CEU Electronic Theses and Dissertations, 2020
| Author | Ravelonanosy, Mahefa Ratsisetraina |
|---|---|
| Title | The concentration of measure and the concentration of distance phenomena |
| Summary | In this thesis, we prove non-concentration of distance for the 2-dimensional integer lattice {o, 1, ..., n} 2 , the two dimensional integer torus Z × Z/(nZ × nZ) aIn this thesis, we prove non-concentration of distance for the 2-dimensional integer lattice {o, 1, ..., n} 2 , the two dimensional integer torus Z × Z/(nZ × nZ) and the Uniform Spanning Tree (UST) of the complete graph K n . On the other hand, we prove that the distance has concentration property for the Hypercube {0, 1} n , the Euclidean space R n with the n-dimensional standard Gaussian measure, the unit sphere S n−1 ⊆ R n with the normalized Lebesgue measure and the ball of radius R of a non-elementary Hyperbolic group. To our knowledge, this last example has not been discussed in the literature and it is the main novel part of this work. We do some Python visualizations for the distribution of distance in the UST of K n , (Z/nZ) 2 and (Z/nZ) 5 , and we present an application of the concentration of distance phenomenon on a transitive metric probability space.nd the Uniform Spanning Tree (UST) of the complete graph K n . On the other hand, we prove that the distance has concentration property for the Hypercube {0, 1} n , the Euclidean space R n with the n-dimensional standard Gaussian measure, the unit sphere S n−1 ⊆ R n with the normalized Lebesgue measure and the ball of radius R of a non-elementary Hyperbolic group. To our knowledge, this last example has not been discussed in the literature and it is the main novel part of this work. We do some Python visualizations for the distribution of distance in the UST of K n , (Z/nZ) 2 and (Z/nZ) 5 , and we present an application of the concentration of distance phenomenon on a transitive metric probability space. |
| Supervisor | Pete Gabor |
| Department | Mathematics MSc |
| Full text | https://www.etd.ceu.edu/2020/ravelonanosy_mahefa.pdf |
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