A sluggish random walk with subdiffusive spread

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• Satya Majumdar (Université Paris Saclay)
• Wednesday 04 September 2024, 09:45-10:30
• Seminar Room 1, Newton Institute.

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SSDW05 - Modelling and Applications of Anomalous Diffusions

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

This talk is part of the Isaac Newton Institute Seminar Series series.

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