Asymptotics of Plancherel measures for symmetric groups

We consider the asymptotics of the Plancherel measures on partitions of as goes to infinity. We prove that the local structure of a Plancherel typical partition in the middle of the limit shape converges to a determinantal point process with the discrete sine kernel.

On the edges of the limit shape, we prove that the joint distribution of suitably scaled 1st, 2nd, and so on rows of a Plancherel typical diagram converges to the corresponding distribution for eigenvalues of random Hermitian matrices (given by the Airy kernel). This proves a conjecture due to Baik, Deift, and Johansson by methods different from the Riemann-Hilbert techniques used in their original papers and from the combinatorial proof given by the second author. Our approach is based on an exact determinantal formula for the correlation functions of the poissonized Plancherel measures in terms of a new kernel involving Bessel functions. Our asymptotic analysis relies on the classical asymptotic formulas for the Bessel functions and depoissonization techniques.

     

Graph Decomposition and Parity

Motivated by a recent extension of the zero‐one law by Kolaitis and Kopparty, we study the distribution of the number of copies of a fixed disconnected graph in the random graph . We use an idea of graph decompositions to give a sufficient condition for this distribution to tend to uniform modulo q. We determine the asymptotic distribution of all fixed two‐component graphs in for all q, and we give infinite families of many‐component graphs with a uniform asymptotic distribution for all q. We also prove a negative result that no recursive proof of the simplest form exists for a uniform asymptotic distribution for arbitrary graphs.     

On a Lower Bound of the Uniform Distance in the Central Limit Theorem for ϕ-Mixing Random Variables

We derive a lower bound of the uniform distance in the central limit theorem for real -mixing random variables under the finiteness of the eighth moments of summands. The main result of the present paper generalizes the corresponding author"s result obtained in 1997 for m-dependent random variables to the case of -mixing random variables.     

Exponential sums,the geometry of hyperplane sections,and some diophantine problems

We estimate exponential sums with additive character along an affine variety given by a system of homogeneous equations, with a homogeneous function in the exponent. The proof uses the results of Deligne’s Weil Conjectures II and a generalization of Lefschetz hyperplane theorem to singular varieties. We apply our estimate to obtain an upperbound for the number of integer solutions of a system of homogeneous equations in a box. Another application is devoted to uniform distribution of solutions of a system of homogeneous congruences modulo a prime in the following sense: the portion of solutions in a box is proportional to the volume of the box, provided the box is not very small.     

Distributional Transformations,Orthogonal Polynomials,and Stein Characterizations

A new class of distributional transformations is introduced, characterized by equations relating function weighted expectations of test functions on a given distribution to expectations of the transformed distribution on the test functions higher order derivatives. The class includes the size and zero bias transformations, and when specializing to weighting by polynomial functions, relates distributional families closed under independent addition, and in particular the infinitely divisible distributions, to the family of transformations induced by their associated orthogonal polynomial systems. For these families, generalizing a well known property of size biasing, sums of independent variables are transformed by replacing summands chosen according to a multivariate distribution on its index set by independent variables whose distributions are transformed by members of that same family. A variety of the transformations associated with the classical orthogonal polynomial systems have as fixed points the original distribution, or a member of the same family with different parameter.     

Large eddy simulation turbulence model with Young measures

We show an alternative proof for the existence of weak solutions to equations describing turbulent flows of fluids. The proof proposed by one of the authors in a previous paper (cf. [A. Świerczewska, Large Eddy Simulation. Existence of Stationary Solutions to a Dynamical Model (submitted for publication). Preprint TU-Darmstadt no. 2314, http://wwwbib.mathematik.tu-darmstadt.de/Math-Net/Preprints/Listen/shadow/pp2314.html]) based on more classical methods. We will use Young measures, which allow us to shorten significantly the limiting procedure in the nonlinear terms and generalize the statement.     

