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Introduction to Random Matrices - Theory and Practice

Giacomo Livan, Marcel Novaes, Pierpaolo Vivo

arXiv:1712.07903v1math-phcond-mat.stat-mech

TL;DR

The book addresses how beginners can enter random matrix theory without being overwhelmed by jargon or inaccessible calculations. It provides an accessible introduction spanning classical ensembles, analytical and numerical methods, and advanced topics, while acknowledging deliberate limits on coverage. Its central supported outcome is that the mean-field approach captures the first three free-energy terms but fails at O(N) contributions.

  • Problem

    Beginning researchers need an accessible introduction to random matrix theory that explains standard and advanced topics without incomprehensible jargon.

  • Method

    The book combines short informal chapters, fully explained calculations, numerical verification, classical random-matrix techniques, and introductions to replica methods and free probability.

  • Results

    The mean-field continuum approach captures the first three terms of the free energy but fails at O(N) contributions because the renormalized self-energy cannot be precisely determined by simple scaling.

  • Takeaways & Limitations

    The book gives interested beginners a route toward reading other random-matrix sources without feeling overwhelmed by jargon or endless unexplained steps.

  • Takeaways & Limitations

    The book deliberately omits ensembles with complex eigenvalues and many other topics because of space limitations, and it is not encyclopedic.

Abstract

from arXiv · show

This is a book for absolute beginners. If you have heard about random matrix theory, commonly denoted RMT, but you do not know what that is, then welcome!, this is the place for you. Our aim is to provide a truly accessible introductory account of RMT for physicists and mathematicians at the beginning of their research career. We tried to write the sort of text we would have loved to read when we were beginning Ph.D. students ourselves. Our book is structured with light and short chapters, and the style is informal. The calculations we found most instructive are spelt out in full. Particular attention is paid to the numerical verification of most analytical results. Our book covers standard material - classical ensembles, orthogonal polynomial techniques, spectral densities and spacings - but also more advanced and modern topics - replica approach and free probability - that are not normally included in elementary accounts on RMT. This book is dedicated to the fond memory of Oriol Bohigas.

Getting Started

The book introduces random matrices through Gaussian ensembles, real eigenvalues, empirical histograms, and the statistical structure of eigenvalue spacings. It develops foundational results including Wigner’s semicircle law, Wigner’s surmise, level repulsion, and the eigenvalue jpdf.

  • Random matrix ensembles: Gaussian random matrices are generated by sampling entries and symmetrizing them, producing real symmetric matrices with real eigenvalues.The symmetrization is Hs = (H + H^T)/2.
  • Random matrix ensembles: GOE, GUE, and GSE ensembles differ through real, complex, or quaternionic entries together with the conditions needed for real eigenvalues.The corresponding matrices are real symmetric, Hermitian, or self-dual, respectively.
  • Spectral densities: For T = 50000 samples and N = 8, normalized histograms collect N × T eigenvalues to display ensemble spectral distributions.The histograms are symmetric, with roughly half of the eigenvalues positive and half negative.
  • Spectral densities: For large N, eigenvalue histograms approach Wigner’s semicircle law, while finite-N calculations provide exact spectral shapes.The book explicitly asks how histogram shapes change as N becomes large and answers with the limiting semicircle distribution.
  • Eigenvalue spacings: Wigner’s surmise for GOE spacings is ¯p(s) = (πs/2)exp(−πs^2/4), showing that very small eigenvalue gaps are unlikely.The spacing distribution is normalized by the mean level spacing before comparison.
  • Eigenvalue correlations: Random-matrix eigenvalues are correlated because their joint probability density does not factorize, producing generic level repulsion unlike independent variables.The eigenvalue repulsion factor makes every eigenvalue feel the presence of all the others.

Classified Material

This section introduces spectral density and classifies random matrix ensembles by entry independence and rotational invariance. The Gaussian ensembles uniquely occupy both categories.

