Random Walk
+Written: June 2023 (Github)
+ +Introduction
+A random walk is the path followed when each individual movement is probablistic. We can simulate this by saying for each time increment, there is a 50% chance of moving up and a 50% chance of moving down. An example of this is given below
+ However the above is one of only many possibilities, we can add more simulations
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+ \[ {N \choose r + 1} - {N \choose r} = \frac{N!}{r! (N-r)!} \frac{N - 2r - 1}{r + 1} = {N \choose r} \frac{N - 2r - 1}{r + 1} \]
+\[ {N \choose r + 1}( \frac{N - 1}{2} - (r + 1) ) - {N \choose r} ( \frac{N - 1}{2} - r ) = {N \choose r + 1} [ \frac{(N - 2r - 1)^2}{2(N - r)} - 1 ] \]
+\[ r_{crt} = \frac{(2N - 3) \pm \sqrt{5 + 4N} }{4} \]
+As \( N \rightarrow \infty \), we can see how far away from the midpoint the inflection point moves.
+\[ \lim_{N \to \infty} (\frac{N - 1}{2} - r_{crt}) = \frac{\sqrt{N}}{2} \]
+So in the large N limit, the inflection point happens from \( \pm \frac{\sqrt{N}}{2} \) of the midpoint. It is natural to expect that most random paths will be within this range. In fact the RMS deviation for a random walk is \( \sqrt{N} \)
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