## 1 Introduction  
Multiple-scattering represents a fundamental scientific problem, with implications in a wide spectrum of practical applications ranging from biological imaging [1] to dense spray diagnostics [2]. However, multiple-scattering problems usually are complicated and have to be solved numerically, which is computationally costly and often does not provide intuitive understanding of the problem. As an example, Monte Carlo simulation [3; 4; 5; 6], which is an extensively used numerical technique, can be prohibitively expensive when probing large optical depth. Considerable efforts have been investigated to overcome this limitation, and one notable technique demonstrated recently [7] involves the use of GPU (Graphical Processing Units) to parallelize and accelerate Monte Carlo simulations.  
Alternatively, theoretical approaches have been developed to derive analytical expressions for the observable quantities involved in multiple-scattering [8]. Notable examples include photon diffusion theory [9] and random walk theory (RWT) [10, 11]. This report focuses on the RWT approach, which can yield analytical solutions to provide insights not easily obtained with numerical methods such as the Monte Carlo simulations.  
In particular, RWT expressions have been derived for such quantities as transmittance, reflectance, and the intensity distributions of transmitted and reflected photons. These expressions not only address the computational limitation of numerical techniques, but also provide physical insights into the underlying phenomena. They have been validated by extensive comparisons against Monte Carlo simulations, and have been found to be useful for a range of practical applications such as the estimation of signal levels and the design of imaging optics. However, RWT analytical expressions are obtained in terms of lattice units and step lengths, and careful attention must be given to their conversion to expressions involving physical units, _viz._, the absorption and scattering coefficients (\(\mu_{\mathrm{s}}\) and \(\mu_{\mathrm{a}}\)), and the anisotropy factor, g. Furthermore, it is desirable to quantitatively understand the applicable range of the RWT expressions, and to investigate the feasibility of extending these expressions to anisotropic media.  
In the case of isotropic scattering, random walk expressions can be related to real macroscopic variables by equating the RWT lattice unit to the real displacement divided by the rms (root mean square) scattering length (equivalent to dividing the macroscopic scattering coefficient by \(\sqrt{2}\) ), while expressing the total photon path length in terms of the mean distance traveled on each step of the walk. In this communication we consider transmittance through slabs and provide corrections to some previously-derived RWT equations [12] that contain typographical errors. Also, we correct some expressions appearing in the appendix of [12], where a factor of \(\sqrt{2}\)  was overlooked and various conversions were affected. We then derive the transmittance expression for isotropic scattering media without absorption and demonstrate an unusual scaling law for anisotropic scattering that arises by using a mixture of rms step lengths and mean path lengths within the RWT formulation [12].