Random words, quantum statistics, central limits, random matrices

Recently Tracy and Widom conjectured [math.CO/9904042] and Johansson proved [math.CO/9906120] that the expected shape \lambda of the semi-standard tableau produced by a random word in k letters is asymptotically the spectrum of a random traceless k by k GUE matrix. In this article we give two arguments for this fact. In the first argument, we realize the random matrix itself as a quantum random variable on the space of random words, if this space is viewed as a quantum state space. In the second argument, we show that the distribution of \lambda is asymptotically given by the usual local limit theorem, but the resulting Gaussian is disguised by an extra polynomial weight and by reflecting walls. Both arguments more generally apply to an arbitrary finite-dimensional representation V of an arbitrary simple Lie algebra g. In the original question, V is the defining representation of g = su(k).Comment: 11 pages. Minor changes suggested by the refere

Orthogonal polynomial ensembles in probability theory

We survey a number of models from physics, statistical mechanics, probability theory and combinatorics, which are each described in terms of an orthogonal polynomial ensemble. The most prominent example is apparently the Hermite ensemble, the eigenvalue distribution of the Gaussian Unitary Ensemble (GUE), and other well-known ensembles known in random matrix theory like the Laguerre ensemble for the spectrum of Wishart matrices. In recent years, a number of further interesting models were found to lead to orthogonal polynomial ensembles, among which the corner growth model, directed last passage percolation, the PNG droplet, non-colliding random processes, the length of the longest increasing subsequence of a random permutation, and others. Much attention has been paid to universal classes of asymptotic behaviors of these models in the limit of large particle numbers, in particular the spacings between the particles and the fluctuation behavior of the largest particle. Computer simulations suggest that the connections go even farther and also comprise the zeros of the Riemann zeta function. The existing proofs require a substantial technical machinery and heavy tools from various parts of mathematics, in particular complex analysis, combinatorics and variational analysis. Particularly in the last decade, a number of fine results have been achieved, but it is obvious that a comprehensive and thorough understanding of the matter is still lacking. Hence, it seems an appropriate time to provide a surveying text on this research area.Comment: Published at http://dx.doi.org/10.1214/154957805100000177 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

A Cover-Merging-Based Algorithm for the Longest Increasing Subsequence in a Sliding Window Problem

A longest increasing subsequence problem (LIS) is a well-known combinatorial problem with applications mainly in bioinformatics, where it is used in various projects on DNA sequences. Recently, a number of generalisations of this problem were proposed. One of them is to find an LIS among all fixed-size windows of the input sequence (LISW). We propose an algorithm for the LISW problem based on cover representation of the sequence that outperforms the existing methods for some class of the input sequences

Subsequences in Bounded Ranges: Matching and Analysis Problems

In this paper, we consider a variant of the classical algorithmic problem of checking whether a given word $v$ is a subsequence of another word $w$. More precisely, we consider the problem of deciding, given a number $p$ (defining a range-bound) and two words $v$ and $w$, whether there exists a factor $w[i:i+p-1]$ (or, in other words, a range of length $p$) of $w$ having $v$ as subsequence (i.\,e., $v$ occurs as a subsequence in the bounded range $w[i:i+p-1]$). We give matching upper and lower quadratic bounds for the time complexity of this problem. Further, we consider a series of algorithmic problems in this setting, in which, for given integers $k$, $p$ and a word $w$, we analyse the set $p$-Subseq$_{k}(w)$ of all words of length $k$ which occur as subsequence of some factor of length $p$ of $w$. Among these, we consider the $k$-universality problem, the $k$-equivalence problem, as well as problems related to absent subsequences. Surprisingly, unlike the case of the classical model of subsequences in words where such problems have efficient solutions in general, we show that most of these problems become intractable in the new setting when subsequences in bounded ranges are considered. Finally, we provide an example of how some of our results can be applied to subsequence matching problems for circular words.Comment: Extended version of a paper which will appear in the proceedings of the 16th International Conference on Reachability Problems, RP 202

Universal Randomness

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 {\it 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.Comment: 34 pages, 6 figure

Run Generation Revisited: What Goes Up May or May Not Come Down

In this paper, we revisit the classic problem of run generation. Run generation is the first phase of external-memory sorting, where the objective is to scan through the data, reorder elements using a small buffer of size M , and output runs (contiguously sorted chunks of elements) that are as long as possible. We develop algorithms for minimizing the total number of runs (or equivalently, maximizing the average run length) when the runs are allowed to be sorted or reverse sorted. We study the problem in the online setting, both with and without resource augmentation, and in the offline setting. (1) We analyze alternating-up-down replacement selection (runs alternate between sorted and reverse sorted), which was studied by Knuth as far back as 1963. We show that this simple policy is asymptotically optimal. Specifically, we show that alternating-up-down replacement selection is 2-competitive and no deterministic online algorithm can perform better. (2) We give online algorithms having smaller competitive ratios with resource augmentation. Specifically, we exhibit a deterministic algorithm that, when given a buffer of size 4M , is able to match or beat any optimal algorithm having a buffer of size M . Furthermore, we present a randomized online algorithm which is 7/4-competitive when given a buffer twice that of the optimal. (3) We demonstrate that performance can also be improved with a small amount of foresight. We give an algorithm, which is 3/2-competitive, with foreknowledge of the next 3M elements of the input stream. For the extreme case where all future elements are known, we design a PTAS for computing the optimal strategy a run generation algorithm must follow. (4) Finally, we present algorithms tailored for nearly sorted inputs which are guaranteed to have optimal solutions with sufficiently long runs