Invariance relations for random walks on simple cubic lattices

In recent work, we have built upon seminal contributions of Montroll and Weiss to develop invariance relations for random walks on k = 2 hexagonal and square-planar lattices. In this study, we extend our approach to determine invariance relations for random walks on k = 3 finite, simple cubic lattices subject to periodic boundary conditions. Use of these invariance relations allows one to calculate estimates of the overall mean walklength before trapping, and we show that the results obtained are in excellent agreement with numerically-exact calculated values. The results and analysis presented here yield insights on the problem of self-avoiding walks on cubic lattices.