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Aspects of the Heisenberg group
The Heisenberg group is a remarkable simple mathematical object, with interesting algebraic, geometric, and probabilistic aspects. It is available in tow flavors: discrete and continuous. The...
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Independence and central limit theorems
Let \( {\mathcal{A}} \) be a \( {\star} \)-algebra over \( {\mathbb{C}} \) with zero \( {0} \), unit \( {1} \), and involution \( {a\mapsto a^*} \) such that \( {(ab)^*=b^*a^*} \) for every \(...
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Spectrum of Markov generators of random graphs
I had the pleasure to upload recently on arXiv and on HAL a collaborative work with Charles Bordenave and Pietro Caputo, entitled Spectrum of Markov generators on sparse random graphs. Let \(...
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Average Characteristic Polynomial
François Viète – Opera Mathematica. The characteristic polynomial \( {\chi} \) of a sequence of complex numbers \( {z_1,\ldots,z_n} \) is defined by \[ \chi(z):=\prod_{k=1}^n(z-z_k). \] If \(...
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Random walks on regular graphs
We will see in this post how the arcsine law appears naturally in the combinatorics of paths of the simple random walk on \( {\mathbb{Z}} \), while the Wigner semicircle law appears naturally in the...
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From Boltzmann to random matrices and beyond
Persi during his talk – Toulouse, March 2014. I have recently uploaded on the French HAL a preprint numbered hal-00987177 (on arXiv it is 1405.1003) modestly entitled From Boltzmann to random matrices...
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Informal aspects of free probability
Dan-Virgil Voiculescu (photo by G. Bergman) This post gives some few bits of free probability theory. The content is mostly taken from the expository text arXiv:1405.1003. Free probability theory was...
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