Bivariate moments of the two-point correlation function for embedded Gaussian unitary ensemble with $k$-body interactions

• URL: http://arxiv.org/abs/2208.11312v2
• Date: Sat, 25 Mar 2023 12:03:54 GMT
• Title: Bivariate moments of the two-point correlation function for embedded Gaussian unitary ensemble with $k$-body interactions
• Authors: V.K.B. Kota
• Abstract summary: Two-point correlation function in eigenvalues of a random matrix ensemble is the ensemble average of the product of the density of eigenvalues at two eigenvalues. Fluctuation measures such as the number variance and Dyson-Mehta $Delta_3$ statistic are defined by the two-point function.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Embedded random matrix ensembles with $k$-body interactions are well established to be appropriate for many quantum systems. For these ensemble the two point correlation function is not yet derived though these ensembles are introduced 50 years back. Two-point correlation function in eigenvalues of a random matrix ensemble is the ensemble average of the product of the density of eigenvalues at two eigenvalues say $E$ and $E^\prime$. Fluctuation measures such as the number variance and Dyson-Mehta $\Delta_3$ statistic are defined by the two-point function and so also the variance of the level motion in the ensemble. Recently, it is recognized that for the embedded ensembles with $k$-body interactions the one-point function (ensemble averaged density of eigenvalues) follows the so called $q$-normal distribution. With this, the eigenvalue density can be expanded by starting with the $q$-normal form and using the associated $q$-Hermite polynomials $He_\zeta(x|q)$. Covariances $\overline{S_\zeta S_{\zeta^\prime}}$ (overline representing ensemble average) of the expansion coefficients $S_\zeta$ with $\zeta \ge 1$ here determine the two-point function as they are a linear combination of the bivariate moments $\Sigma_{PQ}$ of the two-point function. Besides describing all these, in this paper derived are formulas for the bivariate moments $\Sigma_{PQ}$ with $P+Q \le 8$, of the two-point correlation function, for the embedded Gaussian unitary ensembles with $k$-body interactions [EGUE($k$)] as appropriate for systems with $m$ fermions in $N$ single particle states. Used for obtaining the formulas is the $SU(N)$ Wigner-Racah algebra. These formulas with finite $N$ corrections are used to derive formulas for the covariances $\overline{S_\zeta S_{\zeta^\prime}}$ in the asymptotic limit.

Related papers

• Distribution of lowest eigenvalue in $k$-body bosonic random matrix ensembles [0.8999666725996978]
  We numerically study the distribution of the lowest eigenvalue of finite many-boson systems with $k$-body interactions. Our results show the distribution exhibits a smooth transition from Gaussian like for $q$ close to 1 to a modified Gumbel like for intermediate values of $q$ to the well-known Tracy-Widom distribution for $q=0$.
  arXiv  Detail & Related papers  (2024-04-30T20:44:31Z)
• Two species $k$-body embedded Gaussian unitary ensembles: $q$-normal form of the eigenvalue density [0.0]
  Eigenvalue density generated by embedded Gaussian unitary ensemble with $k$-body interactions for two species. EGUE($k:mathbfpi mathbfnu$) formalism and results are extended to two species boson systems.
  arXiv  Detail & Related papers  (2023-06-21T18:55:49Z)
• Exactly solvable piecewise analytic double well potential $V_{D}(x)=min[(x+d)^2,(x-d)^2]$ and its dual single well potential $V_{S}(x)=max[(x+d)^2,(x-d)^2]$ [0.0]
  Two piecewise analytic quantum systems with a free parameter $d>0$ are obtained. Their eigenvalues $E$ for the even and odd parity sectors are determined. vivid pictures unfold showing the tunneling effects between the two wells.
  arXiv  Detail & Related papers  (2022-09-20T03:46:03Z)
• Near-optimal fitting of ellipsoids to random points [68.12685213894112]
  A basic problem of fitting an ellipsoid to random points has connections to low-rank matrix decompositions, independent component analysis, and principal component analysis. We resolve this conjecture up to logarithmic factors by constructing a fitting ellipsoid for some $n = Omega(, d2/mathrmpolylog(d),)$. Our proof demonstrates feasibility of the least squares construction of Saunderson et al. using a convenient decomposition of a certain non-standard random matrix.
  arXiv  Detail & Related papers  (2022-08-19T18:00:34Z)
• Thermalization in many-fermion quantum systems with one- plus random $k$-body interactions [2.28438857884398]
  We study thermalization in finite many-fermion systems with random $k$-body interactions in presence of a mean-field. For higher body rank of interaction $k$, system thermalizes faster.
  arXiv  Detail & Related papers  (2022-06-21T15:24:40Z)
• Learning a Single Neuron with Adversarial Label Noise via Gradient Descent [50.659479930171585]
  We study a function of the form $mathbfxmapstosigma(mathbfwcdotmathbfx)$ for monotone activations. The goal of the learner is to output a hypothesis vector $mathbfw$ that $F(mathbbw)=C, epsilon$ with high probability.
  arXiv  Detail & Related papers  (2022-06-17T17:55:43Z)
• Random matrices in service of ML footprint: ternary random features with no performance loss [55.30329197651178]
  We show that the eigenspectrum of $bf K$ is independent of the distribution of the i.i.d. entries of $bf w$. We propose a novel random technique, called Ternary Random Feature (TRF) The computation of the proposed random features requires no multiplication and a factor of $b$ less bits for storage compared to classical random features.
  arXiv  Detail & Related papers  (2021-10-05T09:33:49Z)
• Exact eigenvalue order statistics for the reduced density matrix of a bipartite system [0.0]
  eigenvalues $lambda_1(m),ldots, lambda_m(m)$ of $rho_A(m)$ are correlated random variables because their sum equals unity. We numerically generate histograms of the ordered set of eigenvalues corresponding to ensembles of over $105$ random complex pure states of the bipartite system.
  arXiv  Detail & Related papers  (2021-10-03T15:17:12Z)
• Spectral properties of sample covariance matrices arising from random matrices with independent non identically distributed columns [50.053491972003656]
  It was previously shown that the functionals $texttr(AR(z))$, for $R(z) = (frac1nXXT- zI_p)-1$ and $Ain mathcal M_p$ deterministic, have a standard deviation of order $O(|A|_* / sqrt n)$. Here, we show that $|mathbb E[R(z)] - tilde R(z)|_F
  arXiv  Detail & Related papers  (2021-09-06T14:21:43Z)
• Optimal Robust Linear Regression in Nearly Linear Time [97.11565882347772]
  We study the problem of high-dimensional robust linear regression where a learner is given access to $n$ samples from the generative model $Y = langle X,w* rangle + epsilon$ We propose estimators for this problem under two settings: (i) $X$ is L4-L2 hypercontractive, $mathbbE [XXtop]$ has bounded condition number and $epsilon$ has bounded variance and (ii) $X$ is sub-Gaussian with identity second moment and $epsilon$ is
  arXiv  Detail & Related papers  (2020-07-16T06:44:44Z)
• Linear Time Sinkhorn Divergences using Positive Features [51.50788603386766]
  Solving optimal transport with an entropic regularization requires computing a $ntimes n$ kernel matrix that is repeatedly applied to a vector. We propose to use instead ground costs of the form $c(x,y)=-logdotpvarphi(x)varphi(y)$ where $varphi$ is a map from the ground space onto the positive orthant $RRr_+$, with $rll n$.
  arXiv  Detail & Related papers  (2020-06-12T10:21:40Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.