Equality of limiting inverse-participation ratios under boundary conditions

Establish whether the limiting values of the proxy inverse participation ratio Y_2^*(N,\beta) are equal for closed and periodic boundary conditions as the system size N diverges.

Background

The paper studies the proxy inverse participation ratio Y_2^*(N,\beta)=\sum_i\langle a_i^2(\omega)\rangle, where a_i(\omega) is the stationary mean mass profile for the disordered mass-exchange model. Numerical data indicate that, for diverging system size, this quantity appears to be insensitive to whether the model has closed or periodic boundary conditions, with the periodic case seemingly reaching its asymptotic value more rapidly.

The authors explicitly state that the equality of the limiting values has not been established by their arguments. The unresolved problem is therefore to prove or disprove equality of the large-system limits of Y_2^* for the two boundary geometries.