Semicircle Law Research Articles

A Renormalisation Group Approach to the Universality of Wigner’s Semicircle Law for Random Matrices with Dependent Entries

We show that if the non-Gaussian part of the cumulants of a random matrix model obeys some scaling bounds in the size of the matrix, then Wigner’s semicircle law holds. This result is derived using the replica technique and an analogue of the renormalisation group equation for the replica effective action.

Local semicircle law under moment conditions: Stieltjes transform, rigidity and delocalization

В данной работе рассматриваются симметричные случайные матрицы $\mathbf X= [X_{jk}]_{j,k=1}^n$ с независимыми одинаково распределенными элементами в верхней треугольной части, имеющими нулевое математическое ожидание, единичную дисперсию и конечный момент порядка $4+\delta$, $\delta>0$. Доказано, что расстояние между преобразованиями Стилтьеса эмпирической спектральной функции распределения собственных значений матрицы $n^{-1/2}\mathbf X$ и полукруговым законом Вигнера имеет порядок $(nv)^{-1}$, где $v$ - расстояние в комплексной области до действительной оси. Также обсуждаются вопросы скорости сходимости к полукруговому закону Вигнера, жесткость собственных значений и делокализация собственных векторов.

Spectral and thermodynamic properties of the Sachdev-Ye-Kitaev model

We study spectral and thermodynamic properties of the Sachdev-Ye-Kitaev model, a variant of the $k$-body embedded random ensembles studied for several decades in the context of nuclear physics and quantum chaos. We show analytically that the fourth and sixth order energy cumulants vanish in the limit of large number of particles $N \to \infty$ which is consistent with a Gaussian spectral density. However, for finite $N$, the tail of the average spectral density is well approximated by a semi-circle law. The specific heat coefficient, determined numerically from the low temperature behavior of the partition function, is consistent with the value obtained by previous analytical calculations. For energy scales of the order of the mean level spacing we show that level statistics are well described by random matrix theory. Due to the underlying Clifford algebra of the model, the universality class of the spectral correlations depends on $N$. For larger energy separations we identify an energy scale that grows with $N$, reminiscent of the Thouless energy in mesoscopic physics, where deviations from random matrix theory are observed. Our results are a further confirmation that the Sachdev-Ye-Kitaev model is quantum chaotic for all time scales. According to recent claims in the literature, this is an expected feature in field theories with a gravity-dual.

Convergence rate of eigenvector empirical spectral distribution of large Wigner matrices

In this paper, we adopt the eigenvector empirical spectral distribution (VESD) to investigate the limiting behavior of eigenvectors of a large dimensional Wigner matrix $$\mathbf {W}_n.$$ In particular, we derive the optimal bound for the rate of convergence of the expected VESD of $$\mathbf{W}_n$$ to the semicircle law, which is of order $$O(n^{-1/2})$$ under the assumption of having finite 10th moment. We further show that the convergence rates in probability and almost surely of the VESD are $$O(n^{-1/4})$$ and $$O(n^{-1/6}),$$ respectively, under finite eighth moment condition. Numerical studies demonstrate that the convergence rate does not depend on the choice of unit vector involved in the VESD function, and the best possible bound for the rate of convergence of the VESD is of order $$O(n^{-1/2}).$$

Turning a Coin over Instead of Tossing It

Given a sequence of numbers (p_n)_{nge 2} in [0, 1], consider the following experiment. First, we flip a fair coin and then, at step n, we turn the coin over to the other side with probability p_n, nge 2, independently of the sequence of the previous terms. What can we say about the distribution of the empirical frequency of heads as nrightarrow infty ? We show that a number of phase transitions take place as the turning gets slower (i. e., p_n is getting smaller), leading first to the breakdown of the Central Limit Theorem and then to that of the Law of Large Numbers. It turns out that the critical regime is p_n=text {const}/n. Among the scaling limits, we obtain uniform, Gaussian, semicircle, and arcsine laws.

