensional random walk with memory in which the step lengths to the left and to the right evolve at each step in order to reduce the wandering of the walker. The feedback is quite efficient and lead to a non-diffusive walk. The time evolution of the displacement is given by an equivalent Markovian dynamical process. The probability density for the position of the walker is the same at any time as for a random walk with shrinking steps, although the two-time correlation functions are quite different. PACS numbers: 02.50.-r, 05.40.-a, 05-40.Fb Submitted to: J. Phys. A: Math. Gen.

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“…1 , which follows from (3.6) and(3.8). Thus the scaled and centered random variable (ℓ t − 1)/(1 − λ) has the same symmetric probability density as X t−1 .…”

mentioning

confidence: 84%

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

Turban¹

2010

J. Phys. A: Math. Theor.

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“…Numerous authors have considered memory effects in the context of classical random walks [30][31][32][33][34][35][36][37][38], and some steps have also been made in the quantum context [39][40][41], which we build upon.…”

Section: Introductionmentioning

confidence: 99%

Quantum walks with memory provided by recycled coins and a memory of the coin-flip history

2013

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Quantum walks have emerged as an interesting approach to quantum information processing, exhibiting many unique properties compared to the analogous classical random walk. Here we introduce a model for a discrete-time quantum walk with memory by endowing the walker with multiple recycled coins and using a physical memory function via a history dependent coin flip. By numerical simulation we observe several phenomena. First in one dimension, walkers with memory have persistent quantum ballistic speed up over classical walks just as found in previous studies of multicoined walks with trivial memory function. However, measurement of the multicoin state can dramatically shift the mean of the spatial distribution. Second, we consider spatial entanglement in a two-dimensional quantum walk with memory and find that memory destroys entanglement between the spatial dimensions, even when entangling coins are employed. Finally, we explore behavior in the presence of spatial randomness and find that in the time regime where single-coined walks localize, multicoined walks do not and in fact a memory function can speed up the walk relative to a multicoin walker with no memory. We explicitly show how to construct linear optics circuits implementing the walks, and discuss prospects for classical simulation.

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“…We use basic tools of first-passage processes to determine the time dependence of the size of the region in which the cookies have been eaten and thence the probability for the walk to first return to the origin. The long-time behavior of the first-passage probability in a closely related model was obtained previously, using quite different methods, in [10,11]. Our approach is also complementary to the tools employed in the probabilistic studies of excited random walks given in Ref.…”

Section: Introductionmentioning

confidence: 63%

“…In the simplest case of the 1-excited walk on the positive half-line, we employed a physical approach, complementary to that of Dickman et al [10,11], to show that the first-passage probability for the walk to hit the origin at time t decays as t −1−q for any q > 0. Thus the 1-excited walk is always recurrent, except for the trivial case of perfect bias p = 1.…”

Section: Discussionmentioning

confidence: 99%

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The excited random walk in one dimension

Antal

Redner

2005

J. Phys. A: Math. Gen.

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We study the excited random walk, in which a walk that is at a site that contains cookies eats one cookie and then hops to the right with probability p and to the left with probability q = 1 − p. If the walk hops onto an empty site, there is no bias. For the 1-excited walk on the half-line (one cookie initially at each site), the probability of first returning to the starting point at time t scales as t −(2−p) . Although the average return time to the origin is infinite for all p, the walk eats, on average, only a finite number of cookies until this first return when p < 1/2. For the infinite line, the probability distribution for the 1-excited walk has an unusual anomaly at the origin. The positions of the leftmost and rightmost uneaten cookies can be accurately estimated by probabilistic arguments and their corresponding distributions have power-law singularities near the origin. The 2-excited walk on the infinite line exhibits peculiar features in the regime p > 3/4, where the walk is transient, including a mean displacement that grows as t ν , with ν > 1 2 dependent on p, and a breakdown of scaling for the probability distribution of the walk.

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Variable survival exponents in history-dependent random walks: hard movable reflector

Cited by 6 publications

References 21 publications

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

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

Quantum walks with memory provided by recycled coins and a memory of the coin-flip history

The excited random walk in one dimension

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