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Entropic CLT and phase transition in high-dimensional Wishart matrices
TLDR
An information theoretic phase transition is proved: high dimensional Wishart matrices are close in total variation distance to the corresponding Gaussian ensemble if and only if d is much larger than $n^3$ and the chain rule for relative entropy is used.
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Upper tails and independence polynomials in random graphs
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Recovery and Rigidity in a Regular Stochastic Block Model
TLDR
R rigidity is proved by showing that with high probability an exact recovery of the community structure is possible and that, in this setting, any suitably good partial recovery can be bootstrapped to obtain a full Recovery of the communities.
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Consistent nonparametric estimation for heavy-tailed sparse graphs
TLDR
This work studies graphons as a non-parametric generalization of stochastic block models, and shows how to obtain compactly represented estimators for sparse networks in this framework and characterize identifiability for these graphons and their underlying position spaces.
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Upper Tails for Edge Eigenvalues of Random Graphs
In this note we prove a precise large deviation principle for the largest and second largest eigenvalues of a sparse Erd\H{o}s-R\'enyi graph. Our arguments rely on various recent breakthroughs in the…
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On Non-localization of Eigenvectors of High Girth Graphs
TLDR
Improved bounds are proved on how localized an eigenvector of a high girth regular graph can be and a construction showing a lower bound of $\log _d(k)/\varepsilon $ up to multiplicative constants is presented.
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Fractal geometry of Airy$_{2}$ processes coupled via the Airy sheet
In last passage percolation models lying in the Kardar-Parisi-Zhang universality class, maximizing paths that travel over distances of order $n$ accrue energy that fluctuates on scale $n^{1/3}$; and…
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Upper Tail Large Deviations for Arithmetic Progressions in a Random Set
Let $X_k$ denote the number of $k$-term arithmetic progressions in a random subset of $\mathbb{Z}/N\mathbb{Z}$ or $\{1, \dots, N\}$ where every element is included independently with probability $p$.…
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High-girth near-Ramanujan graphs with localized eigenvectors
We show that for every prime $d$ and $\alpha\in (0,1/6)$, there is an infinite sequence of $(d+1)$-regular graphs $G=(V,E)$ with girth at least $2\alpha \log_{d}(|V|)(1-o_d(1))$, second adjacency…
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A complete characterization of the evolution of RC4 pseudo random generation algorithm
TLDR
A novel distinguisher for RC4 shows that under certain conditions the equality of two consecutive bytes is more probable than by random association and holds regardless of the amount of initial keystream bytes thrown away during the RC4 PRGA.
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