The theory of critical phenomena in systems at equilibrium is reviewed at an introductory level with special emphasis on the values of the critical point exponents α, β, γ,..., and their interrelations. The experimental observations are surveyed and the analogies between different physical systems - fluids, magnets, superfluids, binary alloys, etc. - are developed phenomenologically. An exact theoretical basis for the analogies follows from the equivalence between classical and quantal `lattice gases' and the Ising and Heisenberg-Ising magnetic models. General rigorous inequalities for critical exponents at and below Tc are derived. The nature and validity of the `classical' (phenomenological and mean field) theories are discussed, their predictions being contrasted with the exact results for plane Ising models, which are summarized concisely. Pade approximant and ratio techniques applied to appropriate series expansions lead to precise critical-point estimates for the three-dimensional Heisenberg and Ising models (tables of data are presented). With this background a critique is presented of recent theoretical ideas: namely, the `droplet' picture of the critical point and the `homogeneity' and `scaling' hypotheses. These lead to a `law of corresponding states' near a critical point and to relations between the various exponents which suggest that perhaps only two or three exponents might be algebraically independent for any system.

https://iopscience.iop.org/article/10.1088/0034-4885/30/2/306/pdf

The theory of equilibrium critical phenomena

Random-matrix theories in quantum physics : common concepts

We review the development of random-matrix theory (RMT) during the last decade. We emphasize both the theoretical aspects, and the application of the theory to a number of fields. These comprise chaotic and disordered systems, the localization problem, many-body quantum systems, the Calogero-Sutherland model, chiral symmetry breaking in QCD, and quantum gravity in two dimensions. The review is preceded by a brief historical survey of the developments of RMT and of localization theory since their inception. We emphasize the concepts common to the above-mentioned fields as well as the great diversity of RMT. In view of the universality of RMT, we suggest that the current development signals the emergence of a new "statistical mechanics": Stochasticity and general symmetry requirements lead to universal laws not based on dynamical principles.

Random Matrix Theories in Quantum Physics: Common Concepts

The theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial). This diverse array of tools, while attesting to the vitality of the field, presents several formidable obstacles to the newcomer, and even the expert probabilist. This rigorous introduction to the basic theory is sufficiently self-contained to be accessible to graduate students in mathematics or related sciences, who have mastered probability theory at the graduate level, but have not necessarily been exposed to advanced notions of functional analysis, algebra or geometry. Useful background material is collected in the appendices and exercises are also included throughout to test the reader's understanding. Enumerative techniques, stochastic analysis, large deviations, concentration inequalities, disintegration and Lie algebras all are introduced in the text, which will enable readers to approach the research literature with confidence.

/pdf/an-introduction-to-random-matrices-1gerxro6r6.pdf

An Introduction to Random Matrices

The field of random matrix theory has seen an explosion of activity in recent years, with connections to many areas of mathematics and physics. However, this makes the current state of the field almost too large to survey in a single book. In this graduate text, we focus on one specific sector of the field, namely the spectral distribution of random Wigner matrix ensembles (such as the Gaussian Unitary Ensemble), as well as iid matrix ensembles. The text is largely self-contained and starts with a review of relevant aspects of probability theory and linear algebra. With over 200 exercises, the book is suitable as an introductory text for beginning graduate students seeking to enter the field.

/pdf/topics-in-random-matrix-theory-3dl0sp9tcn.pdf

Topics in Random Matrix Theory

Statistical ensembles of complex, quaternion, and real matrices with Gaussian probability distribution, are studied. We determine the over‐all eigenvalue distribution in these three cases (in the real case, under the restriction that all eigenvalues are real). We also determine, in the complex case, all the correlation functions of the eigenvalues, as well as their limits when the order N of the matrices becomes infinite. In particular, the limit of the eigenvalue density as N → ∞ is constant over the whole complex plane.

Statistical Ensembles of Complex, Quaternion, and Real Matrices

The reduced density matrices for quantum gases are studied by Banach space techniques. For suitably restricted interactions, they are shown to be analytic functions of the activity. As the volume of the system becomes infinite, they tend in some sense to well‐defined limits for which the same analyticity properties hold. As a consequence, the virial expansion is shown to be convergent in a neighborhood of the origin.

Reduced Density Matrices of Quantum Gases. I. Limit of Infinite Volume

The reduced density matrices of quantum gases are studied by means of a Wiener integral representation described in a previous paper. They are shown to satisfy a cluster property in the form of an absolute integrability condition of the natural quantum analogues of the Ursell functions, considered as functions of the differences of their arguments. Use is made of the natural transposition to the quantum case of the algebraic formalism introduced by Ruelle in the classical case. By‐products are two results on the signs of the coefficients of the Mayer expansion, in the case of Maxwell‐Boltzmann and Fermi‐Dirac statistics, respectively.

Reduced Density Matrices of Quantum Gases. II. Cluster Property

In two previous papers, the reduced density matrices of quantum gases, in the grand canonical formalism and for suitably restricted interaction potentials, have been shown to be analytic vector‐valued functions of the activity in a neighborhood of the origin, to tend in some sense to well‐defined limits as the volume of the system becomes infinite, and to satisfy a cluster decomposition property. The same results are extended here by the same methods to a wider class of potentials, including hard‐core, and allowing attractive interactions, which were excluded previously.

Reduced Density Matrices of Quantum Gases. III. Hard‐Core Potentials

For simple Lie groups, matrix elements of vector operators (i.e. operators which transform according to the adjoint representation of the group) within an irreducible (finite‐dimensional) representation, are studied. By use of the Wigner‐Eckart theorem, they are shown to be linear combinations of the matrix elements of a finite number of operators. The number of linearly independent terms is calculated and shown to be at most equal to the rank l of the group. l vector operators are constructed explicitly, among which a basis can be chosen for this decomposition. Okubo's mass formula arises as a consequence.

Jean Ginibre

Papers

Statistical Ensembles of Complex, Quaternion, and Real Matrices

Reduced Density Matrices of Quantum Gases. I. Limit of Infinite Volume

Reduced Density Matrices of Quantum Gases. II. Cluster Property

Reduced Density Matrices of Quantum Gases. III. Hard‐Core Potentials

Wigner‐Eckart Theorem and Simple Lie Groups