Abstract: Many important quantities in large random structures exhibit a strong concentration phenomenon. However, applying a general concentration inequality usually cannot derive meaningful results because often such quantities are difficult to express as a function of the structures. For example, the rank of a random matrix is difficult to be expressed as a function of its entries and may be challenging to analyze.

 To determine and quantify such a phenomenon, one needs to observe from the insights of the random structures and find some crucial events in which sharp non-asymptomatic estimates are obtainable. In this talk, we will discuss some works in these: the rank of random matrices,  nodal domains of random graphs, reconstruct-ability of random graphs from its local structures, and the distribution of minimal distance for random linear codes, and a classical discrete convex geometry problem in which concentration of measure approach is useful.

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