On the distribution of the length of the longest increasing subsequence of random permutations

Baik, Jinho; Deift, Percy; Johansson, Kurt

doi:10.1090/s0894-0347-99-00307-0

Cited by 1,025 publications

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“…However, the main interest in this system is not the typical value of the longest increasing subsequences, but their fluctuations. Recently it has been shown [2,22] that in the limit of large N the fluctuations of l N scale as N 1/6 , namely,…”

Section: A Combinatoricsmentioning

confidence: 99%

“…In this formulation the problem of statistics of the length of such directed polymers looks somewhat different from the LIS problem considered above. It can be shown however, that in the limit of large times t and large N these two problems become equivalent [2]. For a given (fixed) density ρ of dots instead of the total number of dots N one can measure the length l(t) of the polymer in terms of the size of the square t. Since ρ = 2N/t 2 , assuming that ρ is a parameter which is of the order of one, we note that t ∝ √ N .…”

Section: Directed Polymersmentioning

confidence: 99%

“…Nowadays we have got rather comprehensive list of various systems (both purely mathematical and physical) whose macroscopic statistical properties are described by the same universal Tracy-Widom (TW) distribution function. These systems are: the longest increasing subsequences (LIS) model [2] (Section I.A) zero-temperature lattice directed polymers with geometric disorder [3] the polynuclear growth (PNG) system [4], (Section I.B) the oriented digital boiling model [5], the ballistic decomposition model [6], the longest common subsequences (LCS) [7], the onepoint distribution of the solutions of the KPZ equation [8] (which describes the motion of an interface separating two homogeneous bulk phases) in the long time limit [9,10], and finally finite temperature directed polymers in random potentials with short-range correlations [11][12][13][14]. It should be noted that directed polymers in a quenched random potential have been the subject of intense investigations during the past three decades (see e.g.…”

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confidence: 99%

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Universal randomness

Dotsenko

2011

Phys.-Usp.

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During last two decades it has been discovered that the statistical properties of a number of microscopically rather different random systems at the macroscopic level are described by the same universal probability distribution function which is called the Tracy-Widom (TW) distribution. Among these systems we find both purely methematical problems, such as the longest increasing subsequences in random permutations, and quite physical ones, such as directed polymers in random media or polynuclear crystal growth. In the extensive Introduction we discuss in simple terms these various random systems and explain what the universal TW function is. Next, concentrating on the example of one-dimensional directed polymers in random potential we give the main lines of the formal proof that fluctuations of their free energy are described the universal TW distribution. The second part of the review consist of detailed appendices which provide necessary self-contained mathematical background for the first part.

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Section: A Combinatoricsmentioning

confidence: 99%

Section: Directed Polymersmentioning

confidence: 99%

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Universal randomness

Dotsenko

2011

Phys.-Usp.

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“…[36]) but its connection with RMT is recent [37,38]. In a recent paper, Baik, Deift and Johansson [36] have proved the following remarkable result:…”

Section: Distribution Of the Length Of The Longest Increasing Subsequmentioning

confidence: 99%

“…We thank J. Baik for sending us [36] prior to publication. The first author acknowledges helpful conversations with Bruno Nachtergaele and Marko Robnik.…”

Section: Acknowledgmentsmentioning

confidence: 99%

Universality of the distribution functions of random matrix theory

Tracy¹,

Widom²

2000

CRM Proceedings and Lecture Notes

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In probability theory and statistics a common first approximation to many random processes is a sequence X 1 , X 2 , X 3 , . . . of independent and identically distributed (iid) random variables. Let F denote their common distribution. To motivate the material below, we take these random variables and construct a particularly simpleThe order statistics are the eigenvalues orderedand the distribution of the largest eigenvalue, λ max (N ) = λ N , is Prob (λ max (N ) ≤ x) = Prob (X 1 ≤ x, . . . , X N ≤ x) = F (x) N .

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Coincidences and Patterns in Sequences

Dasgupta

2005

Encyclopedia of Statistical Sciences

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This entry provides a contemporary exposition at a moderately quantitative level of the distribution theory associated with sequences and patterns in iid multinomial trials, the birthday problem, and the matching problem. The section on patterns includes the classic distribution theory for waiting time for runs and more general patterns, and their means and moments. It also includes the central limit theorem and a.s. properties of the longest run, and modern applications to DNA sequence alignment. The section on birthdays reviews the Poisson approximation in the classical birthday problem, with new bounds on the total variation distance. It also includes a number of variants of the classic birthday problem, including the case of unequal probabilities, similar triplets, and the Bayesian version with Dirichlet priors on birthday probabilities. A new problem called the strong birthday problem with application to criminology is introduced. The section on matching covers the Poisson asymptotics, errors of Poisson approximations, and some variants such as near matches. It also briefly reviews the recent extraordinary developments in the area of longest increasing subsequences in random permutations. The entry provides a large number of examples, many not well known, to help a reader have a feeling for these questions at an intuitive level

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On the distribution of the length of the longest increasing subsequence of random permutations

Cited by 1,025 publications

References 51 publications

Universal randomness

Universal randomness

Universality of the distribution functions of random matrix theory

Coincidences and Patterns in Sequences

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