On maximal relative projection constants

This article focuses on the maximum of relative projection constants over all m-dimensional subspaces of the N-dimensional coordinate space equipped with the max-norm. This quantity, called maximal relative projection constant, is studied in parallel with a lower bound, dubbed quasimaximal relative projection constant. Exploiting alternative expressions for these quantities, we show how they can be computed when N is small and how to reverse the Kadec–Snobar inequality when N does not tend to infinity. Precisely, we first prove that the (quasi)maximal relative projection constant can be lower-bounded by cm, with c arbitrarily close to one, when N is superlinear in m. The main ingredient is a connection with equiangular tight frames. By using the semicircle law, we then prove that the lower bound cm holds with c<1 when N is linear in m.

Universality for general Wigner-type matrices

We consider the local eigenvalue distribution of large self-adjoint Ntimes N random matrices mathbf {H}=mathbf {H}^* with centered independent entries. In contrast to previous works the matrix of variances s_{i j}=mathbbm {E} |h_{i j}|^2 is not assumed to be stochastic. Hence the density of states is not the Wigner semicircle law. Its possible shapes are described in the companion paper (Ajanki et al. in Quadratic Vector Equations on the Complex Upper Half Plane. arXiv:1506.05095). We show that as N grows, the resolvent, mathbf {G}(z)=(mathbf {H}-z)^{-1}, converges to a diagonal matrix, mathrm {diag}(mathbf {m}(z)), where mathbf {m}(z)=(m_1(z),dots ,m_N(z)) solves the vector equation -1/m_i(z) = z,+,sum _j s_{i j} m_j(z) that has been analyzed in Ajanki et al. (Quadratic Vector Equations on the Complex Upper Half Plane. arXiv:1506.05095). We prove a local law down to the smallest spectral resolution scale, and bulk universality for both real symmetric and complex hermitian symmetry classes.

An information complexity index for probability measures on ℝ with all moments

We prove that, each probability meassure on [Formula: see text], with all moments, is canonically associated with (i) a ∗-Lie algebra; (ii) a complexity index labeled by pairs of natural integers. The measures with complexity index [Formula: see text] consist of two disjoint classes: that of all measures with finite support and the semi-circle-arcsine class (the discussion in Sec. 4.1 motivates this name). The class [Formula: see text] coincides with the [Formula: see text]-measures in the finite support case and includes the semi-circle laws in the infinite support case. In the infinite support case, the class [Formula: see text] includes the arcsine laws, and the class [Formula: see text] appeared in central limit theorems of quantum random walks in the sense of Konno. The classes [Formula: see text], with [Formula: see text], do not seem to be present in the literature. The class [Formula: see text] includes the Gaussian and Poisson measures and the associated ∗-Lie algebra is the Heisenberg algebra. The class [Formula: see text] includes the non-standard (i.e. neither Gaussian nor Poisson) Meixner distributions and the associated ∗-Lie algebra is a central extension of [Formula: see text]. Starting from [Formula: see text], the ∗-Lie algebra associated to the class [Formula: see text] is infinite dimensional and the corresponding classes include the higher powers of the standard Gaussian.

Transfer Matrix Approach to 1d Random Band Matrices: Density of States

We study the special case of $n\times n$ 1D Gaussian Hermitian random band matrices, when the covariance of the elements is determined by the matrix $J=(-W^2\triangle+1)^{-1}$. Assuming that $n\ge CW\log W\gg 1$, we prove that the averaged density of states coincides with the Wigner semicircle law up to the correction of order $W^{-1}$.

Wigner matrices, the moments of roots of Hermite polynomials and the semicircle law

In the present paper we give two alternate proofs of the well known theorem that the empirical distribution of the appropriately normalized roots of the nth monic Hermite polynomial Hn converges weakly to the semicircle law, which is also the weak limit of the empirical distribution of appropriately normalized eigenvalues of a Wigner matrix. In the first proof–based on the recursion satisfied by the Hermite polynomials–we show that the generating function of the moments of roots of Hn is convergent and it satisfies a fixed point equation, which is also satisfied by c(z2), where c(z) is the generating function of the Catalan numbers Ck. In the second proof we compute the leading and the second leading term of the kth moments (as a polynomial in n) of Hn and show that the first one coincides with Ck/2, the (k/2)th Catalan number, where k is even and the second one is given by −(22k−1−(2k−1k)). We also mention the known result that the expectation of the characteristic polynomial (pn) of a Wigner random matrix is exactly the Hermite polynomial (Hn), i.e. Epn(x)=Hn(x), which suggests the presence of a deep connection between the Hermite polynomials and Wigner matrices.

