  A statistical mechanical framework for analyzing random linear vector
channels is presented in a large system limit. The framework is based on the
assumptions that the left and right singular value bases of the rectangular
channel matrix $\bH$ are generated independently from uniform distributions
over Haar measures and the eigenvalues of $\bH^{\rm T}\bH$ asymptotically
follow a certain specific distribution. These assumptions make it possible to
characterize the communication performance of the channel utilizing an integral
formula with respect to $\bH$, which is analogous to the one introduced by
Marinari {\em et. al.} in {\em J. Phys. A} {\bf 27}, 7647 (1994) for large
random square (symmetric) matrices. A computationally feasible algorithm for
approximately decoding received signals based on the integral formula is also
provided.
