We study a certain random growth model in two dimensions closely related to the one-dimensional totally asymmetric exclusion process. The results show that the shape fluctuations, appropriately scaled, converges in distribution to the Tracy-Widom largest eigenvalue distribution for the Gaussian Unitary Ensemble (GUE).

Abstract. We describe an implementation of the generalized Schur algorithm for the superfast solution of real positive definite Toeplitz systems of order n + 1, where n = 2ν. Our implementation uses the split-radix fast Fourier transform algorithms for real data of Duhamel. We are able to obtain the nth Szegő polynomial using fewer than 8n log2 2 n real arithmetic operations without explicit use of the bit-reversal permutation. Since Levinson’s algorithm requires slightly more than 2n2 operations to obtain this polynomial, we achieve crossover with Levinson’s algorithm at n = 256. Key words. Toeplitz matrix, Schur’s algorithm, split-radix Fast Fourier Transform

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Dedicated to V. I. Arnold on his 65th birthday Abstract. Let a second order Fuchsian differential equation with only univalued solutions have finite singular points at z1,..., zn with exponents (ρ1,1, ρ2,1),...,(ρ1,n, ρ2,n). Let the exponents at infinity be (ρ1,∞, ρ2,∞). Then for fixed generic z1,..., zn, the number of such Fuchsian equations is equal to the multiplicity of the irreducible sl2 representation of dimension |ρ2, ∞ − ρ1,∞ | in the tensor product of irreducible sl2 representations of dimensions |ρ2,1 − ρ1,1|,..., |ρ2,n − ρ1,n|. To show this we count the number of critical points of a suitable function which plays the crucial role in constructions of the hypergeometric solutions of the sl2 KZ equation and of the Bethe vectors in the sl2 Gaudin model. As a byproduct of this study we conclude that the set of Bethe vectors is a basis in the space of states for the sl2 inhomogeneous Gaudin model. 1.

Consider random Young diagrams with fixed number n of boxes, distributed according to the Plancherel measure Mn. That is, the weight Mn(λ) of a diagram λ equals dim 2 λ/n!, where dim λ denotes the dimension of the irreducible representation of the symmetric group Sn indexed by λ. As n → ∞, the boundary of the (appropriately rescaled) random shape λ concentrates near a curve Ω (Logan– Shepp 1977, Vershik–Kerov 1977). In 1993, Kerov announced a remarkable theorem describing Gaussian fluctuations around the limit shape Ω. Here we propose a reconstruction of his proof. It is largely based on Kerov’s unpublished work notes, 1999.

Abstract. It is proved that the limiting distribution of the length of the longest weakly increasing subsequence in an inhomogeneous random word is related to the distribution function for the eigenvalues of a certain direct sum of Gaussian unitary ensembles subject to an overall constraint that the eigenvalues lie in a hyperplane. 1.

Abstract. The Gibbs distribution for n particles of the Coulomb gas on the unit circle at inverse temperature β is given by E β n(f) = 1 · · · f(e

We calculate the triple correlations for the truncated divisor sum λR(n). The λR(n) behave over certain averages just as the prime counting von Mangoldt function Λ(n) does or is conjectured to do. We also calculate the mixed (with a factor of Λ(n)) correlations. The results for the moments up to the third degree, and therefore the implications for the distribution of primes in short intervals, are the same as those we obtained (in the first paper with this title) by using the simpler approximation ΛR(n). However, when λR(n) is used, the error in the singular series approximation is often much smaller than what ΛR(n) allows. Assuming the Generalized Riemann Hypothesis (GRH) for Dirichlet L-functions, we obtain an Ω±-result for the variation of the error term in the prime number theorem. Formerly, our knowledge under GRH was restricted to Ω-results for the absolute value of this variation. An important ingredient in the last part of this work is a recent result due to Montgomery and Soundararajan which makes it possible for us to dispense with a large error term in the evaluation of a certain singular series average. We believe that our results on the sums λR(n) and ΛR(n) can be employed in diverse problems concerning primes. 1.

Dunkl operators are parametrized differential-difference operators on R N which are related to finite reflection groups. They can be regarded as a generalization of partial derivatives and play a major role in the study of Calogero-Moser-Sutherland-type quantum many-body systems. Dunkl operators lead to generalizations of various analytic structures, like the Laplace operator, the Fourier transform, Hermite polynomials and the heat semigroup. In this paper we investigate some probabilistic aspects of this theory in a systematic way. For this, we introduce a concept of homogeneity of Markov processes on R N which generalizes the classical notion of processes with independent, stationary increments to the Dunkl setting. This includes analogues of Brownian motions and Cauchy processes. The generalizations of Brownian motions have the c`adl`ag property and form, after symmetrization with respect to the underlying reflection groups, diffusions on the Weyl chambers. A major part of the p...

The purpose of this article is threefold. First, it provides the reader with a few useful and efficient tools which should enable her/him to evaluate nontrivial determinants for the case such a determinant should appear in her/his research. Second, it lists a number of such determinants that have been already evaluated, together with explanations which tell in which contexts they have appeared. Third, it points out references where further such determinant evaluations can be found.