  • Spectral density: The normalized counting function represents each eigenvalue as a spike and reproduces the fraction of eigenvalues in any interval.Its integral over [a,b] gives the fraction between a and b, using the Dirac delta representation.
  • Spectral density: The average counting function ⟨n(x)⟩=ρ(x), called the average spectral density, describes the continuous profile formed by many eigenvalue sets.For T = 4 sets of N = 8 spikes, the ensemble average produces the profile shown in Fig. 3.1.
  • Spectral density: Wigner’s semicircle law states that the large-N Gaussian spectral density has a semicircular or semielliptical shape.The eigenvalue histograms concentrate between spectral edges whose scale grows with N unless eigenvalues are rescaled by βN.
  • Ensemble classification: Wigner matrices have independent entries, whereas rotationally invariant ensembles assign equal probability to matrices related by similarity transformations.Rotational invariance makes eigenvectors statistically unimportant and constrains the entry density to functions of traces of matrix powers.
  • Ensemble classification: Only the Gaussian ensemble lies in both the independent-entry and rotationally invariant classes.The three incarnations are GOE, GUE, and GSE; outside the Gaussian case, independence and high rotational symmetry cannot both be retained.

The fluid semicircle

The Coulomb gas technique rewrites the Gaussian eigenvalue problem as a static fluid of logarithmically repelling particles confined by a quadratic potential. In the large-N limit, a smooth density profile enables a continuum free-energy calculation.

  • Coulomb-gas interpretation: The Coulomb gas technique interprets Gaussian eigenvalues as particles with logarithmic repulsion and a quadratic confining potential.The logarithmic interaction reflects the two-dimensional electrostatic potential for charges constrained to a line.
  • Coulomb-gas interpretation: Exponentiating the eigenvalue interaction product produces a canonical partition function with Gibbs-Boltzmann weight e^(-βN^2V[x]).The resulting fluid is static because V[x] contains no kinetic term.
  • Large-N limit: The large-N calculation seeks the free energy F=−(1/β)lnZ_N,β and simplifies through the simultaneous thermodynamic and zero-temperature limit.At zero temperature, equilibrium positions are found by minimizing the free energy.
  • Continuum description: The continuum description replaces discrete eigenvalue configurations with a non-negative, normalized, smooth counting function n(x).For finite N it is a collection of spikes; for large N the analysis assumes it becomes smooth and introduces a functional integral over admissible profiles.
  • Continuum description: The continuum logarithmic energy is divergent at coincident positions, so a position-dependent self-energy cutoff is introduced to obtain a finite result.The divergence corresponds to the i=j contribution from neighboring charges becoming arbitrarily close.

5. Evaluate the integral IN[n(x)] for large N

The calculation rewrites the Gaussian model as a charged-fluid partition function and evaluates it through a continuum, entropy, and saddle-point approximation. Comparison with the finite-N result shows agreement through the leading terms, while the O(N) term remains uncertain because of the self-energy constant.

  • Continuum description: The multiple integral counts microscopic configurations compatible with a density profile, whose logarithm yields the fluid’s entropy.The resulting entropy appears in the action used for the large-N evaluation.
  • Continuum description: The functional integral is evaluated with a Laplace or saddle-point approximation because its exponent contains the large parameter N.The saddle-point action is constructed after introducing a functional delta-function representation.
  • Self-energy: The short-distance cutoff ∆(x) is chosen from a density-based self-energy argument, but the argument is not rigorous and leaves its constant c undetermined.The missing constant affects an O(N) contribution but is described as inconsequential for the following analysis.
  • Cross-check: For β = 2, Barnes G-function asymptotics match the finite-N partition-function asymptotics, validating the mean-field calculation through the first three free-energy terms.The remaining discrepancy occurs at O(N), where the renormalized self-energy cannot be fixed by the scaling argument.
  • Statistical-mechanics interpretation: The partition function is reinterpreted as that of a 2D charged fluid confined to a line, whose large-N behavior is governed by free-energy minimization.The functional-integral representation over normalized densities is evaluated in the next saddle-point treatment.

