Random matrix ensembles with column/row constraints: I

Abstract

We analyze statistical properties of a complex system subjected to conditions which manifests through specific constraints on the column/row sum of the matrix elements of its Hermitian operators. The presence of additional constraints besides real-symmetric nature leads to new correlations among their eigenfunctions, hinders a complete delocalization of dynamics and affects the eigenvalues too. The statistical analysis of the latter indicates the presence of a new universality class analogous to that of a special type of Brownian ensemble appearing between Poisson and Gaussian orthogonal ensemble.

Disorder perturbed flat bands. II. Search for criticality

“…Another feature different from the Anderson transition is the following: with increasing disorder, the spectral statistics increasing disorder [19]. Notwithstanding these differences, the complexity parameter formulation predicts an Anderson analog of a weakly disordered flat band and also reveals its connection of to a wide range of other ensembles [27,28,39] of the same global constraint class; the prediction is verified by a numerical analysis discussed later in the paper. Although the theoretical analysis presented here is based on the Gaussian disorder in flat bands but it can also be extended to other type of disorders [18].…”

“…It also reveals an important analogy in the localization to delocalization crossover in finite systems: notwithstanding the difference in the number of critical points as well as equilibrium limits, the statistics of a disordered flat band can be mapped to that of a single parametric Brownian ensemble [20] as well as multi-parametric Anderson ensemble [19] (see section VI). The analogy of these ensembles to other multiparametric ensembles intermediate between Poisson and Wigner-Dyson is already known [18,20,27,39]. In fact it seems a wide range of localization → delocalization transition can be modeled by a single parameter Brownian ensemble appearing between Poisson and GOE (Rosenzweig-Porter ensemble) [28].…”

Localization to ergodic transitions: is Rosenzweig–Porter ensemble the hidden skeleton?

“…However as shown in [22] by an exact analytical study, the eignevalue correlations of the ensembles with pairwise matrix elements correlations can be mapped to Rosenzweig-Porter ensembles. A recent study [23] also reveals the connection of Rosenzweig-porter ensemble to the random matrix ensembles with column/row constraints which appear in diverse areas e.g bosonic Hamiltonians such as phonons, and spin-waves in Heisenberg and XY ferromagnets, antiferromagnets, and spin-glasses, euclidean random matrices, random reactance networks, financial systems and Internet related Google matrix etc. In fact, the applicability of Rosezweig-Porter ensemble or equivalently the Brownian ensemble between Poisson and Gaussian orthogonal/ unitary ensembles, is wide ranging.…”

Statistical analysis of chiral structured ensembles: Role of matrix constraints

“…The latter is a characteristic of column-row constraint matrix [25] whereas in case 5 there are two pairs of such localized eigenfunctions (Fig 6b).…”