A sluggish random walk with subdiffusive spread
Aniket Zodage, Rosalind J. Allen, M. R. Evans, Satya N. Majumdar
Abstract
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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> <mml:mi>k</mml:mi> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> </mml:math> from the origin as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> <mml:mi>k</mml:mi> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> </mml:math> for large <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> <mml:mi>k</mml:mi> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> </mml:math> . We show that the typical position after time t scales as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> <mml:mi>k</mml:mi> <mml:msup> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> <mml:mi>α</mml:mi> </mml:msup> </mml:math> with α > 0.