Saddle-point-of-view

The saddle-point treatment seeks the equilibrium density minimizing the Coulomb-gas action. Solving the resulting integral equation requires determining both the density and its support, leading to Wigner’s semicircle law and its numerical checks.

  • Saddle-point formulation: The large-N Coulomb gas is dominated by an O(N^2) energetic contribution, while entropy is sub-leading because the particles interact all-to-all.This scaling distinguishes the model from standard short-range systems.
  • Saddle-point formulation: The equilibrium density n⋆(x) minimizes the action over normalized, non-negative functions, with a Lagrange multiplier enforcing normalization.The intensive free energy equals the action evaluated at this saddle-point density.
  • Determining the support: The density’s support cannot be the whole real line, so the solution is sought on an interval [a,b] whose endpoints are fixed by free-energy minimization.For the quadratic potential, a single interval is physically motivated by the single minimum of the confining well.
  • Solving the integral equation: Differentiating the singular integral equation in the weak sense and applying Tricomi’s theorem produces a density depending parametrically on a and b.The density solves the equation between a and b before the endpoints are optimized.
  • The final touch: For a = −b, the resulting equilibrium density is Wigner’s semicircle law, and numerical checks verify that it solves the integral equation.The book also identifies the semicircular density as the equilibrium profile and normalized eigenvalue histogram for a large Gaussian matrix.
  • Epilogue: The spectral density is highly non-universal, although many ensembles share the semicircle law at large N under suitable conditions.Wigner ensembles with sufficiently fast-decaying entry distributions are given as an example.

Time for a change

This section derives eigenvalue densities from matrix-entry densities by changing variables from matrix entries to eigenvalues and eigenvectors. The Jacobian is the Vandermonde determinant, and invariant ensembles permit integration over eigenvectors to obtain eigenvalue jpdfs.

  • Motivation: The chapter answers how to derive the eigenvalue jpdf from the jpdf of entries for real symmetric matrices.The change of variables uses the diagonalization H = OX O^T.
  • Change of variables: The orthogonal-group volume element dO supplies the measure over eigenvector degrees of freedom in the transformation H → {x,O}.The dimensions match because N(N+1)/2 = N + N(N−1)/2.
  • The Jacobian: The Jacobian of the matrix-to-eigenvalue/eigenvector transformation depends only on the eigenvalues and equals the Vandermonde determinant.For hermitian and quaternion self-dual matrices, the Vandermonde is raised to β = 2 and 4, respectively.
  • Invariant models: For invariant entry distributions, integrating over eigenvectors yields a jpdf of eigenvalues alone because the entry density is independent of eigenvector components.The orthogonal-group integral then contributes only a constant volume factor.
  • Invariant models: The invariant-ensemble theorem gives the jpdf for ordered eigenvalues, while unordered eigenvalues require an absolute value and differ in normalization by N!.The distinction follows from ordering conventions and the need to account for permutations.
  • Normalization: The derivation must fix eigenvector signs or phases to make the eigen-decomposition one-to-one, reducing the orthogonal volume integral by 2^N.The analogous unitary reduction is by (2π)^N.

Meet Vandermonde

The Vandermonde determinant encodes eigenvalue repulsion and can be represented using determinants of polynomials. Its Jacobian derivation follows directly from differentiating the diagonalization of a real symmetric matrix.

  • Vandermonde properties: The Vandermonde is a completely antisymmetric polynomial, changing sign whenever two eigenvalues are exchanged.For N = 3, it is the product (x2−x1)(x3−x1)(x3−x2).
  • Polynomial representation: Replacing monomial rows by degree-k polynomials changes the determinant only by the product of their leading coefficients.Lower-order polynomial terms do not affect the Vandermonde structure.
  • Polynomial representation: Hermite and Laguerre orthogonal polynomials provide determinant representations of the Vandermonde useful for later calculations.The book postpones extensive use of this property to Chapter 10.
  • Jacobian derivation: The resulting Jacobian is the Vandermonde determinant, and this proof does not require rotational invariance of the ensemble.Analogous complex hermitian and quaternion self-dual proofs are noted separately.
  • Jacobian derivation: For real symmetric matrices, differentiating H = OX O^T and using δΩ = O^TδO reduces the Jacobian calculation to δX and an antisymmetric δΩ.The orthogonal transformations relating δH and δĤ do not alter the relevant Jacobian structure.

