Nonlinear random matrix statistics, symmetric functions and hyperdeterminants

Abstract

Nonlinear statistics (i.e. statistics of permanents) on the eigenvalues of invariant random matrix models are considered for the three Dyson's symmetry classes β = 1, 2, 4. General formulas in terms of hyperdeterminants are found for β = 2. For specific cases and all βs, more computationally efficient results are obtained, based on symmetric functions expansions. As an application, we consider the case of quantum transport in chaotic cavities extending results from [D.V. Savin, H.-J. Sommers and W. Wieczorek, Phys. Rev. B 77, 125332 (2008)].

Asymptotics of Selberg-like integrals: The unitary case and Newton's interpolation formula

“…31 that integral (32) can be restated in terms of monomial symmetric functions First, two of the authors remarked in Ref.…”

“…31 and we are interested in the following integrals 31 and we are interested in the following integrals…”

Asymptotics of Selberg-like integrals by lattice path counting

“…Exact results appeared for small n in [11,12,13,14] and for general n in [15,16]. In this last work the present author used a result of Kaneko [2] and Kadell [17] which gives the average value of any Schur function of the eigenvalues, an approach that was later taken further in [18,19]. For large numbers of channels, a generating function for the average value of ( 4) was presented in [20], and an explicit expression appeared in [13].…”

Untitled

“…(A.2) in principle allows for a complete characterization of statistical properties of experimental observables. For most recent analytical results, we refer to [30,31,32,33,46,47,48,49,50,51,52,53,54,55].…”

“…In particular, the study of higher moments of the transmission matrix τ n = Tr[T n ] has recently seen many analytical progresses 5 [24,25,30,31,32,33,48,49,53]. In particular, we now have two different (but equivalent) formulae for higher moments for β = 2 and arbitrary N 1 , N 2 :…”