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This document contains supplementary material for the main articles in ourRandom Cayley Graphs project.  We prove refined results about simple random walks on the integers and on thecycle. We are primarily interested in the entropy of these random walks atcertain times and how this entropy changes when the time changes slightly.Additionally, we prove some large deviation and exit time estimates.  We prove some results on the size of discrete lattice balls and how this sizechanges when the radius changes slightly. We do this in a general $L_q$ norm,with $q \in [1,\infty]$.  We also prove some other technical results deferred from the main papers.  We hope that some of the results, particularly the simple random walkestimates, will be useful in their own right for other researchers.

Supplementary Material for Random Cayley Graphs Project