Resolve(nt) the semicircle

The resolvent is a complex function that encodes the spectral density, turning a singular integral problem into an algebraic equation for Gaussian ensembles. Solving that equation recovers the density and its support in the large-N limit.

  • Resolvent basics: The resolvent is a complex function from which the spectral density can be calculated.Its advantage is replacing a singular integral equation with an algebraic equation, at the cost of using complex analysis.
  • Resolvent basics: The imaginary part of the resolvent represents the delta-function definition of the spectral density.This connects the resolvent to the eigenvalue density through the Sokhotski-Plemelj formula.
  • Large-N interpretation: In the large-N limit, eigenvalue poles merge into a real-line cut whose support is the spectral density.The averaged resolvent is defined outside this cut, such as outside the Gaussian ensemble’s spectral interval.
  • Further properties: The resolvent also has 1/z asymptotics and generates the moments of the matrix.Its trace definition expresses it as a sum over diagonal resolvent elements and eigenvector components.
  • Gaussian ensemble: The Gaussian-ensemble resolvent satisfies an algebraic equation obtained from a saddle-point formulation after a derivative term becomes subleading.The resulting equation can be solved directly as a quadratic equation.
  • Gaussian ensemble: For the Gaussian ensemble, the solved resolvent gives zero density outside the spectral support and a nonzero density inside it.The correct quadratic branch depends on the sign of x, and numerical integration checks the branch choices.

One pager on eigenvectors

The chapter studies eigenvector statistics in invariant ensembles and introduces localization through the inverse participation ratio. Component distributions can be derived from the unit-norm constraint, while IPR scaling distinguishes extended from localized eigenvectors.

  • Eigenvector statistics: Invariant matrix models decouple eigenvalues and eigenvectors, leaving eigenvector components constrained primarily by unit norm.This permits a joint probability density and marginal distributions for individual components.
  • Eigenvector statistics: The component statistics of GUE and GOE eigenvectors lead, after rescaling, to limiting component densities including the Porter-Thomas distribution.The derivation does not depend on Gaussianity and applies to orthogonal or unitary ensembles.

10.1 β = 2 is easier

For β = 2 rotationally invariant ensembles, orthogonal polynomials and the reproducing kernel turn difficult eigenvalue integrations into determinant calculations. This yields finite-N spectral densities and, for GUE, the semicircle law asymptotically.

  • 10.1 β = 2 is easier: The finite-N spectral density problem involves an N−1-fold integral whose integrand does not factorize.The chapter focuses first on β = 2 because the calculation is easier there.
  • Orthogonal polynomials: The Vandermonde determinant is rewritten using polynomials chosen orthonormal with respect to the potential weight.This choice produces the kernel, a central object in the calculation.
  • Reproducing kernel: The kernel’s reproducing property allows repeated integrations of determinants while reducing the kernel-matrix dimension one step at a time.This is the Dyson-Gaudin integration lemma and creates a determinant-based integration procedure.
  • Determinantal structure: The resulting determinantal structure solves not only the one-point marginal but also any k-point correlation function for these ensembles.The construction requires suitable orthonormal polynomials with respect to V(x).
  • GUE example: For GUE, Hermite polynomials give the finite-N spectral density, which is numerically compared with a generated eigenvalue histogram.The large-N limit recovers the semicircle law through Hermite-polynomial asymptotics.

Meet Andr´eief

The Andréief identity converts multiple integrals of products of determinants into finite determinants, making β = 2 random-matrix calculations tractable. Its applications include partition functions, eigenvalue-count probabilities, and Hankel-determinant relations.

