The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

One-dimensional Bose-gas One-dimensional Heisenberg magnet Massive Thirring model Classical r-matrix Fundamentals of inverse scattering method Algebraic Bethe ansatz Quantum field theory integral models on a lattice Theory of scalar products Form factors Mean value of operator Q Assymptotics of correlation functions Temperature correlation functions Appendices References.

Quantum Inverse Scattering Method and Correlation Functions

We study a certain random groeth 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.

Shape Fluctuations and Random Matrices

Thank you for downloading reflection groups and coxeter groups. Maybe you have knowledge that, people have search numerous times for their favorite books like this reflection groups and coxeter groups, but end up in malicious downloads. Rather than enjoying a good book with a cup of coffee in the afternoon, instead they cope with some infectious bugs inside their computer. reflection groups and coxeter groups is available in our digital library an online access to it is set as public so you can download it instantly. Our book servers spans in multiple countries, allowing you to get the most less latency time to download any of our books like this one. Kindly say, the reflection groups and coxeter groups is universally compatible with any devices to read.

Reflection Groups And Coxeter Groups

Free Random Variables

We prove the existence of a limit shape and give its explicit description for certain probability distribution on signatures (or highest weights for unitary groups). The distributions have representation theoretic origin—they encode decomposition on irreducible characters of the restrictions of certain extreme characters of the infinite-dimensional unitary group $U(\infty)$ to growing finite-dimensional unitary subgroups $U(N)$. The characters of $U(\infty)$ are allowed to depend on $N$. In a special case, this describes the hydrodynamic behavior for a family of random growth models in $(2+1)$-dimensions with varied initial conditions.

/pdf/limit-shapes-for-growing-extreme-characters-of-u-44o1tgcn6n.pdf

Limit shapes for growing extreme characters of U(

A combination of direct and inverse Fourier transforms on the unitary group $U(N)$ identifies normalized characters with probability measures on $N$-tuples of integers. We develop the $N\to\infty$ version of this correspondence by matching the asymptotics of partial derivatives at the identity of logarithm of characters with Law of Large Numbers and Central Limit Theorem for global behavior of corresponding random $N$-tuples. 
As one application we study fluctuations of the height function of random domino and lozenge tilings of a rich class of domains. In this direction we prove the Kenyon-Okounkov's conjecture (which predicts asymptotic Gaussianity and exact form of the covariance) for a family of non-simply connected polygons. 
Another application is a central limit theorem for the $U(N)$ quantum random walk with random initial data.

Fourier transform on high-dimensional unitary groups with applications to random tilings

In this paper we show that a variety of interacting particle systems with multiple species can be viewed as random walks on Hecke algebras. This class of systems includes the asymmetric simple exclusion process (ASEP), M-exclusion TASEP, ASEP(q,j), stochastic vertex models, and many others. As an application, we study the asymptotic behavior of second class particles in some of these systems.

Interacting particle systems and random walks on Hecke algebras

In this paper, we introduce a procedure that, given a solution to the Yang–Baxter equation as input, produces a stochastic (or Markovian) solution to (a possibly dynamical version of) the Yang–Baxter equation. We then apply this “stochasticization procedure” to obtain three new, stochastic solutions to several different forms of the Yang–Baxter equation. The first is a stochastic, elliptic solution to the dynamical Yang–Baxter equation; the second is a stochastic, higher rank solution to the dynamical Yang–Baxter equation; and the third is a stochastic solution to a dynamical variant of the tetrahedron equation.

/pdf/stochasticization-of-solutions-to-the-yang-baxter-equation-1v2tx7opzk.pdf

Stochasticization of Solutions to the Yang–Baxter Equation

The asymptotics of the first rows and columns of random Young diagrams corresponding to extremal characters of the infinite symmetric group is studied. We consider rows and columns with linear growth in $n$, the number of boxes of random diagrams, and prove the central limit theorem for them in the case of distinct Thoma parameters. We also establish a more precise statement relating the growth of rows and columns of Young diagrams to a simple independent random sampling model. 
After this paper was completed, the author learned that the central limit theorem has been also proved in the work of M\'eliot (arXiv:1105.0091v1) by a different method.

Alexey Bufetov

Papers

Limit shapes for growing extreme characters of U(

Fourier transform on high-dimensional unitary groups with applications to random tilings

Interacting particle systems and random walks on Hecke algebras

Stochasticization of Solutions to the Yang–Baxter Equation

The central limit theorem for extremal characters of the infinite symmetric group