Student Probability

I will cover the method of establishing superconcentration via hypercontractive inequalities with two examples: Talagrand's Gaussian L1-L2 inequality and the KKL inequality for Boolean functions. The basic definitions and identities involving semigroups and the dirichlet form will be covered for…

I will introduce the book "Superconcentration and Related Topics" by Sourav Chatterjee, which will be the topic of this quarter's seminar.  Superconcentration occurs when classical concentration of measure gives suboptimal bounds on the order of fluctuations. I will go over several examples…

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I plan to cover Newman-Piza's theorem that in two dimensions, the FPP passage time T(0, x) has super-constant fluctuation (subject to some assumptions about the weight distribution). I will follow the original paper of…

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I will sketch Auffinger and Damron’s proof of the universal relation \chi = 2 \ksi – 1 assuming the existence of the fluctuation exponent \chi and the wandering exponent \ksi in the sense of Chatterjee. The talk will be based on section 4.3 of the notes and the papers …

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I will talk about fluctuations in first passage percolations. I will cover the linear and sublinear variance bounds in Sections 3.1 and 3.2 of “Fifty years of FFP”. This material is also reviewed in the survey “Fluctuations in First Passage Percolation” by Philippe Sosoe: …

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First- and last-passage percolation are models in which weights are placed on the edges of a graph (usually Z^d), and where the objects of interest are the shortest (respectively longest) paths from the origin to various points on the graph. I'll describe both models and describe…

Abstract: The set of all n by n orthogonal (or unitary) matrices form a compact topological group. This means that orthogonal matrices have a uniform distribution called Haar measure. A natural question is, what do typical orthogonal or unitary matrices “look like”? We will answer two version of…

Abstract: This quarter, we've explored the empirical distribution of the eigenvalues for general Wigner matrices, showing convergence in distribution to the semicircle law for Hermitian matrices. As Christian described, the special case with Gaussian entries allows us to say more and get an…