Why is the correlation of two random markov chains so large?
Hi: you can't cumulate two sets of random numbers and then calculate a correlation between them. The two samples are non-stationary ( since you are cumulating them )  and what you get is referred to as "spurious correlation" or "non-sensical correlation" in econometrics. ( some would say statistics but the econometricians really figured out what was happening ).  I wouldn't do it justice trying to explain it here but, essentially, what you are doing is calculating two series whose means are changing over time so the correlation calculation is illegal statistically. Granger and Newbold figured out the statistics behind why this is in 1976 so if you want all the details, just google for "spurious regression". The econometrics literature on this is enormous and, even though what you are doing has maybe zero to do with econometrics, you are basically generating two series that are each random walks and then correlating them which is a garbage in garbage type of scenario.  I hope this helps.