2021/10/31 by Alexios P. Polychronakos
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #Combinatorics #Computer science #Diffusion and Search Dynamics #Embedding #Generating function #Geometry #Logarithm #Mathematical analysis #Mathematics #Path length #Planar #Propagator #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #cond-mat.stat-mech #hep-th #math-ph #math.CO #math.MP
paper · pdf · doi:10.1103/physreve.105.024102
published as Phys. Rev. E 105, 024102, 2 February 2022 · Version to appear in Phys.Rev.E; includes additional references and discussion
arxiv created 2022/01/18 · openalex publication_date 2022/02/02 · arxiv updated 2022/02/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We derive the length and area generating function of planar height-restricted forward-moving discrete paths of increments ±1 or 0 with arbitrary starting and ending points, the so-called Motzkin meanders, and the more general length-area generating functions for Motzkin paths with markers monitoring the number of passages from the two height boundaries ("floor" and "ceiling") and the time spent there. The results are obtained by embedding Motzkin paths in a two-step anisotropic Dyck path process and using propagator, exclusion statistics, and bosonization techniques. We also present a cluster expansion of the logarithm of the generating functions that makes their polynomial structure explicit. These results are relevant to the derivation of statistical mechanical properties of physical systems such as polymers, vesicles, and solid-on-solid interfaces.