  • Andréief identity: The Andréief identity replaces a multiple integral involving two determinants with a determinant of single integrals.This is especially useful for unitary invariant ensembles, where the squared Vandermonde is a determinant product.
  • Andréief identity: A difficult 20-fold integral can thereby become a 20×20 determinant suitable for scientific software or closed-form evaluation.The identity is presented as a general way to turn an otherwise difficult calculation into a manageable one.
  • Hankel determinants: For β = 2, the resulting determinants are Hankel determinants because their entries depend on sums of matrix indices.This structure follows from the two equal determinant factors in the integrand.
  • Related identities: Related de Bruijn identities use Pfaffians, which are sums over pairings of an even-dimensional skew-symmetric matrix.These identities extend the determinant-based toolkit beyond the Andréief formula.
  • Applications: The method gives exact finite-size results for eigenvalue-count probabilities, including the 9×9 GUE example.The calculation reduces the probability to ratios of Hankel determinants evaluated exactly with symbolic software.
  • Further connections: The chapter connects Hankel determinants to Toda-lattice equations and Painlevé functions.These links are described as underlying further results in integrable systems.

Classical Ensembles: Wishart-Laguerre

The chapter develops finite-N spectral-density calculations for the GOE and GSE using skew-orthogonal-polynomial and Pfaffian methods, with numerical checks. It also introduces the Wishart-Laguerre ensemble and its large-N spectral-density question.

  • Classical Ensembles: Wishart-Laguerre: Gaussian finite-N densities are checked numerically against eigenvalue histograms from 50000 matrices of size N = 8 for GOE, GUE, and GSE.The comparison uses the theoretical densities corresponding to the three Gaussian ensembles.
  • Classical Ensembles: Wishart-Laguerre: Wishart matrices are constructed as W = HH† from an N × M Gaussian matrix H with M ≥ N, yielding correlated entries and rotational invariance.The real, complex, and quaternion cases correspond to β = 1, 2, and 4, respectively.
  • Classical Ensembles: Wishart-Laguerre: Wishart matrices are positive semidefinite, so their N eigenvalues are non-negative; the large-N limiting-density problem leads to the Marˇcenko-Pastur density.The chapter also notes the Anti-Wishart case, where N − M eigenvalues are exactly zero.

Meet Marˇcenko and Pastur

The Marˇcenko-Pastur law gives the large-N average spectral density of Wishart-Laguerre ensembles at fixed rectangularity ratio. The chapter derives it with the resolvent method and checks it against numerical Wishart histograms.

  • 14.1 The Marˇcenko-Pastur density: For N,M →∞ with c = N/M ≤ 1 fixed, the Wishart-Laguerre average density has a scaling form whose function ρMP is independent of β.This function is the Wishart analogue of the Gaussian ensemble’s semicircle density.
  • 14.1 The Marˇcenko-Pastur density: The Marˇcenko-Pastur density is supported between ζ− = (1−c−1/2)^2 and ζ+ = (1+c−1/2)^2.These edge points define the interval on which the stated scaling function applies.
  • 14.1 The Marˇcenko-Pastur density: As c → 1, the origin becomes a hard edge and the density becomes singular there, indicating eigenvalue accumulation near zero for square H.For c < 1, the support lies on the positive semi-axis with two soft edges.
  • 14.2 Do it yourself: the resolvent method: The Wishart Coulomb-gas derivation must enforce xi > 0, handled by a penalty −µ ∑i ln(xi) whose strength is taken to zero after the calculation.The positive-domain constraint is essential because the eigenvalues are non-negative.
  • 14.2 Do it yourself: the resolvent method: The resolvent derivation fixes the normalization constant by requiring the density to integrate to one, recovering the Marˇcenko-Pastur edge points.The calculation obtains K = 1/γ and then x± → (1 ± 1/√c)^2.
  • 14.2 Do it yourself: the resolvent method: Numerical diagonalization of 5000 Wishart matrices of size N = 100 is compared with the Marˇcenko-Pastur density for two rectangularity ratios and all β values.The comparison is presented as a numerical check of the derived density.

