Pastur Law Research Articles

Uniform approximation of eigenvalues in Laguerre and Hermite 𝛽-ensembles by roots of orthogonal polynomials

We derive strong uniform approximations for the eigenvalues in general Laguerre and Hermite β \beta -ensembles by showing that the maximal discrepancy between the suitably scaled eigenvalues and roots of orthogonal polynomials converges almost surely to zero when the dimension converges to infinity. We also provide estimates of the rate of convergence. In the special case of a normalized real Wishart matrix W ( I n , s ) / s W(I_n,s)/s , where n n denotes the dimension and s s the degrees of freedom, the rate is ( log ⁡ n / s ) 1 / 4 (\log n/s)^{1/4} , if n , s → ∞ n,s\to \infty with n ≤ s n\leq s , and the rate is log ⁡ n / n \sqrt {\log n/n} , if n , s → ∞ n,s\to \infty with n ≤ s ≤ n + K n\leq s\leq n+K . In the latter case we also show the a.s. convergence of the ⌊ n t ⌋ \lfloor nt \rfloor largest eigenvalue of W ( I n , s ) / s W(I_n,s)/s to the corresponding quantile of the Marc̆enko-Pastur law.

Random matrices and the spectrum of N-flation

N-flation is a promising embedding of inflation in string theory in which many string axionscombine to drive inflation. We characterize the dynamics of a general N-flation modelwith nondegenerate axion masses. Although the precise mass of a single axiondepends on compactification details in a complicated way, the distribution of massescan be computed with very limited knowledge of microscopics: the shape of themass distribution is an emergent property. We use random matrix theory to showthat a typical N-flation model has a spectrum of masses distributed according tothe Marčenko–Pastur law. This distribution depends on a single parameter, thenumber of axions divided by the dimension of the moduli space. We use thisresult to describe the inflationary dynamics and phenomenology of a generalN-flation model. We produce an ensemble of models and use numerical integration totrack the axions’ evolution and the resulting scalar power spectrum. For realisticinitial conditions, the power spectrum is considerably more red than in single-fieldm2ϕ2 inflation. We conclude that random matrix models of N-flation are surprisingly tractableand have a rich phenomenology that differs in testable ways from that of single-fieldm2ϕ2 inflation.