Convergence of Clock Processes and Aging in Metropolis Dynamics of a Truncated REM

We study the aging behavior of a truncated version of the Random Energy Model evolving under Metropolis dynamics. We prove that the natural time-time correlation function defined through the overlap function converges to an arcsine law distribution function, almost surely in the random environment and in the full range of time scales and temperatures for which such a result can be expected to hold. This establishes that the dynamics ages in the same way as Bouchaud’s REM-like trap model, thus extending the universality class of the latter model. The proof relies on a clock process convergence result of a new type where the number of summands is itself a clock process. This reflects the fact that the exploration process of Metropolis dynamics is itself an aging process, governed by its own clock. Both clock processes are shown to converge to stable subordinators below certain critical lines in their time-scale and temperature domains, almost surely in the random environment.     

Asymptotic normality in random graphs with given vertex degrees

We consider random graphs with a given degree sequence and show, under weak technical conditions, asymptotic normality of the number of components isomorphic to a given tree, first for the random multigraph given by the configuration model and then, by a conditioning argument, for the simple uniform random graph with the given degree sequence. Such conditioning is standard for convergence in probability, but much less straightforward for convergence in distribution as here. The proof uses the method of moments, and is based on a new estimate of mixed cumulants in a case of weakly dependent variables. The result on small components is applied to give a new proof of a recent result by Barbour and Röllin on asymptotic normality of the size of the giant component in the random multigraph; moreover, we extend this to the random simple graph.     

A uniform estimate for fourier transforms of measures on polynomial curves in ℝ n

We prove a uniform estimate for the decrease of the fourier transforms of smooth measures on polynomial curves in ⁿ . Our estimate improves the estimate recently obtained byW. Stadje.     

Mean‐Field Interaction of Brownian Occupation Measures II: A Rigorous Construction of the Pekar Process

We consider mean‐field interactions corresponding to Gibbs measures on interacting Brownian paths in three dimensions. The interaction is self‐attractive and is given by a singular Coulomb potential. The logarithmic asymptotics of the partition function for this model were identified in the 1980s by Donsker and Varadhan in terms of the Pekar variational formula, which coincides with the behavior of the partition function of the polaron problem under strong coupling. Based on this, in 1986 Spohn made a heuristic observation that the strong coupling behavior of the polaron path measure, on certain time scales, should resemble a process, named as the Pekar process, whose distribution could somehow be guessed from the limiting asymptotic behavior of the mean‐field measures under interest, whose rigorous analysis remained open. The present paper is devoted to a precise analysis of these mean‐field path measures and convergence of the normalized occupation measures towards an explicit mixture of the maximizers of the Pekar variational problem. This leads to a rigorous construction of the aforementioned Pekar process and hence is a contribution to the understanding of the “mean‐field approximation” of the polaron problem on the level of path measures. The method of our proof is based on the compact large deviation theory developed by Mukherjee and Varadhan in 2016; its extension to the uniform strong metric for the singular Coulomb interaction was carried out by König and Mukherjee in 2015, as well as an idea inspired by a partial path exchange argument appearing in 1997 in work by Bolthausen and Schmock.© 2017 Wiley Periodicals, Inc.     

Uniform recursive trees: Branching structure and simple random downward walk

As models for spread of epidemics, family trees, etc., various authors have used a random tree called the uniform recursive tree. Its branching structure and the length of simple random downward walk (SRDW) on it are investigated in this paper. On the uniform recursive tree of size n, we first give the distribution law of ζn,m, the number of m-branches, whose asymptotic distribution is the Poisson distribution with parameter . We also give the joint distribution of the numbers of various branches and their covariance matrix. On Ln, the walk length of SRDW, we first give the exact expression of P(Ln=2). Finally, the asymptotic behavior of Ln is given.     

