Scaling behavior for random walks with memory of the largest distance from the origin

Abstract

We study a one-dimensional random walk with memory. The behavior of the walker is modified with respect to the simple symmetric random walk only when he or she is at the maximum distance ever reached from his or her starting point (home). In this case, having the choice to move farther or to move closer, the walker decides with different probabilities. If the probability of a forward step is higher then the probability of a backward step, the walker is bold, otherwise he or she is timorous. We investigate the asymptotic properties of this bold-timorous random walk, showing that the scaling behavior varies continuously from subdiffusive (timorous) to superdiffusive (bold). The scaling exponents are fully determined with a new mathematical approach based on a decomposition of the dynamics in active journeys (the walker is at the maximum distance) and lazy journeys (the walker is not at the maximum distance).

Solvable random-walk model with memory and its relations with Markovian models of anomalous diffusion

“…The above expression indeed differs from (15) or (18). The limit distribution g(x) in the present model does not reduce, to our knowledge, to a standard distribution.…”

“…The diffusion front is a function which is practically unknown for reinforced walk models and almost never studied for other non-Markovian models which are solvable for the MSD [13][14][15] (but see Ref. [57] for a discussion on the socalled elephant walk).…”

“…Note that (14), (15), and (16) are universal relations, i.e., the corresponding limiting distributions P (n,t) do not depend on q nor on the details of the memory kernel but only on the fact that F (τ ) ∼ τ −β at large τ . Alternatively, expressions (15) and (16) can be written using special functions.…”

“…Alternatively, expressions (15) and (16) can be written using special functions. In this paper we define…”

“…This type of motion, widely studied in * boyer@fisica.unam.mx Markovian contexts [11,12], is characterized by a mean-square displacement (MSD) of the particle which does not follow the Smoluchowski-Einstein law of Brownian motion, but a slower one, of the form X 2 t ∝ t μ with μ an exponent <1. Few random-walk models with long-range memory are analytically tractable [13][14][15]. For a single particle, pathdependent diffusion is notoriously difficult to formalize and many results on reinforced random walks are based on numerical simulations.…”