University Library
  • Login
A gateway to Melbourne's research publications
Minerva Access is the University's Institutional Repository. It aims to collect, preserve, and showcase the intellectual output of staff and students of the University of Melbourne for a global audience.
View Item 
  • Minerva Access
  • Science
  • School of Mathematics and Statistics
  • School of Mathematics and Statistics - Theses
  • View Item
  • Minerva Access
  • Science
  • School of Mathematics and Statistics
  • School of Mathematics and Statistics - Theses
  • View Item
JavaScript is disabled for your browser. Some features of this site may not work without it.

    Interacting Quarter-Plane Lattice Walk Problems: Solutions and Proofs

    Thumbnail
    Download
    Final thesis file (1.943Mb)

    Supporting thesis file (316.0Mb)

    Citations
    Altmetric
    Author
    Xu, Ruijie
    Date
    2020
    Affiliation
    School of Mathematics and Statistics
    Metadata
    Show full item record
    Document Type
    PhD thesis
    Access Status
    Open Access
    URI
    http://hdl.handle.net/11343/265884
    Description

    © 2020 Ruijie Xu

    Abstract
    Lattice walk problems in the quarter-plane have been widely studied in recent years. The main objective is to calculate the number of configurations, that is the number of $n$-step walks ending at certain points or alternatively, the generating function of the walks. In combinatorics, physics and probability theory, other properties such as asymptotic behavior and the algebra of the generating functions are also of interest. In this thesis we focus on solving quarter-plane lattice walks with interactions via the kernel method. We assign interaction $a$ to the $x$-axis, $b$ to the $y$-axis and $c$ to the origin. We denote $q_{n,k,l,h,u,v}$ as the number of $n$-step paths that start at $(0,0)$ and end at point $(k,l)$, and which visit vertices on the horizontal boundary (except the origin) $h$ times, vertices on the vertical boundary (except the origin) $v$ times and the origin $u$ times. Then $Q(x,y,t)=\sum_{n,k,l,h,u,v} q_{n,k,l,h,u,v}x^k y^l t^n a^h b^v c^u$ is the generating function of the walk. Our aim is to find whether the interactions will affect the solubility of quarter-plane lattice walk models and the properties of the generation functions $Q(x,y,t)$. In particular, we give solutions to all $23$ quarter-plane lattice walks associated with finite groups. We solve $21$ of them explicitly, writing an expression for the generating functions $Q(x,y,t)$. The other $2$ are only partly solved. We discuss when the generating function $Q(x,y,t)$ become D-algebraic, D-finite and algebraic by the variations of $a,b,c$. We also compare the solutions obtained by the kernel method and the solutions obtained by an analytic method based on homomorphisms on a Riemann surface. We combine the solutions in another form with the solution of the kernel method, and prove the properties of the generating function $Q(x,y,t)$. We observe that the solution obtained using the kernel method and other analytic methods are consistent with one another.
    Keywords
    lattice walks; quarter-plane; random walks; enumeration; algebraic combinatorics; generating functions; kernel method; algebraic, D-finite, D-algebraic; Weierstrass elliptic function

    Export Reference in RIS Format     

    Endnote

    • Click on "Export Reference in RIS Format" and choose "open with... Endnote".

    Refworks

    • Click on "Export Reference in RIS Format". Login to Refworks, go to References => Import References


    Collections
    • Minerva Elements Records [53102]
    • School of Mathematics and Statistics - Theses [194]
    Minerva AccessDepositing Your Work (for University of Melbourne Staff and Students)NewsFAQs

    BrowseCommunities & CollectionsBy Issue DateAuthorsTitlesSubjectsThis CollectionBy Issue DateAuthorsTitlesSubjects
    My AccountLoginRegister
    StatisticsMost Popular ItemsStatistics by CountryMost Popular Authors