High-dimensional phenomena : dilations, tensor products and geometry of L₁
High-dimensional phenomena : dilations, tensor products and geometry of L₁
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高维现象：L₁ 的膨胀、张量积和几何
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
T. Tkocz
中科院分区：
文献类型：
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作者：
T. Tkocz
本论文的目的是研究与高维现象有关的几个分析、几何和概率问题。
第一个问题是研究具有一定对称性的集合的膨胀的概率测度的行为。我们表明，对于标准的高斯测度复向量空间，圆柱体是最佳的意义上说，根据膨胀，高斯测度增长不超过其他域具有足够的对称性更迅速的圆柱体。我们还证明了一个类似的结果在真实的情况下的威布尔分布和伽玛分布。因此，我们得到这些分布的矩的最佳比较。
第二个问题源于对复合周期量子系统的研究。它询问某些随机矩阵的行为，当它们的大小趋于无穷大时。我们证明了两个大的随机酉矩阵的张量积的谱是渐近泊松的，这是我们对对角矩阵的期望。对于大量的2 2随机酉矩阵的张量积也得到了同样的结论。
第三个问题涉及L1上算子的可逆性。我们构造了一个具有任意大维数核的局部可逆算子的例子。该结构是组合的，依赖于扩展图和计算机科学关于`1上的限制等距属性的最新结果。我们还建立了一些Sobolev型不等式，并找到了一类在大子空间上全局可逆的卷积算子
The purpose of this dissertation is to study several problems related to high-dimensional phenomena in analysis, geometry and probability.
The first problem examines the behaviour of probability measures of dilations of sets possessing certain symmetries. We show that for the standard Gaussian measure on complex vector space, cylinders are optimal in the sense that, under dilations, the Gaussian measure grows no more rapidly for cylinders than for other domains possessing enough symmetries. We also prove an analogous result in the real case for Weibull and Gamma distributions. As a consequence, we derive optimal comparison of moments for these distributions.
The second problem stems from the study of composite periodic quantum systems. It asks about the behaviour of certain random matrices when their size tends to infinity. We show that the spectrum of the tensor product of two large random unitary matrices is asymptotically Poissonian; what we would expect for diagonal matrices. The same conclusion is established for the tensor product of a large number of 2 2 random unitary matrices.
The third problem concerns the invertibility of operators on L1. We construct an example of a locally invertible operator with kernel of arbitrarily large dimension. The construction is combinatorial, relying on expander graphs and recent results from computer science about the restricted isometry property on `1. We also establish some Sobolev-type inequalities and find a certain large class of convolution operators which are globally invertible on large subspaces