An introduction to large deviations with applications in physics

Random matrix theory has found many applications in physics, statistics and engineering since its inception. Although early developments were motivated by practical experimental problems, random matrices are now used in fields as diverse as Riemann hypothesis, stochastic differential equations, condensed matter physics, statistical physics, chaotic systems, numerical linear algebra, neural networks, multivariate statistics, information theory, signal processing and small-world networks. This article provides a tutorial on random matrices which provides an overview of the theory and brings together in one source the most significant results recently obtained. Furthermore, the application of random matrix theory to the fundamental limits of wireless communication channels is described in depth.

A Busy March Meeting Is Brewing in Seattle

Some randomness is present in most phenomena, ranging from biomolecules and nanodevices to financial markets and human organizations. However, it is not easy to gain an intuitive understanding of such stochastic phenomena, because their modeling requires advanced mathematical tools, such as sigma algebras, the Itô formula and martingales. Here, we discuss a simple finite difference algorithm that can be used to gain understanding of such complex physical phenomena. In particular, we simulate the motion of an optically trapped particle that is typically used as a model system in statistical physics and has a wide range of applications in physics and biophysics, for example, to measure nanoscopic forces and torques.

This volume presents a representative work of Chinese probabilists on probability theory and its applications in physics. Interesting results of jump Markov processes are discussed, as well as Markov interacting processes with noncompact states, including the Schlogal model taken from statistical physics. The main body of this book is self-contained and can be used in a course in stochastic processes for graduate students. The book consists of four parts. In Parts 1 and 2, the author introduces the general theory for jump processes. New contributions to the classical problems: uniqueness, recurrence and positive recurrence are studied. Then, probability metrics and coupling methods, stochastically monotonicity, reversibility, large deviations and the estimates of L squared-spectral gap are discussed. Part 3 begins with the study of equilibrium particle systems. This contains the criteria of the reversibility, the construction of Gibbs states and the particle systems on lattice fractals. The final part emphasizes the reaction-diffusion processes which come from non-equilibrium statistical physics. Topics include constructions, existence of stationary distributions, ergodicity, phase transitions and hydrodynamic limits for the processes.

This manuscript is the lecture notes of B. Barak’s course in the Les Houches ‘Statistical Physics and Machine Learning’ summer school in 2022. It surveys various proxies for computational hardness in random planted problems, from the low-degree likelihood ratio to statistical query complexity and the Franz–Parisi criterion, as well as the various relationships between those criteria. We also present a few aspects of the study of deep learning, from both a theoretical and empirical point of view.

In recent years statistical physics has proven to be a valuable tool to probe into large dimensional inference problems such as the ones occurring in machine learning. Statistical physics provides analytical tools to study fundamental limitations in their solutions and proposes algorithms to solve individual instances. In these notes, based on the lectures by Marc Mézard in 2022 at the summer school in Les Houches, we will present a general framework that can be used in a large variety of problems with weak long-range interactions, including the compressed sensing problem, or the problem of learning in a perceptron. We shall see how these problems can be studied at the replica symmetric level, using developments of the cavity methods, both as a theoretical tool and as an algorithm.

It is natural to model the spectral behaviour of disordered quantum systems by the typical spectral properties of random operators. Depending on the underlying physics different kinds of ensembles have been introduced. Two types of ensembles, random Schrödinger operators and random matrices, have proved to be of particular interest because of their wide ranges of applicability and because of their rich mathematical structures. On first sight these two types of random operators may appear to be close relatives. However, as it turns out, their typical spectral properties differ significantly. Moreover, the methods that have been developed for their respective analysis have little in common. One may say that over the years two different cultures have evolved leading to two almost disjoint mathematical communities. It was the goal of this workshop to stimulate exchange between these two communities by highlighting important recent developments in both areas. The workshop brought together 44 researchers from 9 different countries. This report contains the extended abstracts of the 30 lectures that were delivered during the meeting. The transition from ordered to disordered system was the main topic of four lectures. There were 14 talks about random Schrödinger operators. One of the major topics was the theory of Anderson localization/delocalization. The density of states, especially the investigation of Lifshitz tails was the subject of several contributions. The third topic in this field concerns the theory of level statistics which is a very active field at the moment. This area is much inspired by the theory of random matrices. Twelve of the lectures were related to the theory of random matrices. In these talks a variety of ensembles, results, open problems, and methods of proof were discussed that yield a somewhat representative picture of the current state of the art in this field. The ensembles considered include the two classical cases of Wigner ensembles (independent entries) and of invariant ensembles (invariant under appropriate changes of basis). Some of the lectures dealt with more general classes of ensembles that are motivated by applications in physics and statistics. Most of the results that were presented are concerned with local correlations of eigenvalues, with the distribution of the largest eigenvalue and with (central) limit theorems for spectral quantities. One talk was devoted to the Asymmetric Simple Exclusion Process, generalizing a much celebrated result that provided a connection to random matrix theory. Finally, it is our happy task to thank all the participants for the lively discussions, the staff of Oberwolfach for providing such perfect and pleasant working conditions and Bernd Metzger for collecting and editing the extended abstracts.

