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Random Matrices, the Ulam Problem, Directed Polymers & Growth Models, and Sequence Matching

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Author:  Satya N. Majumdar 














Description 
In these lecture notes I will give a pedagogical introduction to some common aspects of 4 different problems: (i) random matrices (ii) the longest increasing subsequence problem (also known as the Ulam problem) (iii) directed polymers in random medium and growth models in (1 1) dimensions and (iv) a problem on the alignment of a pair of random sequences. Each of these problems is almost entirely a subfield by itself and here I will discuss only some specific aspects of each of them. These 4 problems have been studied almost independently for the past few decades, but only over the last few years a common thread was found to link all of them. In particular all of them share one common limiting probability distribution known as the TracyWidom distribution that describes the asymptotic probability distribution of the largest eigenvalue of a random matrix. I will mention here, without mathematical derivation, some of the beautiful results discovered in the past few years. Then, I will consider two specific models (a) a ballistic deposition growth model and (b) a model of sequence alignment known as the Bernoulli matching model and discuss, in some detail, how one derives exactly the TracyWidom law in these models. The emphasis of these lectures would be on how to map one model to another. Some open problems are discussed at the end. 

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