2007/01/10 by Satya N. Majumdar, Majumdar, Satya N.
Computer Science · Decision Sciences · Mathematics · Physics and Astronomy · #Data Management and Algorithms #Disordered Systems and Neural Networks (cond-mat.dis-nn) #FOS: Mathematics #FOS: Physical sciences #Fuzzy and Soft Set Theory #Graph Theory and Algorithms #Mathematical Physics (math-ph) #Probability (math.PR) #Statistical Mechanics (cond-mat.stat-mech) #cond-mat.dis-nn #cond-mat.stat-mech #math-ph #math.MP #math.PR
paper · pdf · doi:10.48550/arxiv.cond-mat/0701193
Les Houches lecture notes for the summer school on "Complex Systems" (Les Houches, July, 2006); 39 pages, 11 .eps figures included
arxiv created 2007/01/10 · openalex publication_date 2007/01/10 · arxiv updated 2009/12/01 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
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 sub-field 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 Tracy-Widom 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 Tracy-Widom 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.