Random matrix theory is at the intersection of linear algebra, probability theory and integrable systems, and has a wide range of applications in physics, engineering, multivariate statistics and beyond. This volume is based on a Fall 2010 MSRI program which generated the solution of long-standing questions on universalities of Wigner matrices and beta-ensembles and opened new research directions especially in relation to the KPZ universality class of interacting particle systems and low-rank perturbations. The book contains review articles and research contributions on all these topics, in addition to other core aspects of random matrix theory such as integrability and free probability theory. It will give both established and new researchers insights into the most recent advances in the field and the connections among many subfields.

A random field is the representation of the joint probability distribution for a set of random variables. Markov fields, in particular, have a long standing tradition as the theoretical foundation of many applications in statistical physics and probability. For strictly positive probability densities, a Markov random field is also a Gibbs field, i.e., a random field supplemented with a measure that implies the existence of a regular conditional distribution. Markov random fields have been used in statistical physics, dating back as far as the Ehrenfests. However, their measure theoretical foundations were developed much later by Dobruschin, Lanford and Ruelle, as well as by Hammersley and Clifford. Aside from its enormous theoretical relevance, due to its generality and simplicity, Markov random fields have been used in a broad range of applications in equilibrium and non-equilibrium statistical physics, in non-linear dynamics and ergodic theory. Also in computational molecular biology, ecology, structural biology, computer vision, control theory, complex networks and data science, to name but a few. Often these applications have been inspired by the original statistical physics approaches. Here, we will briefly present a modern introduction to the theory of random fields, later we will explore and discuss some of the recent applications of random fields in physics, biology and data science. Our aim is to highlight the relevance of this powerful theoretical aspect of statistical physics and its relation to the broad success of its many interdisciplinary applications.

These lecture notes provide a comprehensive, self-contained introduction to the analysis of Wishart matrix moments. This study may act as an introduction to some particular aspects of random matrix theory, or as a self-contained exposition of Wishart matrix moments. Random matrix theory plays a central role in statistical physics, computational mathematics and engineering sciences, including data assimilation, signal processing, combinatorial optimization, compressed sensing, econometrics and mathematical finance, among numerous others. The mathematical foundations of the theory of random matrices lies at the intersection of combinatorics, non-commutative algebra, geometry, multivariate functional and spectral analysis, and of course statistics and probability theory. As a result, most of the classical topics in random matrix theory are technical, and mathematically difficult to penetrate for non-experts and regular users and practitioners. The technical aim of these notes is to review and extend some important results in random matrix theory in the specific context of real random Wishart matrices. This special class of Gaussian-type sample covariance matrix plays an important role in multivariate analysis and in statistical theory. We derive non-asymptotic formulae for the full matrix moments of real valued Wishart random matrices. As a corollary, we derive and extend a number of spectral and trace-type results for the case of non-isotropic Wishart random matrices. We also derive the full matrix moment analogues of some classic spectral and trace-type moment results. For example, we derive semi-circle and Marchencko-Pastur-type laws in the non-isotropic and full matrix cases. Laplace matrix transforms and matrix moment estimates are also studied, along with new spectral and trace concentration-type inequalities.

Wang-Landau Algorithm in a Two-Dimensional Spin–1 Blume-Capel Model

The paper presents algorithms for simulation rare events in stochastic systems based on the theory of large deviations. Here, this approach is used in conjunction with the tools of optimal control theory to estimate the probability that some observed states in a stochastic system will exceed a given threshold by some upcoming time instant. Algorithms for obtaining controlled extremal trajectory (A-profile) of the system, along which the transition to a rare event (threshold) occurs most likely under the influence of disturbances that minimize the action functional, are presented. It is also shown how this minimization can be efficiently performed using numerical-analytical methods of optimal control for linear and nonlinear systems. These results are illustrated by an example for a precipitation-measured monsoon intraseasonal oscillation (MISO) described by a low-order nonlinear stochastic model.

Entrée to Paris got Cécile DeWitt into physics. Working with great physicists kept her in the field.

We report on the exact treatment of a random-matrix representation of a bond-percolation model on a square lattice in two dimensions with occupation probability p. The percolation problem is mapped onto a random complex matrix composed of two random real-valued matrices of elements +1 and -1 with probability p and 1-p, respectively. We find that the onset of percolation transition can be detected by the emergence of power-law divergences due to the coalescence of the first two extreme eigenvalues in the thermodynamic limit. We develop a universal finite-size scaling law that fully characterizes the scaling behavior of the extreme eigenvalue's fluctuation in terms of a set of universal scaling exponents and amplitudes. We make use of the relative entropy as an index of the disparity between two distributions of the first and second-largest extreme eigenvalues to show that its minimum underlies the scaling framework. Our study may provide an inroad for developing new methods and algorithms with diverse applications in machine learning, complex systems, and statistical physics.

In this paper, some applications of statistical physics (SP) concepts and techniques in applied geosciences are reviewed. The domain includes hydrology, oil and gas industry, nuclear or CO 2 waste geological disposal, massive energy storage, heat recovery from geothermal formations and many other applications. Several scales are considered: we start from applications of SP at the molecular scale to understand the effect of extreme confinements concerning the fluid transport in nanopores in clay rocks. The paper ends with coarse graining techniques that are employed to build operational models relevant at the practical scale of several kilometers, including strongly fractured geological environments. Perspectives are proposed regarding some issues about the practical use of these often over-parameterized models in connection to random matrix and graph theories and the associated quenched disorder problems and “big data” issues.