Alladi Ramakrishnan Hall

Statistical analysis of complex systems with dynamical constraints: A random matrix approach

Suchetana Sadhukhan

Department of Physics, Indian Institute of Technology (IIT) Kharagpur

The talk will be concerned with a specific physical constraint on the dynamics of a complex system which appears as column/row constraint on the matrix elements of the generator of the dynamics in a physically motivated basis. We shall consider the effect of a combination of matrix constraints e.g. Hermiticity and time-reversal symmetry besides column/row sum rule, as well as the ensemble constraints (e.g. disorder), on the matrix ensembles. The study reveals that the presence of additional constraints besides real-symmetric nature introduce new correlations among the matrix elements which, as a consequence, influence their distribution and manifest in their eigenvalues and eigenfunctions too. As physical properties, in principle, can be explained in terms of eigenvalues and eigenfunctions, therefore, we focus on the effects of constraints on the joint eigenvalue-eigenfunction distribution. The latter turns out to be analogous to that of a special type of critical Brownian ensemble without such constraint, intermediate between Poisson and Gaussian orthogonal ensemble. Another important finding of our work is the lack of single basis state localization for a typical eigenvector which is in contrast with an unconstrained
real-symmetric matrix. Our numerical results show a rich behavior hidden beneath the spectral statistics and also confirm our analytical predictions. Later, our detailed analysis of criticality in Brownian ensemble reveals the appearance of a new scale-invariant spectral statistics, non-stationary along the spectrum, associated with multifractal eigenstates and different from the two end-points if the transition parameter becomes size-independent. We found the number of such critical points during transition is governed by a competition between the average perturbation strength and the local spectral density.