15.1 Meet Edwards and Jones

The Edwards-Jones formula expresses the average spectral density of generic real symmetric random matrices using only the entry jpdf, avoiding an explicit eigenvalue jpdf. Its practical evaluation becomes tractable mainly as N →∞ and requires handling a logarithmic average.

  • 15.1 Meet Edwards and Jones: The Edwards-Jones formula computes the average spectral density of a generic real symmetric random-matrix ensemble from the jpdf of its upper-triangular entries.It therefore starts from entry-level information rather than an eigenvalue jpdf.
  • 15.1 Meet Edwards and Jones: The formula is valid in principle for finite N, but calculations can generally be completed only in the N →∞ limit.The large-N limit supplies several simplifying approximations.
  • 15.1 Meet Edwards and Jones: The central obstacle is the logarithm inside the disorder average, which prevents directly exchanging the H and y integrations in the required order.The chapter identifies annealed averaging and the replica trick as two strategies for addressing this obstacle.
  • 15.1 Meet Edwards and Jones: The method introduces a partition function Z(x), whose logarithm represents the free energy of an associated statistical-mechanics model.The model contains both the random matrix H and auxiliary variables y.
  • 15.1 Meet Edwards and Jones: Annealed averaging moves the average inside the logarithm, simplifying the calculation but producing a result described as not entirely justifiable.The chapter uses this approach to obtain the GOE semicircle law for training purposes.

Replicas for GOE

The chapter applies the Edwards-Jones framework with replicas to the GOE, using replica continuation and a replica-symmetric saddle point. The calculation recovers Wigner’s semicircle law in the large-N limit.

  • Replicas for GOE: The GOE calculation applies the Edwards-Jones formula in its quenched form and uses the replica trick to handle the logarithmic disorder average.Replicating the auxiliary integral permits exchanging the integrations over the matrix and auxiliary variables before continuing toward n = 0.
  • Replicas for GOE: After rescaling, the GOE spectral-density edges remain finite rather than growing with N.The text states that the corresponding edges do not grow with N and identifies the result with the semicircle law.
  • Replicas for GOE: The replica procedure reverses the mathematically required order of the n → 0 and N →∞ limits, and the text explicitly states that this is not mathematically justified.The calculation proceeds formally despite this limitation.
  • Replicas for GOE: The replica calculation assumes replica symmetry, supported by research indicating that the replica-symmetric high-temperature solution is exact although not formally proven.The saddle-point functions are taken to depend only on the norm of the replica vector.
  • Replicas for GOE: The large-N saddle-point evaluation yields Wigner’s semicircle law for the GOE.The derivation uses the Edwards-Jones representation together with the replicated partition function and its saddle point.

Born to be free

The chapter introduces freeness as the random-matrix analogue of statistical independence and develops free probability tools for computing spectra of sums. It applies the R-transform method to GOE and Wishart matrices, with numerical verification, and notes that semicircle densities remain stable under free addition.

  • Freeness: Freeness generalizes statistical independence to random matrices, where independent entries alone do not eliminate angular correlations.It is expressed through vanishing traces of centered noncommutative matrix-polynomial products and generalizes moment factorization.
  • Freeness: Freeness has no counterpart in conventional probability theory and captures effects arising from matrix non-commutativity.
  • Free addition: For large random matrices, the chapter seeks the average spectral density of sums from different ensembles.
  • Free addition: The R-transform makes free addition algorithmic: add the component R-transforms, recover the Blue function and resolvent, then derive the eigenvalue density.The construction begins from ensemble resolvents and uses the Blue function as their functional inverse, after removing its singular part to obtain the R-transform.
  • GOE–Wishart example: For a GOE–Wishart sum, the method yields a third-degree resolvent equation whose appropriate solution is selected by its positive imaginary part and numerically verified.The examples vary p, which quantifies the relative weight between the two ensembles.
  • Free addition: The semicircle distribution is stable under free addition: freely summing M matrices with semicircle spectral densities again produces a semicircle density.
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