Exact spatiotemporal dynamics of lattice random walks in hexagonal and honeycomb domains

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2 Citations (Scopus)


A variety of transport processes in natural and man-made systems are intrinsically random. To model their stochasticity, lattice random walks have been employed for a long time, mainly by considering Cartesian lattices.
However, in many applications in bounded space the geometry of the domain may have profound effects on the dynamics and ought to be accounted for. We consider here the cases of the six-neighbor (hexagonal) and three-neighbor (honeycomb) lattices, which are utilized in models ranging from adatoms diffusing in metals and excitations diffusing on single-walled carbon nanotubes to animal foraging strategy and the formation of territories in scent-marking organisms. In these and other examples, the main theoretical tool to study the dynamics of lattice random walks in hexagonal geometries has been via simulations. Analytic representations have in most cases been inaccessible, in particular in bounded hexagons, given the complicated “zigzag” boundary
conditions that a walker is subject to. Here we generalize the method of images to hexagonal geometries and obtain closed-form expressions for the occupation probability, the so-called propagator, for lattice random walks
both on hexagonal and honeycomb lattices with periodic, reflective, and absorbing boundary conditions. In the periodic case, we identify two possible choices of image placement and their corresponding propagators. Using
them, we construct the exact propagators for the other boundary conditions, and we derive transport-related statistical quantities such as first-passage probabilities to one or multiple targets and their means, elucidating the
effect of the boundary condition on transport properties.
Original languageEnglish
Article number 054139
JournalPhys. Rev. E
Issue number5
Publication statusPublished - 30 May 2023

Bibliographical note

Funding Information:
L.G. acknowledges funding from the Biotechnology and Biological Sciences Research Council (BBSRC) Grant No. BB/T012196/1 and the Natural Environment Research Council (NERC) Grant No. NE/W00545X/1, while D.M. and S.S. acknowledge funding from Engineering and Physical Sciences Research Council (EPSRC) DTP studentships with References No. 2610858 and No. 2123342, respectively.

Publisher Copyright:
© 2023 authors. Published by the American Physical Society.

Structured keywords

  • Engineering Mathematics Research Group


  • Random walks
  • Lattice


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