System dependent random matrix ensembles: an unavoidable tool for many body theoriesMPIM

by Pragya Shukla (IIT Kharagpur/MPIM)

Monday Aug 11, 2025, 2:30 PM → 3:30 PM Europe/Berlin

MPIM, Vivatsgasse, 7 - Lecture Hall (Max Planck Institute for Mathematics)

Number theory lunch seminar

The ignorance of the detail in a complex system e.g one with many
body interactions introduces a degree of randomness in its matrix
representation (even in absence of disorder) and it can be modeled by a
random matrix (with some or all random entries). While the statistical
behavior of many body systems in ergodic regime can be well-modeled
by the stationary random matrix ensembles, the non-ergodic regime
requires consideration of system-dependent random matrix ensembles:
those which take into account the physical constraints on the system
e.g. local interactions, dimensionality, symmetry, local conservation
laws. For example, the combined e ect of nearest neighbor interactions
and dimensionality can lead to a sparse random matrix representation
of the Hamiltonian in a basis of interest, with sparsity and nature of
randomness dependent on system-specific details.

The statistical analysis of various many body systems requires,
therefore, a thorough probing of a wide range of random matrix
ensembles which is not an easy task. It is highly desirable, if possible,
to identify a common mathematical structure among the ensembles
and analyze it to gain information about the physical properties. Our
successful search in this direction leads to Brownian ensembles as the
hidden skeleton for a wide range of systems. A complete investigation
of Brownian ensembles can then help us in the spectral analysis of a
wide range of many body systems.