Arctic Octahedron in Three-Dimensional Rhombus Tilings and Related Integer Solid Partitions

Three-dimensional integer partitions provide a convenient representation of codimension-one three-dimensional random rhombus tilings. Calculating the entropy for such a model is a notoriously difficult problem. We apply transition matrix Monte Carlo simulations to evaluate their entropy with high precision. We consider both free- and fixed-boundary tilings. Our results suggest that the ratio of free- and fixed-boundary entropies is _free/ _fixed=3/2, and can be interpreted as the ratio of the volumes of two simple, nested, polyhedra. This finding supports a conjecture by Linde, Moore, and Nordahl concerning the arctic octahedron phenomenon in three-dimensional random tilings.     

Asymptotics of the Spectra of One-Dimensional Natural Vibrations in Media Consisting of Solid and Fluid Layers

Doklady Physics - The spectra of one-dimensional natural vibrations of two-phase layered media with a periodic structure are studied. The first phase is an isotropic (elastic or viscoelastic)...     

Asymptotics of the Eta Invariant

We prove an asymptotic bound on the eta invariant of a family of coupled Dirac operators on an odd dimensional manifold. In the case when the manifold is the unit circle bundle of a positive line bundle over a complex manifold, we obtain precise formulas for the eta invariant.     

Asymptotics of the Airy-Kernel Determinant

The authors use Riemann-Hilbert methods to compute the constant that arises in the asymptotic behavior of the Airy-kernel determinant of random matrix theory.     

Asymptotics of the S-matrix and unitarisation

The manner in which the elastic scattering amplitude obeys unitarity, how it enters the circle of unitarity, and what its asymptotic limit is, remains a problem for models which include terms that rise fast with s. We have checked that the features of cross sections which come from unitarisation are present for most unitarisation schemes, e.g. those that saturate the profile function or those that describe multiple exchanges via an analytic formula. We have also obtained a scheme which interpolates between different classes of the unitarisation and found corresponding non-linear equations. Considering different forms of energy dependence of the scattering amplitude, and a variety of unitarisation schemes, we show that, in order to reproduce the data, the fits choose an amplitude that corresponds to an asymptotic value S = 0.     

Asymptotics of the Mean-Field Heisenberg Model

We consider the mean-field classical Heisenberg model and obtain detailed information about the total spin of the system by studying the model on a complete graph and sending the number of vertices to infinity. In particular, we obtain Cramér- and Sanov-type large deviations principles for the total spin and the empirical spin distribution and demonstrate a second-order phase transition in the Gibbs measures. We also study the asymptotics of the total spin throughout the phase transition using Stein’s method, proving central limit theorems in the sub- and supercritical phases and a nonnormal limit theorem at the critical temperature.     

Entropy of Partitions on Quantum Logic

Partition and entropy of partitions in quantum logic are introduced and their properties are investigated. The results are generalized to the general case of T-norm and T-conorm.     

Plane Partitions with Two-Periodic Weights

We study scaling limits of skew plane partitions with periodic weights under several boundary conditions. We compute the correlation kernel of the limiting point process in the bulk and near turning points on the frozen boundary. The system develops pairs of turning points (points where three different phases meet), which are separated by “semi-frozen” regions. We show that the point process at such a turning point is a pair of non-trivially correlated GUE minor processes. In the limit when all weights become the same, i.e. in the homogeneous case, such a pair of turning points collapses into a single turning point and the local process becomes the GUE minor process. We also study an intermediate regime when the weights are periodic but all converge to 1. In this regime the limit shape and correlations in the bulk are the same as in the case of homogeneous weights, and periodicity is not visible in the bulk. However, the process at turning points is still not the GUE minor process.     

Markov Partitions for dispersed billiards

Markov Partitions for some classes of billiards in two-dimensional domains on ² or two-dimensional torus are constructed. Using these partitions we represent the microcanonical distribution of the corresponding dynamical system in the form of a limit Gibbs state and investigate the character of its approximations by finite Markov chains.Dedicated to the memory of Rufus Bowen     

The Entropy of Partitions on MV-Algebras

Partitions of MV-algebras are studied. Using the notion of a state (as a probabilisticmeasure) on MV-algebras, we introduce the entropy of partitions. We show asuitable method for the refinement of partitions and the subadditivity of theentropy with respect to this refinement.     

Random Skew Plane Partitions and the Pearcey Process

We study random skew 3D partitions weighted by q ^vol and, specifically, the q → 1 asymptotics of local correlations near various points of the limit shape. We obtain sine-kernel asymptotics for correlations in the bulk of the disordered region, Airy kernel asymptotics near a general point of the frozen boundary, and a Pearcey kernel asymptotics near a cusp of the frozen boundary.     

On Asymptotics for the Airy Process

The Airy process t→A(t), introduced by Prähofer and Spohn, is the limiting stationary process for a polynuclear growth model. Adler and van Moerbeke found a PDE in the variables s ₁,s ₂, and t for the probability Pr(A(0)≤s ₁, A(t)≤s ₂). Using this they were able, assuming the truth of a certain conjecture and appropriate uniformity, to obtain the first few terms of an asymptotic expansion for this probability as t→∞, with fixed s ₁ and s ₂. We shall show that the expansion can be obtained by using the Fredholm determinant representation for the probability. The main ingredients are formulas obtained by the author and C. A. Tracy in the derivation of the Painlevé II representation for the distribution function F ₂ plus a few others obtained in the same way.     

Matrix Kernels for Measures on Partitions

We consider the problem of computation of the correlation functions for the z-measures with the deformation (Jack) parameters 2 or 1/2. Such measures on partitions are originated from the representation theory of the infinite symmetric group, and in many ways are similar to the ensembles of Random Matrix Theory of β=4 or β=1 symmetry types. For a certain class of such measures we show that correlation functions can be represented as Pfaffians including 2×2 matrix valued kernels, and compute these kernels explicitly. We also give contour integral representations for correlation kernels of closely connected measures on partitions. Supported by US-Israel Binational Science Foundation (BSF) Grant No. 2006333.     

Conditional Entropy of Partitions on Quantum Logic

A construction of conditional entropy of partitions on quantum logic is given, and the properties of conditional entropy are investigated.     

Limit Shapes for Gibbs Ensembles of Partitions

We explicitly compute limit shapes for several grand canonical Gibbs ensembles of partitions of integers. These ensembles appear in models of aggregation and are also related to invariant measures of zero range and coagulation-fragmentation processes. We show, that all possible limit shapes for these ensembles fall into several distinct classes determined by the asymptotics of the internal energies of aggregates.     

Affine Crystal Graphs And Two-Colored Partitions

We define the cores and the quotients of the elements in the crystal graph of the basic representation of or , and then obtain new combinatorial interpretations of the character and the string function formula. Mathematics Subject Classifications (2000). 17B37, 05A17.