Random Walks on Finite Fields and Heisenberg Groups
dc.contributor.advisor | Benson, Chal | en_US |
dc.contributor.author | Zhu, Yi | en_US |
dc.contributor.department | Mathematics | en_US |
dc.date.accessioned | 2011-06-24T15:35:40Z | |
dc.date.available | 2011-06-24T15:35:40Z | |
dc.date.issued | 2011 | en_US |
dc.description.abstract | Let H be a finite group and [mu] a probability measure on H. This data determines an invariant random walk on H beginning from the identity element. The probability distribution for the state of the random walk after n steps is given by the n'th convolution power of the probability measure [mu]. The random walk and measure [mu] are said to be ergodic if the support of this distribution is the entire group for n sufficiently large. In this case a specialization of the Markov Ergodic Theorem ensures that the distribution after n steps converges point-wise to the uniform distribution. One employs the total variation distance on probability measures to analyze the rate of convergence to equilibrium. Suppose now that a finite group K acts on H by automorphisms. We say that the action pair K : H is ergodic when the K-invariant probability measure [mu] supported on some K-orbit is ergodic. We call, moreover, K : H a Gelfand action pair when the convolution algebra of K-invariant functions on H is commutative. Specializing the theory of spherical functions to the context of Gelfand action pairs we obtain a version of the Diaconis-Shahshahani Upper Bound Lemma, controlling the total variation distance to equilibrium for the random walk determined by [mu]. The main results in this thesis concern invariant random walks on finite fields and three dimensional Heisenberg groups over finite fields. Let F be a finite field of odd characteristic and K a subgroup of the multiplicative group for F with even order. We obtain a necessary and sufficient condition for ergodicity of the action pair K : F and an explicit summation formula for the upper bound on total variation distance to equilibrium guaranteed by the Upper Bound Lemma. Let F[~] be a quadratic extension field for F and U denote the kernel of the norm mapping from F[~] to F. An application of our field theoretic criterion for ergodicity shows that U : F[~] is an ergodic action pair and we specialize our upper bound result to this context. Forming the three dimensional Heisenberg group H = F[~] x F over F the action of U on F[~] induces an action of U on H by automorphisms. Benson and Ratcliff have shown that U : H is a Gelfand action pair and determined the associated spherical functions. We prove that the pair U : H is ergodic and make explicit the bound given by the Upper Bound Lemma. | en_US |
dc.description.degree | M.A. | en_US |
dc.format.extent | 92 p. | en_US |
dc.format.medium | dissertations, academic | en_US |
dc.identifier.uri | http://hdl.handle.net/10342/3603 | |
dc.language.iso | en_US | |
dc.publisher | East Carolina University | en_US |
dc.subject | Mathematics | en_US |
dc.subject | Gelfand pairs | en_US |
dc.subject | Heisenberg group | en_US |
dc.subject | Random walks (Mathematics) | en_US |
dc.subject | Spherical functions | en_US |
dc.subject.lcsh | Finite fields (Algebra) | |
dc.subject.lcsh | Invariants | |
dc.title | Random Walks on Finite Fields and Heisenberg Groups | en_US |
dc.type | Master's Thesis | en_US |
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