  The transport properties on the two-dimensional surface of coupled multilayer
heterostructures are studied in the integer quantum Hall states. We emphasize
the criticality of the surface state and the phase coherent transport
properties in the thermodynamic limit. A new, stable numerical algorithm for
large scale conductance calculations in the transfer matrix approach is
discussed in detail. It is then applied to a directed network model describing
the quantum mechanical tunneling and impurity scattering of the multilayer edge
states. We calculate the two-probe conductance in the direction parallel to the
external magnetic field, its fluctuations and statistical distributions as a
function of the interlayer tunneling strength. Using finite size scaling, the
asymptotic scaling functions of the ensemble averaged conductance and the
conductance fluctuations are calculated for a fixed aspect ratio and found to
be in remarkable agreement with the analytical results obtained using the
supersymmetric nonlinear $\sigma$-model approach. The conductance distribution
is determined in the quasi-one-dimensional metallic, insulating, as well as the
crossover regime where comparisons are made to that at the single-layer quantum
Hall transition. We present, for the first time, a detailed study of the level
statistics in the eigenvalue spectrum of the transfer matrix. Coexistence of
metallic and insulating statistics is observed in the crossover regime, which
is attributed to the emergence of a finite range level repulsion in the
crossover regime, separating the metallic (Wigner-surmise) behavior at small
level spacings from the insulating (uncorrelated or Poisson) behavior at large
level spacings.
