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Random Matrices and the Six-Vertex Model

Random Matrices and the Six-Vertex Model

Pavel Bleher; Karl Liechty

American Mathematical Society
2013
sidottu
This book provides a detailed description of the Riemann-Hilbert approach (RH approach) to the asymptotic analysis of both continuous and discrete orthogonal polynomials, and applications to random matrix models as well as to the six-vertex model. The RH approach was an important ingredient in the proofs of universality in unitary matrix models. This book gives an introduction to the unitary matrix models and discusses bulk and edge universality. The six-vertex model is an exactly solvable two-dimensional model in statistical physics, and thanks to the Izergin-Korepin formula for the model with domain wall boundary conditions, its partition function matches that of a unitary matrix model with nonpolynomial interaction. The authors introduce in this book the six-vertex model and include a proof of the Izergin-Korepin formula. Using the RH approach, they explicitly calculate the leading and subleading terms in the thermodynamic asymptotic behavior of the partition function of the six-vertex model with domain wall boundary conditions in all the three phases: disordered, ferroelectric, and antiferroelectric.
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Random Growth Models

Random Growth Models

American Mathematical Society
2018
sidottu
The study of random growth models began in probability theory about 50 years ago, and today this area occupies a central place in the subject. The considerable challenges posed by these models have spurred the development of innovative probability theory and opened up connections with several other parts of mathematics, such as partial differential equations, integrable systems, and combinatorics. These models also have applications to fields such as computer science, biology, and physics.This volume is based on lectures delivered at the 2017 AMS Short Course ``Random Growth Models'', held January 2-3, 2017 in Atlanta, GA.The articles in this book give an introduction to the most-studied models; namely, first- and last-passage percolation, the Eden model of cell growth, and particle systems, focusing on the main research questions and leading up to the celebrated Kardar-Parisi-Zhang equation. Topics covered include asymptotic properties of infection times, limiting shape results, fluctuation bounds, and geometrical properties of geodesics, which are optimal paths for growth.
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Random Matrices

Random Matrices

American Mathematical Society
2019
sidottu
Random matrix theory has many roots and many branches in mathematics, statistics, physics, computer science, data science, numerical analysis, biology, ecology, engineering, and operations research. This book provides a snippet of this vast domain of study, with a particular focus on the notations of universality and integrability. Universality shows that many systems behave the same way in their large scale limit, while integrability provides a route to describe the nature of those universal limits. Many of the ten contributed chapters address these themes, while others touch on applications of tools and results from random matrix theory.This book is appropriate for graduate students and researchers interested in learning techniques and results in random matrix theory from different perspectives and viewpoints. It also captures a moment in the evolution of the theory, when the previous decade brought major break-throughs, prompting exciting new directions of research.
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Random Explorations

Random Explorations

Gregory F. Lawler

AMERICAN MATHEMATICAL SOCIETY
2023
nidottu
The title Random Explorations has two meanings. First, a few topics of advanced probability are deeply explored. Second, there is a recurring theme of analyzing a random object by exploring a random path.This book is an outgrowth of lectures by the author in the University of Chicago Research Experiences for Undergraduate (REU) program in 2020. The idea of the course was to expose advanced undergraduates to ideas in probability research.The book begins with Markov chains with an emphasis on transient or killed chains that have finite Green's function. This function, and its inverse called the Laplacian, is discussed next to relate two objects that arise in statistical physics, the loop-erased random walk (LERW) and the uniform spanning tree (UST). A modern approach is used including loop measures and soups. Understanding these approaches as the system size goes to infinity requires a deep understanding of the simple random walk so that is studied next, followed by a look at the infinite LERW and UST. Another model, the Gaussian free field (GFF), is introduced and related to loop measure. The emphasis in the book is on discrete models, but the final chapter gives an introduction to the continuous objects: Brownian motion, Brownian loop measures and soups, Schramm-Loewner evolution (SLE), and the continuous Gaussian free field. A number of exercises scattered throughout the text will help a serious reader gain better understanding of the material.
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Random Matrices, Frobenius Eigenvalues, and Monodromy

Random Matrices, Frobenius Eigenvalues, and Monodromy

Nicholas M. Katz; Peter Sarnak

AMERICAN MATHEMATICAL SOCIETY
1999
nidottu
The main topic of this book is the deep relation between the spacings between zeros of zeta and $L$-functions and spacings between eigenvalues of random elements of large compact classical groups. The authors draw upon many disparate areas of mathematics from algebraic geometry, moduli spaces, mondromy, equidistribution, and the Weil conjectures to probability theory and the compact classical groups.
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Random Riches

Random Riches

Routledge
2016
sidottu
Gambling is a fascinating subject which for many centuries has attracted public interest. Yet, despite its ubiquity, gambling (or gaming) leads a marginal existence within the boundaries of scholarly research. Providing a longue duree survey, this volume promotes a historical understanding of the subject enriched with a diverse academic approach that draws upon sociology, economics and psychology. Each chapter in the collection is the work of a renowned scholar with a long standing interest in gambling research. The contributions offer historical analyses of the medieval origins of the 'Gambler State' and of mathematical risk calculation. They cast light on the roles of different stakeholders in gambling including the playing public, business, and the state. They provide a controversial discussion of the alleged 'pathological' nature of chance games and the reasons for either regulating or freeing them from state control. Last but not least, two authors deal with country-by-country specifics in gaming cultures and gambling markets. Taken as a whole, the chapters in this volume chart the development of European gambling culture from the medieval to modern times. In so doing it provides essential context for both historical and current debates about the nature of gambling and lotteries, addiction to gambling, poverty and social degradation on the fringes of the welfare state.
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Random Heterogeneous Materials

Random Heterogeneous Materials

Salvatore Torquato

Springer-Verlag New York Inc.
2013
nidottu
The interdisciplinary subject of random heterogeneous materials has experienced remarkable growth since the publication of the well-known monograph Statistical Con­ tinuum Theories by Beran ( 1968). Many of these advances, especially those concerning the statistical characterization of the microstructure and its effect on the physical prop­ erties of the material, have not been treated fully in any book. One of the intents of the present book is to fill this gap. This book also distinguishes itself in that it provides a unified rigorous framework to characterize the microstructures and macroscopic properties of the widely diverse types of heterogeneous materials found in nature and synthetic products. Emphasis is placed on providing foundational theoretical methods that can simultaneously yield results of practical utility. This book treats a wide breadth of topics, but the choice of subjects naturally reflects my own interests. The sheer enormity of the field has prevented me from covering many important topics. I apologize to those colleagues, known and unknown, who may not find enough of their own work cited in the ensuing pages.
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