Consider a one dimensional random walk, in which the probability of moving left along a line is $q=1/2$ and the probability of moving right is $p=1/2$. The square root error $\langle d_N \rangle$, which is the expectation of the absolute distance traveled after $N$ steps is known to be $\sqrt{2N/\pi}.$ I am interested in finding the square root error of a modified version of this problem, in which there are three distinct possibilities. You can move to the left, you can move to the right, or you can stay put on a given turn. There will now be three probabilities, the probability $p$ of moving left, the probability $q$ of moving right, and the probability $r$ of not moving at all. The particular problem of interest to me is when $p=q$. So we will have that $p+q+r=1$. Has this problem been studied, and what would be the reference; what would be the solution?
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1$\begingroup$ You should find stuff if you Google 'lazy random walk'. $\endgroup$– JP McCarthyJan 16, 2023 at 7:38
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1$\begingroup$ @JP McCarthy - Thanks $\endgroup$– EGMEJan 17, 2023 at 8:33
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Because of the central-limit-theorem, for large $N$ the absolute distance $d_N$ converges in distribution as $$P_N(d_N/\sqrt N)\to p(|X|),$$ where $X$ is a Gaussian random variable with mean zero and variance $2p=1-r$. Since $\mathbb{E}(|X|)=2\sqrt{p/\pi}$ we conclude that $$\mathbb{E}(d_N)\to \sqrt\frac{2(1-r)N}{\pi}.$$ So this "lazy" random walk differs from the simple random walk by a rescaling of the number of steps by a factor $1-r$.
answered Jan 16, 2023 at 9:50
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$\begingroup$ Thank you. Do you happen to know where the term "square root error" comes from? I am looking for a reference if there is one. Or, alternatively, what is the common term for $\langle d_N \rangle$? I thank you in advance. $\endgroup$– EGMEJan 17, 2023 at 9:31
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1$\begingroup$ it is common to call $\langle d_N\rangle$ the "average distance" after $N$ steps; "square root error" refers to the displacement $s_N$, which has zero average and variance ${\rm var}\,(s_N)=\mathbb{E}(s_N^2)=(1-r)N$, so rms value $\sqrt{(1-r)N}$. $\endgroup$ Jan 17, 2023 at 9:59
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$\begingroup$ Is this average distance relative to the displacement average which is zero? Do you know a reference where all these concepts are defined? My understanding comes from reading papers which are not specifically about this. Thanks in advance $\endgroup$– EGMEJan 17, 2023 at 10:06
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$\begingroup$ it's elementary, really: if $x_N$ is the position of the random walker after $N$ steps, then $d_N=|x_N-x_0|$ and $s_N=x_N-x_0$. $\endgroup$ Jan 17, 2023 at 10:56