Algebraic Area of Lattice Random Walks and Exclusion Statistics
We focus on the enumeration of closed lattice random walks according to their algebraic area, with connections to quantum exclusion statistics, as well as the combinatorics of generalized Dyck and Motzkin paths. First, taking the closed square lattice walks as an example, we review the concept of the algebraic area and its connections to the Hofstadter model. Then, we introduce two approaches for the algebraic area enumeration. The first approach relies on the computation of the secular determinant of the Hofstadter Hamiltonian and its relation to the exclusion statistics. Precisely, the coefficients of the secular determinant are interpreted in terms of partition functions with exclusion parameter $g=2$. The algebraic area enumeration is obtained in terms of the associated cluster coefficients. The second approach involves a direct computation of the trace of the $n$th power of the Hofstadter matrix, where $n$ is the length of the walks. We study the combinatorics of periodic Dyck paths and obtain explicit expressions for counting Dyck paths with a fixed number of up steps starting from each floor, which provide a combinatorial interpretation to the factor in the algebraic area enumeration formula obtained in the first approach. Then, we study the closed random walks on a honeycomb lattice and establish a correspondence to a system of particles obeying a mixture of $g=1$ (fermions) and $g=2$ exclusion statistics, together with the connection to the combinatorics of periodic Motzkin paths. Furthermore, we extend the algebraic area concept to closed cubic lattice walks and map the enumeration onto the cluster coefficients of three types of particles obeying $g=1$, $g=1$, and $g=2$ exclusion statistics, respectively, with the constraint that the numbers of $g=1$ exclusion particles of the two types are equal.
Jury : Cyril Banderier, Cyril Furtlehner, Brian Hopkins (rapporteur), Stéphane Ouvry (directeur de thèse), Valentina Ros, Clément Sire, Stephan Wagner (rapporteur).