Limiting spectral distribution of block matrices with Toeplitz block structure

We study two specific symmetric random block Toeplitz (of dimension k×kk×k) matrices, where the blocks (of size n×nn×n) are (i) matrices with i.i.d. entries and (ii) asymmetric Toeplitz matrices. Under suitable assumptions on the entries, their limiting spectral distributions (LSDs) exist (after scaling by nk) when (a) kk is fixed and n→∞n→∞ (b) nn is fixed and k→∞k→∞ (c) nn and kk go to ∞∞ simultaneously. Further, the LSDs obtained in (a) and (b) coincide with those in (c) when nn or respectively kk tends to infinity. This limit in (c) is the semicircle law in Case (i). In Case (ii), the limit is related to the limit of the random symmetric Toeplitz matrix as obtained by Bryc et al. (2006) and Hammond and Miller (2005).