Self-similarity in the circular unitary ensemble

Self-similarity in the circular unitary ensemble, Discrete Analysis, 2016:9, 15 pp. The distribution of the eigenvalues of random matrices is a central topic of research. It includes famous results such as Wigner's semicircle law from 1955, which states that the distribution function of a large random symmetric matrix tends to a distribution whose density function, when suitably normalized, has a semicircular graph. (This result is true for several models of random symmetric matrices: in particular, it is true if all entries on and below the diagonal are independent Gaussians with mean 0 and variance 1, with the entries above the diagonal determined by symmetry.) It also received a huge boost in 1973 as a result of an observation by Hugh Montgomery and Freeman Dyson that the correlation between pairs of nearby zeros of the Riemann zeta function appears to be the same, after normalization, as the correlation between pairs of eigenvalues of a random unitary or Hermitian matrices. (Roughly speaking, the pair correlation tells you what you expect the distribution of the remaining points to be given where one point is. In both cases there is a repulsive effect: zeros of the zeta function do not like to be too close to each other, and neither do eigenvalues of these random matrices.) This paper concerns random unitary matrices. The eigenvalues of a unitary matrix lie on the unit circle in the complex plane, and the distribution is rotation invariant (since if one multiplies a random unitary matrix by a scalar of modulus 1, its distribution is unaffected). The main result of this paper is that it also satisfies a self-similarity property, which states, roughly, that the distribution of eigenvalues in an interval is similar to the distribution in the entire circle. To be more precise about it, suppose we take a random $nm\times nm$ unitary matrix, take the eigenvalues of that matrix that have arguments in the interval $[-\pi/m,\pi/m]$, and raise each one to the power $m$ (so that the interval $[-\pi/m,\pi/m]$ is stretched to the interval $[-\pi,\pi]$). Then the resulting distribution of points will closely resemble the distribution of eigenvalues of a random $n\times n$ unitary matrix. Closeness is measured in the $L_1$-_Wasserstein distance_, which is defined as follows. If $\mu$ and $\nu$ are two probability distributions on a metric space $M$, then their $L_1$-Wasserstein distance is the infimum of $\mathbb E\, d(X,Y)$ over all pairs of random variables $X$ and $Y$ such that $X$ has distribution $\mu$ and $Y$ has distribution $\nu$. Intuitively, it is the total amount by which you have to "move" $\mu$ to turn it into $\nu$ if you do so as economically as possible. This result proves a conjecture of Coram and Diaconis, in a slightly modified form. (In the original conjecture, one chooses not an interval of fixed length but an interval from one eigenvalue to the one $n$ places along as you go round the circle, and that interval is stretched and randomly rotated.) ^{_{[Article image](https://www.flickr.com/photos/swallowtailgardenseeds/15625678138/in/pool-publicdomain/) by Swallowtail Garden Seeds}}

Local semicircle law under weak moment conditions

Symmetric random matrices are considered whose upper triangular entries are independent identically distributed random variables with zero mean, unit variance, and a finite moment of order 4 + δ, δ > 0. It is shown that the distances between the Stieltjes transforms of the empirical spectral distribution function and the semicircle law are of order lnn/nv, where v is the distance to the real axis in the complex plane. Applications concerning the convergence rate in probability to the semicircle law, localization of eigenvalues, and delocalization of eigenvectors are discussed.

Emergent geometry from random multitrace matrix models

A novel scenario for the emergence of geometry in random multitrace matrix models of a single hermitian matrix $M$ with unitary $U(N) $ invariance, i.e. without a kinetic term, is presented. In particular, the dimension of the emergent geometry is determined from the critical exponents of the disorder-to-uniform-ordered transition whereas the metric is determined from the Wigner semicircle law behavior of the eigenvalues distribution of the matrix $M$. If the uniform ordered phase is not sustained in the phase diagram then there is no emergent geometry in the multitrace matrix model.

