1 / 9

Recurrent character of biased random walk on regular lattices

Recurrent character of biased random walk on regular lattices. Hyun Keun Lee (KIAS→SKKU) Nonequilibrium Statistical Physics of Complex Systems KIAS, July 1~4 (2008) in collaboration with Hyunggyu Park. Return-to-the-origin problem. May a random walker return to the origin, eventually?

kris
Download Presentation

Recurrent character of biased random walk on regular lattices

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Recurrent character of biased random walk on regular lattices Hyun Keun Lee (KIAS→SKKU) Nonequilibrium Statistical Physics of Complex Systems KIAS, July 1~4 (2008) in collaboration with Hyunggyu Park

  2. Return-to-the-origin problem • May a random walker return to the origin, eventually? • For (pure) random walk on regular lattice, • d : dimension • Renewal of stochastic system

  3. -1 0 -1 2b+3 +1 0 Biased random walk • Nearest-neighbor stochastic hopping biased to the origin • Probability: • b(>=1): bias strength • ? b c above which the walk becomes recurrent

  4. What already known in tree • For a tree T, b c = br(T) • If T is random, b c = br(T) = <branch#> • More or less, it is intuitive. • Is this applicable to the walk on regular lattice where various kinds of loops exist? • the answer is “yes”. How? * regular lattice = lattice covered by a repeating block

  5. Decomposition of walk on regular lattices • lattice walk = patched walks on trees ..,T1|T2,T2,T2|T1,T1|T2|.. T1T2 • lattice origin = the root of each tree • all lattice points appear in each tree • distance to the origin = distance to the root • every edge between two lattice points showing different distances to the origin appears in the trees at least once

  6. Useful (in)equalities • For each Ti, it holds that1≤ • : # of points at the nth level from the root • Thus, the walk on each Ti is recurrent as long as b>1 . • Therefore, the lattice walk (series of the patched walks on Ti s) remains recurrent.

  7. Generalization I • In case of cubic lattice • z-axis and x-y planes • Similar ones for x- and y-axis • Then, T1,…,T6 trees • Note that ⇒br(Ti)=1 for all i • For d-dim hypercube lattice • xd-axis and x1-x2…-xd-1planes • Consider all combinations⇒T1,…,Td ! trees • still holds.

  8. Generalization II • Independence of lattice detail • |di-dj |= 0 or 1 for nearest neighbors i and j=> It is always possible to construct the tree set which holds the four conditions. • ..,Ti|Tj,Tj,Tj|na,na|Tj|Tk,Tk|.. • nas just cause time-delay • still holds in any case.

  9. Conclusion • Random walk biased to the origin on regular lattices is always recurrent irrespective of bias strength, lattice detail, and dimension. • as long as b>1 • No sufficient room in any type of regular lattice, that can drive the biased random walker out of the origin.

More Related