Consistency of Probability Measure Quantization by Means of Power Repulsion–Attraction Potentials

This paper is concerned with the study of the consistency of a variational method for probability measure quantization, deterministically realized by means of a minimizing principle, balancing power repulsion and attraction potentials. The proof of consistency is based on the construction of a target energy functional whose unique minimizer is actually the given probability measure \(\omega \) to be quantized. Then we show that the discrete functionals, defining the discrete quantizers as their minimizers, actually \(\Gamma \)-converge to the target energy with respect to the narrow topology on the space of probability measures. A key ingredient is the reformulation of the target functional by means of a Fourier representation, which extends the characterization of conditionally positive semi-definite functions from points in generic position to probability measures. As a byproduct of the Fourier representation, we also obtain compactness of sublevels of the target energy in terms of uniform moment bounds, which already found applications in the asymptotic analysis of corresponding gradient flows. To model situations where the given probability is affected by noise, we further consider a modified energy, with the addition of a regularizing total variation term and we investigate again its point mass approximations in terms of \(\Gamma \)-convergence. We show that such a discrete measure representation of the total variation can be interpreted as an additional nonlinear potential, repulsive at a short range, attractive at a medium range, and at a long range not having effect, promoting a uniform distribution of the point masses.     

Exact and limiting distributions of the number of successions in a random permutation

Traditionally the distributions of the number of patterns and successions in a random permutation ofn integers 1,2, ..., andn were studied by combinatorial analysis. In this short article, a simple way based on finite Markov chain imbedding technique is used to obtain the exact distribution of successions on a permutation. This approach also gives a direct proof that the limiting distribution of successions is a Poisson distribution with parameter =1. Furthermore, a direct application of the main result, it also yields the waiting time distribution of a succession.This work was supported in part by the Natural Sciences and Engineering Research Council of Canada under Grant NSERC A-9216, and National Science Council of Republic of China under Grant 85-2121-M-259-003.     

The probability of avoiding consecutive patterns in the Mallows distribution

We use combinatorial and probabilistic techniques to study growth rates for the probability that a random permutation from the Mallows distribution avoids consecutive patterns. The Mallows distribution is a q‐analogue of the uniform distribution weighting each permutation π by , where is the number of inversions in π and q is a positive, real‐valued parameter. We prove that the growth rate exists for all patterns and all q > 0, and we generalize Goulden and Jackson's cluster method to keep track of the number of inversions in permutations avoiding a given consecutive pattern. Using singularity analysis, we approximate the growth rates for length‐3 patterns, monotone patterns, and non‐overlapping patterns starting with 1, and we compare growth rates between different patterns. We also use Stein's method to show that, under certain assumptions on q and σ, the number of occurrences of a given pattern σ is well approximated by the normal distribution.     

Derangements and Tensor Powers of Adjoint Modules for \mathfrak{s}\mathfrak{l}_n

We obtain the decomposition of the tensor space as a module for , find an explicit formula for the multiplicities of its irreducible summands, and (when n 2k) describe the centralizer algebra = ( ) and its representations. The multiplicities of the irreducible summands are derangement numbers in several important instances, and the dimension of is given by the number of derangements of a set of 2k elements.     

Central Limit Theorem for Random Strict Partitions

We consider the set of all partitions of a number n into distinct summands (the so-called strict partitions) with the uniform distribution on it and study fluctuations of a random partition near its limit shape, for large n. The use of geometrical language allows us to state the problem in terms of the limit behavior of random step functions (Young diagrams). A central limit theorem for such functions is proven. Our method essentially uses the notion of large canonical ensemble of partitions. Bibliography: 7 titles.     

Tails of waiting times and their bounds

Tails of distributions having the form of the geometric convolution are considered. In the case of light-tailed summands, a simple proof of the famous Cramér asymptotic formula is given via the change of probability measure. Some related results are obtained, namely, bounds of the tails of geometric convolutions, expressions for the distribution of the 1st failure time and failure rate in regenerative systems, and others. In the case of heavy-tailed summands, two-sided bounds of the tail of the geometric convolution are given in the cases where the summands have either Pareto or Weibull distributions. The results obtained have the property that the corresponding lower and upper bounds are tailed-equivalent. This revised version was published online in June 2006 with corrections to the Cover Date.     

On Waring’s problem for several algebraic forms

We reconsider the classical problem of representing a finite number of forms of degree d in the polynomial ring over n + 1 variables as scalar combinations of powers of linear forms. We define a geometric construct called a grove, which, in a number of cases, allows us to determine the dimension of the space of forms which can be so represented for a fixed number of summands. We also present two new examples, where this dimension turns out to be less than what a naïve parameter count would predict.