Delocalization for a class of random block band matrices

We consider Ntimes N Hermitian random matrices H consisting of blocks of size Mge N^{6/7}. The matrix elements are i.i.d. within the blocks, close to a Gaussian in the four moment matching sense, but their distribution varies from block to block to form a block-band structure, with an essential band width M. We show that the entries of the Green’s function G(z)=(H-z)^{-1} satisfy the local semicircle law with spectral parameter z=E+mathbf {i}eta down to the real axis for any eta gg N^{-1}, using a combination of the supersymmetry method inspired by Shcherbina (J Stat Phys 155(3): 466–499, 2014) and the Green’s function comparison strategy. Previous estimates were valid only for eta gg M^{-1}. The new estimate also implies that the eigenvectors in the middle of the spectrum are fully delocalized.

Asymptotic expansion of the expected spectral measure of Wigner matrices

We compute an asymptotic expansion with precision 1/n of the moments of the expected empirical spectral measure of Wigner matrices of size n with independent centered entries. We interpret this expansion as the moments of the addition of the semicircle law and 1/n times an explicit signed measured with null total mass. This signed measure depends only on the second and fourth moments of the entries.

Central limit theorem for moments of spectral measures of wigner matrices

Spectral measures of Wigner matrices are investigated. The Wigner semicircle law for spectral measures is proved. Regard this as the law of large number, the central limit theorem moments of spectral measures is also derived. The proof is based on moment method and combinatorial method.

The local semicircle law for random matrices with a fourfold symmetry

We consider real symmetric and complex Hermitian random matrices with the additional symmetry hxy = hN−y,N−x. The matrix elements are independent (up to the fourfold symmetry) and not necessarily identically distributed. This ensemble naturally arises as the Fourier transform of a Gaussian orthogonal ensemble. It also occurs as the flip matrix model — an approximation of the two-dimensional Anderson model at small disorder. We show that the density of states converges to the Wigner semicircle law despite the new symmetry type. We also prove the local version of the semicircle law on the optimal scale.

The singular values of the GUE (less is more)

Some properties that nominally involve the eigenvalues of the Gaussian Unitary Ensemble (GUE) can instead be phrased in terms of singular values. By discarding the signs of the eigenvalues, we gain access to a surprising decomposition: the singular values of the GUE are distributed as the union of the singular values of two independent ensembles of Laguerre type. This independence is remarkable given the well-known phenomenon of eigenvalue repulsion. The structure of this decomposition reveals that several existing observations about large [Formula: see text] limits of the GUE are in fact manifestations of phenomena that are already present for finite random matrices. We relate the semicircle law to the quarter-circle law by connecting Hermite polynomials to generalized Laguerre polynomials with parameter [Formula: see text]. Similarly, we write the absolute value of the determinant of the [Formula: see text] GUE as a product [Formula: see text] independent random variables to gain new insight into its asymptotic log-normality. The decomposition also provides a description of the distribution of the smallest singular value of the GUE, which in turn permits the study of the leading order behavior of the condition number of GUE matrices. The study is motivated by questions involving the enumeration of orientable maps, and is related to questions involving powers of complex Ginibre matrices. The inescapable conclusion of this work is that the singular values of the GUE play an unpredictably important role that had gone unnoticed for decades even though, in hindsight, so many clues had been around.

Special values of the Riemann zeta function via arcsine random variables

In this paper, using arcsine variables, we get a new elementary proof of ζ(2)=π2/6, known as the Basel problem, and the Euler formula. Using exponential variables, we get an Euler-like formula. We can also solve the Basel problem by using Wigner’s semicircle law and the Legendre generating function.

Zeros of exceptional Hermite polynomials

We study the zeros of exceptional Hermite polynomials associated with an even partition λ. We prove several conjectures regarding the asymptotic behaviour of both the regular (real) and the exceptional (complex) zeros. The real zeros are distributed as the zeros of usual Hermite polynomials and, after contracting by a factor 2n, we prove that they follow the semi-circle law. The non-real zeros tend to the zeros of the generalized Hermite polynomial Hλ, provided that these zeros are simple. It was conjectured by Veselov that the zeros of generalized Hermite polynomials are always simple, except possibly for the zero at the origin, but this conjecture remains open.