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Let $L_{m,m}$ be a $2D$ lattice. Also, suppose that there is a random walker located in position $(0,0)$. The random walker goes right, left, up, or down randomly in each step and cannot get out of $L_{m,m}$.

What is the expected number of steps that the random walker needs to reach $(m,m)$ with high probability?

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  • $\begingroup$ What does the walker do when she hits the boundary? Stay there? Bounce off? $\endgroup$ – kimchi lover Apr 4 at 11:56
  • $\begingroup$ @kimchilover Bounce off $\endgroup$ – Hasan Heydari Apr 4 at 13:49
  • $\begingroup$ So edit your question to include this detail. $\endgroup$ – kimchi lover Apr 4 at 14:17

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