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Lattice walks in cones: Combinatorial and probabilistic aspects

Resource type
Thesis type
(Thesis) Ph.D.
Date created
2019-12-06
Authors/Contributors
Abstract
Lattice walks in cones have many applications in combinatorics and probability theory. While walks restricted to the first quadrant have been well studied, the case of non-convex cones and three-dimensional walks has been systematically approached recently. In this thesis, we extend the analytic method of the study of walks and its discrete harmonic functions in the quarter plane to the three-quarter plane applying the strategy of splitting the domain into two symmetric convex cones. This method is composed of three main steps: write a system of functional equations satisfied by the generating function, which may be simplified into one single equation under symmetry conditions; transform the functional equation into a boundary value problem; and solve this problem using conformal mappings. We obtain explicit expressions for the generating functions of walks and its associated harmonic functions. The advantage of this method is the uniform treatment of models corresponding to different step sets. In a second part of this thesis, we explore the asymptotic enumeration of three-dimensional excursions confined to the positive octant. The critical exponent is related to the smallest eigenvalue of a Dirichlet problem in a spherical triangle. Combinatorial properties of the step set are related to geometric and analytic properties of the associate spherical triangle.
Document
Identifier
etd20644
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Copyright is held by the author.
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This thesis may be printed or downloaded for non-commercial research and scholarly purposes.
Scholarly level
Supervisor or Senior Supervisor
Thesis advisor: Mishna, Marni
Thesis advisor: Raschel, Kilian
Member of collection
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